Baer *-Rings (Reprint of the 1972 Edition with errata list and later developments indicated)

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander A. Kupiainen G. Lebeau F.-H. Lin B.C. Ngô M. Ratner D. Serre Ya.G. Sinai N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan

195

For further volumes: http://www.springer.com/series/138

Sterling K. Berberian

Baer ∗-Rings

Reprint of the 1972 Edition with errata list and later developments indicated

S. K. Berberian Prof. Emer. Mathematics The University of Texas at Austin

ISSN 0072-7830 ISBN 978-3-540-05751-2 e-ISBN 978-3-642-15071-5 DOI 10.1007/978-3-642-15071-5 Springer Heidelberg Dordrecht London New York Library of Congress Catalog Card Number: 72189105 AMS Subject Classifications (1970): Primary 16A34, Secondary 46L10, 06A30, 16A28, 16A30 c Springer-Verlag Berlin Heidelberg 1972, 2nd printing 2011  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Kap

Preface This book is an elaboration of ideas of Irving Kaplansky introduced in his book Rings of operators ([52], [54]). The subject of Baer *-rings has its roots in von Neumann's theory of 'rings of operators' (now called von Neumann algebras), that is, *-algebras of operators on a Hilbert space, containing the identity operator, that are closed in the weak operator topology (hence also the name W*-algebra). Von Neumann algebras are blessed with an excess of structure-algebraic, geometric, topological-so much, that one can easily obscure, through proof by overkill, what makes a particular theorem work. The urge to axiomatize at least portions of the theory of von Neumann algebras surfaced early, notably in work of S. W. P. Steen [84], I. M. Gel'fand and M. A. Naimark [30], C. E. Rickart 1741, and von Neumann himself [53]. A culmination was reached in Kaplansky's AW*-algebras [47], proposed as a largely algebraic setting for the intrinsic (nonspatial) theory of von Neumann algebras (i. e., the parts of the theory that do not refer to the action of the elements of the algebra on the vectors of a Hilbert space). Other, more algebraic developments had occurred in lattice theory and ring theory. Von Neumann's study of the projection lattices of certain operator algebras led him to introduce continuous geometries (a kind of lattice) and regular rings (which he used to 'coordinatize' certain continuous geometries, in a manner analogous to the introduction of division ring coordinates in projective geometry). Kaplansky observed [47] that the projection lattice of every 'finite' A W*-algebra is a continuous geometry. Subsequently [51], he showed that certain abstract lattices were also continuous geometries, employing 'complete *-regular rings' as a basic tool. A similar style of ring theoryemphasizing *-rings, idempotents and projections, and annihilating ideals-underlies both enterprises. Baer a-rings, introduced by Kaplansky in 1955 lecture notes [52], are a common generalization of A W*-algebras and complete *-regular rings. The definition is simple: A Baer *-ring is a ring with involution in which the right annihilator of every subset is a principal right ideal generated by a projection. The A W*-algebras are precisely the Baer

*-rings that happen to be C*-algebras; the complete *-regular rings are the Baer *-rings that happen to be regular in the sense of von Neumann. Although Baer *-rings provided a common setting for the study of (1) certain parts of the algebraic theory of von Neumann algebras, and (2) certain lattices, the two themes were not yet fully merged. In A W*-algebras, one is interested in '*-equivalence7of projections; in complete *-regular rings, 'algcbraic equivalence'. The finishing touch of unification came in the revised edition of Kaplansky's notes [54]: one considers Baer *-rings with a postulated equivalence relation (thereby covering *-cquivalence and algebraic equivalence simultaneously). "Operator algebra" would have been a conceivable subtitle for the present book, alluding to the roots of the subject in the theory of operator algebras and to the fact that the subject is a style of argument as well as a coherent body of theorems; the book falls short of earning the subtitle because large areas of the algebraic theory of operator algebras are omitted (for example, general linear groups and unitary groups, module theory, derivations and automorphisms, projection lattice isomorphisms) and because the theory elaborated here-*-equivalence in Baer *-ringsdoes not develop Kaplansky's theory in its full generality. My reason for limiting the scope of the book to *-equivalence in Baer *-rings is that the reduced subject is more fully developed and is more attuned to the present state of the theory of Hilbert space operator algebras; the more general theories (as far as they go) are beautifully exposed in Kaplansky's book, and need no re-exposition here. Perhaps the most important thing to be explained in the Preface IS the status of functional analysis in the exposition that follows. The subject of Baer a-rings is essentially pure algebra, with historic roots in operator algebras and lattice theory. Accordingly, the exposition is written with two principles in mind: (1) if all the functional analysis is stripped away (by hands more brutal than mine), what remains should stand firmly as a substantial piece of algebra, completely accessible through algebraic avenues; (2) it is not very likely that the typical reader of this book will be unacquainted with, or uninterested in, Banach algebras. Interspersed with the main development are examples and applications pertaining to C*-algebras, AW*-algebras and von Neumann algebras. In principle, the reader can skip over all such matters. One possible exception is the theory of commutative A W*-algebras (Section 7). Thc situation is as follows. Associated with every Baer *-ring there is a complete Boolean algebra (the set of central projections in the ring); the Stone representation space of a complete Boolean algebra is an extremally disconnected, compact topological space (briefly, a Stonian space); Stonian spaces are precisely the compact spaces 9"for which the

algebra C ( 3 ) of continuous, complex-valued functions on 3 is a commutative A W*-algebra. These algebras play an important role in the dimension theory and reduction theory of finite rings (Chapters 6 and 7). They can be approached either through the theory of commutative Banach algebras (as in the text) or from general topology. The choice is mainly one of order of development; give or take some terminology, commutative A W*-algebras are essentially a topic in general topology. The reader can avoid topological considerations altogether by restricting attention to factors, i.e., rings in which 0 and 1 are the only central projections (this amounts to restricting !T to be a singleton). However, the chapter on reduction theory (Chapter 7) then disappears, the objects under study (finite factors) being already irreducible. There is ample precedent for limiting attention to the factorial case the first time through; this is in fact how von Neumann wrote out the theory of continuous geometries [71], and the factorial case dominates the early literature of rings of operators. Baer *-rings are a compromise between operator algebras and lattice theory. Both the operator-theorist ("but this is too general!") and the lattice-theorist ("but this can be generalized!") will be unhappy with the compromise, since neither has any need to feel that the middle ground makes his own subject easier to understand; but uncommitted algebraists may find them enjoyable. 1 personally believe that Baer *-rings have the didactic virtue just mentioned, but the issue is really marginal; the test that counts is the test of intrinsic appeal. The subject will flourish if and only if students find its achievements exciting and its problems provocative. Exercises are graded A-D according to the following mnemonics: A ("Above"): can be solved using preceding material. B ("Below"): can be solved using subsequent material. C ("Complements"): can be solved using outside references. D ("Discovery"): open questions. I am indebted to the University of Texas at Austin, and Indiana University at Bloomington, for making possible the research leave at Indiana University in 1970-71 during which this work took form. Austin, Texas October, 1971

Sterling K. Berberian

Interdependence of Chapters

Contents Part 1: General Theory Chapter 1. Rickart *.Rings. Baer *.Rings. AW*-Algebras: Generalities and Examples . . . . . . . . . . . . $ 1. *-Rings . . . . . . . . . . . # 2. *-Rings with Proper Involution . Q; 3. Rickart *-Rings . . . . . . . # 4. Baer *.Rings . . . . . . . . . # 5. Weakly Rickart *-Rings . . . .

$ 6. $ 7. $ 8. $ 9. $ 10.

3

. . . . .

. . . . . . . . . .

. . . . . . . . . . Central Cover . . . . . . . . . . . . . Commutative AW*.Algebras . . . . . . . Commutative Rickart C*-Algebras . . . . Commutative Weakly Rickart C*-Algebras. C*-Sums . . . . . . . . . . . . . . .

Chapter 2 . Comparability of Projections

. . . . . . . . . . . 55

9; 11. Orthogonal Additivity of Equivalence . . . . . $ 1 2. A General Schroder-Bernstein Theorem . . . . $ 13. The Parallelogram Law (P) and Related Matters $ 14. Generalized Comparability . . . . . . . . .

. . . . 55 . . . . 59

. . . . 62 . . . . 77

Part 2: Structure Theory Chapter 3. Structure Theory of Baer *.Rings . . . . . . . . . .

87

# 15. Decomposition into Types . . . . . . . . . . . . . . 88 $ 16. Matrices . . . . . . . . . . . . . . . . . . . . . 97

9 17. Finite

and Infinite Projections . . . . . . . . . . . . 101 $ 18. Rings of Type I; Homogeneous Rings . . . . . . . . . 110 # 19. Divisibility of Projections in Continuous Rings . . . . . 119

Chapter 4. Additivity of Equivalence . . . . . . . . . . . . . 122 $ 20. General Additivity of Equivalence . . . . . . . . . . 122 $ 21. Polar Decomposition . . . . . . . . . . . . . . . . 132

XI1

Contents

Chapter 5. Ideals and Projectiolls

. . . . . . . . . . . . . . 136

S; 22. Ideals and p-Ideals . . . . . . . . . . . . . . . . . 136 $ 23. The Quotient Ring Modulo a Restricted Ideal . . . . . 142 Ej 24 . Maximal-Restricted Ideals. Weak Centrality . . . . . . 146 Part 3 : Finite Rings Chapter 6. Dimension in Finite Baer *-Rings . . . . . . . . . 153

5 25. S; 26. $ 27. $ 28 . S; 29. S; 30. $ 31 . S; 32. Ej 33. Ej 34.

Statement of the Results . . . . . . . . . . . . . . . Simple Projections . . . . . . . . . . . . . . . . . First Properties of a Dimension Function . . . . . . . Type I,.. Complete Additivity and Uniqueness of Dimension . . . . . . . . . . . . . . . . . . . . Type I,., Existence of a Dimension Function . . . . . . Type IT,,, Dimension Theory of Fundamental Projections Type IT,,, Existence of a Completely Additive Dimension Function . . . . . . . . . . . . . . . . . . . . . Type IT,,. Uniqueness of Dimension . . . . . . . . . . Dimension in an Arbitrary Finite Baer *-Ring with GC . Modularity. Continuous Geometry . . . . . . . . . .

.. .. .

153 154 160 165 166 170 178 180 181 184

Chapter 7. Reduction of Finite Bacr *-Rings . . . . . . . . . . 186 Ej 35. Ej 36. Ej 37. Ej 38. $ 39. $ 40. S; 41. $ 42. S; 43. $ 44 . $ 4 5.

lntroduction . . . . . . . . . . . . . . . . . . . . Strong Semisimplicity . . . . . . . . . . . . . . . . Description of the Maximal p-Ideals of A: The Problcm . Multiplicity Analysis of a Projection . . . . . . . . . Description of the Maximal p-Ideals of A: The Solution . Dimension in A/[ . . . . . . . . . . . . . . . . . AII Theorem: Type 11 Case . . . . . . . . . . . . . AII Theorem : Type I, Case . . . . . . . . . . . . . AII Theorem: Type I Case . . . . . . . . . . . . . . Summary of Results . . . . . . . . . . . . . . . . AIM Theorem for a Finite AW*-Algebra . . . . . . .

186 186 188 189 191 193 195 196 199 201 202

Chapter 8. The Regular Ring of a Finite Baer *-Ring . . . . . . 210 $ 4 6. $ 47. 5 48. $ 49 . $ 50. Ej 51.

Preliminaries . . . . . . . . . Construction of the Ring C . . . First Properties of C . . . . . . C has no New Partial lsometries . Positivity in C . . . . . . . . . Cayley Transform . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

210 213 218 223 224 227

Contents

XITI

$ 52. Regularity of C . . . . . . . . . . . . . . . . . . 232 5 53. Spectral Theory in C . . . . . . . . . . . . . . . . 238 3 54. C has no New Bounded Elements . . . . . . . . . . . 243

Chapter 9. Matrix Rings over Baer *-Rings . . . . . . . . . . 248

5 55. Introduction . . . 5 56. Generalities . . . 3 57. Parallelogram Law

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and Generalized Comparability . . . $ 58. Finiteness . . . . . . . . . . . . . . . . . . . . . S 59. Simple Projections . . . . . . . . . . . . . . . . . 5 60. Type I1 Case . . . . . . . . . . . . . . . . . . . . 5 61. Type 1 Case . . . . . . . . . . . . . . . . . . . . 5 62. Summary of Results . . . . . . . . . . . . . . . .

248 250 254 256 257 259 260 262

Hints. Notes and References . . . . . . . . . . . . . . . . . 264 Bibliography

. . . . . . . . . . . . . . . . . . . . . . . 287

Supplementary Bibliography . . . . . . . . . . . . . . . . . 291 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Part 1: General Theory

Chapter 1

Rickart *-Rings, Baer *-Rings, A W*-Algebras: Generalities and Examples

All rings considered in this book are associative, and, except in a few of the excercises, they are equipped with an involution in the sense of the following definition:

-

Definition 1. A *-ring (or involutive ring, or ring with involution) is a ring with an involution x x* : (x*)* = x,

+

(x +y)* = x* y*,

(X y)* =y* x*

.

When A is also an algebra, over a field with involution A - A* (the identity involution is allowed), we assume further that (Ax)*= A* x* and call A a *-algebra. {The complex *-algebras are especially important special cases, but the main emphasis of the book is actually on *-rings.) The decision to limit attention to *-rings is crucial; it shapes the entire enterprise. {For example, functional-analysts contemplating the voyage are advised to leave their Banach spaces behind; the subject of this book is attuned to Hilbert space (the involution alludes to thc adjoint operation for Hilbert space operators).) From the algebraic point of view, the intrinsic advantage of *-rings over rings is that projections are vastly easier to work with than idempotents. For the rest of the section, A denotes a *-ring.

Definition 2. An element e e A is called a projection if it is selfadjoint (e* =e) and idempotent (e2= e). We write A for the set of all projections in A ; more generally, if S is any subset of A we write 3 = S n2 . If x and y are self-adjoint, then (xy)*=yx shows that xy is selfadjoint if and only if x and y commute (xy =y x). It follows that if r and f are projections, then ef' is a projection iff e and f' commute. A central feature of the theory is the ordering of projections: S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

4

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

Definition 3. For projections e, f, we write e 5 f (therefore e f = f e = e ) .

in case e= e f

Proposition 1. (1) The relation e l f is a partial ordering of projections. (2) e l f i f f ' e A c j ' A iff A e c A f . (3) e = f if eA= f A fi A e = A f . Proof: (2) If e < f then e= ef E Af, hence Ae c AJ: Conversely, if A e c A f then e = e e ~ A ec A f ; say e = x J ' ; then e f = x j:f=x f=e, thus e l f . (3) In view of (2), e A =f A means e = ef ( = f e) and f =f r , thus e= f. (1) Immediate from (2)and (3). I Definition 4. Projections e, f are called orthogonal if ef =O (equivalently, f e =O). Proposition 2. (1) I f e, f are orthogonal projections, then e+ f is a projection. (2) If e, f are projections with e f, then f -e is a projection orthogonal to e and I J: Proof. Trivial. {See Exercises 1 and 2 for partial converses.)

I

In general, extra conditions on A are needed to make 2 a lattice (such assumptions are invoked from Section 3 onward); a drastic condition that works is commutativity:

Proposition 3. If e, ~ ' € 2 commute, then e nf and e u f exist and aregiven by the,formulas e n f = e f and e u f = e + f - e f . Proof. Set g = ef, h = e +f - ef. The proof that g and h have the properties required of inf (e, f ) and sup {e,f ) is routine. I To a remarkable degree (see Section 15), certain *-rings may be classified through their projection-sets; this classification entails the following relation in the set of projections:

-

Definition 5. Projections e, , f in A are said to be equivalent (relative to A), written e f ; in case there exists W E A such that w* w =e and ww*= f. Proposition 4. With notation as in Definition 5, one can suppose, without loss of'generality, that W E fAe. Proof. Set v=ww*w=we= f w . Then v ~ , f A rand v*v=(ew*)(we) =e3=ee,v v * = ( f w ) ( w * f ) =f 3 = f . I

Definition 6. An element ~ E such A that w w* w = w is called a partial isometry. Proposition 5. An element w e A is a partial isometry if and only (f e= w* w is a projection such that w e = w. Then f = w w* is also a projection (thus e f ) and f w = w. Moreover, e is the smallest projection such that we= w, and f the smallest such that f w = w.

-

Proof: If w has the indicated property, then ww* w = we= w, thus w is a partial isometry. Conversely, if w w* w = w then, setting e = w* w, wc havc we = w and e2 = (w* w) (w* w) = w*(w w* w) = w* w = e = e*. It follows from w* w w* = w* that w* is also a partial isometry, and, setting f =(w*)* w* = w w*, we have w* f = w*, f w = w. If g is any projection such that w g = w, then w* wg= w* w, eg = e, thus e < g. Similarly, f is minimal in the property f w = w. I Definition 7. With notation as in Proposition 5, e is called the initial projection and f the finalprojection of the partial isometry w. The equivalence e f is said to be implemented by w.

-

Proposition 6. Let e, fbeprojections in A. Then e - f i f andonly i f there exists a partial isometry with initial projection e and,final projection f:

-

Proof. The "if' part is noted in Proposition 5. Conversely, suppose e f . By Proposition 4, there exists W E f A e with w* w = e and w w* =f'; since w= we= ww* w, w is a partial isometry. I The term "equivalence" is justified by the following proposition:

-

Proposition 7. T h e relation e f' is an equivalence relation in A" (1) e - e , (2) e f implies f e , (3) e - f and f - g imply e - g . Moreover, (4) e -- 0 ifand only i f e = 0 , ( 5 ) e f implies h e h f fbr every central projection h.

-

-

-

-

Proof: (1) e*e=ee*=e2=e. (2) This is clear from Definition 5. (3) By Proposition 6, there exist partial isometries w, v such that w* w = e, w w* =f and v* v =J; v v* = y. Setting u = v w, it results from f w = w and v f = v that u*u=e and uu*=g. (4) If e 0 then, by Proposition 4, there exists w€OAe= { 0 ) with w*w=e, thus e=O.

-

6

Chapter 1. Rickart *-Rings, Raer *-Rings, A W*-Algebras

(5) If w* w=e, ww* =,f and h is a projection in the center of A, I then (wh)*(wh)=h w* w = h e and (wh)(wh)*= hJ: The next proposition shows that equivalence is finitely additive; to a large extent, the first four chapters are a struggle to extend this result to families of arbitrary cardinality:

-

Proposition 8. If' e,, . . ., e, are orthogonul projections, and are orthogonal projections such that ei ,fi ,for i= I, . .., n, then e,+...+e,--J;+...+.f,

l;,...,f,

.

Proof: Let wi be a partial isometry with wT wi= ei, wi w; =,fi, and set w = w , + ... + w,. Since wi ei = w, =f j w,, it is routine to check that w is a partial isometry implementing the desired equivalence. {Incidentally, we, = wi= f i w for all i.) I If two projections are equivalent, what happens 'under' one of them is reflected in what happens under the other:

Proposition 9. I f e

-

f via the partial isometry w, then the,formula

defines a *-isomorphism cp : eAe +f A f . In particular, cp is an order-prcserving bijection of' the set of projections e onto the set of projections 2 f ; cp preserves orthogonality and equivalence; for every projection g ~ e one , has g - cp(g). Proof: Since W E fAe, c p ( x )f ~A f for all X E ~ A PObviously . ip is additive: cp(x+y)=cp(x)+cp(y). cp is multiplicative: if x,y ~ e A e then cp(xy)= w x y w* = ~ > x e j ~ w * =wxw*wyw*=cp(x)cp(Jl). ip is injective: if x ~ e A eand w x w*=O, then O=w*(wxw*)w=exe=x. cpissurjective:if y ~ , f A , f then ; w * y w ~ e A eand cp(w*y w)= ww*yww* =.fuf =y. For all x e e A e , cp(x*)= wx* w* = ( w x w*)* =(cp(x))*. Thus cp is a *-isomorphism. Note that the projections in eAc are precisely the projections ~ E A with g 5 e. If g 5 e, then g cp(g) is implemented by the partial isometry wg. It is clear from the definitions that cp preserves order, orthogonality, and equivalence. I

-

The classification theory requires an ordering of projections more subtle than e 5 f :

Definition 8. For projections e, j in A, we write e 5 J; and say that e is dominated by f , in case e - g 5 f , that is, e is equivalent to a sub-

projection off. {This means (Proposition 6) that there exists a partial isometry w with W*W = e and w w* ~ f . )

Proposition 10. The relation e 5 ,f has the,following proper tie^ ( 1 ) e l f implies e 5 j ; (2) e f implies e 5 J; (3) e d f and f d g imply e s g .

-

Proof. (3) Choose partial isometries w and v such that w* w = e, w w * = f ' < f and v * v = f , vv*=yr
Theorem 1. I f A is a *-ring such that A" is conditionally complete with respect to the ordering e 2 f ; then e 5 f' and f 5 e imply e f:

-

The proof is based on a general lattice-theoretic result:

Lemma. If L is a complete lattice und cp: L + L is an order-preserving mapping, then cp has at least one fixedpoint. Proof. The hypothesis on cp is that x I y implies cp(x)I cp(y). Write O=infL. Let S = ( x € L : cp(x)>x) (e.g., OES) and define u=supS; we show that cp(a)=a. For all x e S , we have a 2 x , therefore cp(a)2 q ( x )2 x ; thus q ( a )2 s u p s = a. Then cp(cp(a))2 cp(a), thus c p ( a ) ~ Shence , q ( a )I supS=a. I Proof of Theorem I . For any projection e, write [0, e] for the set of all projections g I e, that is, [0, e ] =(eAe)". The hypothesis on A is that [0, el is a complete lattice for each projection e. Suppose e 5 f and f 5 e. Let w and v be partial isometries with w*w=e, w w * = f r < f and v * v = f , vv*=e'<e. The plan ofthe proof is to construct an order-preserving mapping c p : [0, f ] -t [0, f 1, to which the Lemma is applied; the mapping cp is taken to be the composite of four mappings cp,, . . ., cp,, defined as follows. Define by the formula cp, (g)= v y v*; by Proposition 9, c p , is an order-preserving mapping of [0, f ] onto 10, e'] c [O, e l . Define

cp2 : [O, el

+

10, el

by cp,(g) = e - y; thus cp, is order-reversing. Similarly, define

cp, : [O> el

+

[O, f l

8

Chapter 1. Rickarl *-Rings, Baer *-Rings, A W*-Algebras

by cp,(g)=f -g. Finally, define cp: LO, f l

+

10, f l

to be the composite cp = cp40 cp,o cp, ocp, (thus cp is order-preserving); explicitly, cp(g)=f - w(e-vgv*)w* for all g 5 f: Since [0, f ] is complete, the Lemma yields a projection go c f' such that cp(y,) = go, thus setting x = w(e-vgov*), this reads x x * = f -go; since w* w=e, one has x* x = e - 1;g0v*, thus Also, setting y = vg,, one calculates y* y = go and yy* = v go v*, thus

(**I Combining (*) and (**), f

YO""YO~*.

- e by Proposition 8.

I

Recalling the classical set-theoretic result, one expects that countable lattice operations should suffice for a theorem of Schroder-Bernstein type; a result of this sort is proved in Section 12. Exercises 1A. Let A be a *-ring in which 2 x = 0 implies x=O, and let e,f be projections in A. (i) If ,f- e is a projection, then e 5 f. (ii) If e + f is a projection, then ef=O.

2A. Let A be a *-ring in which x* x+y* y = 0 implies x = y = 0 , and let e,f be projections in A. Then (i) e l f iff f - e = x * x for some X E A . Also (ii) el. f iff ,f-e is a projection, and (iii) e+ f is a projection iff ef =O. 3A. If e ,,..., en are orthogonal projections, and f l,..., f, are orthogonal projections such that e,sji ( i = l , ...,n), then el+...+ems f; +...+,f,.

4A. Let A be a *-ring, let e,f be projections in A such that e-f, and suppose el,...,en are orthogonal projections with e=e, +...+en. Then there exist orthogonal projections fl, . . .,f, with f = f, +...+ f, and e,- f, (i= I,. ..,n). 5A. If e,f are projections in a *-ring A such that e-,f, then A e and Af are isomorphic left A-modules. (See Exercise 8 for a converse.)

6A. Pursuing Exercise 5, let A be any ring and let e,f be idempotents in A . The following conditions are equivalent: (a) A e and Af are isomorphic left A-modules; (b) there exist X E fAe, y ~ e A f such that y x = e , x y = f; (c) there

exist x, YEA such that y x = e , x y = f. (Such idempotents are sometimes called algebraically equivalent.) 7C. If A is a symmetric *-ring and f' is any idempotent in A, then fA= eA for a suitable projection e. {When A has a unity element, symmetry means that I +a*a is invertible for every UEA; when A is unitless, symmetry means that -a*a is quasiregular for every UEA (XEA is quasiregular if there exists YEA with x+y-xy=0).)

-

8A. If e, f are projections in a *-ring A, then algebraic equivalence in the sense of Exercise 6 implies e f in the sense of Definition 5, provided A satisfies the following condition (called the weak square-root axiom): for each XGA, there exists rE {x*x)" (the bicommutant of x*x [§ 3, Def. 51) such that x* x = r* r(= r r*). 9A. If A is a ring with unity, and e, f are idempotents in A such that Ae= Af, then e and f are similar (that is, e=xfx-' for a suitable invertible element x). 10A. If, in a ring A, e and f are algebraically equivalent idempotents (in the sense of Exercise 6), then the subrings eAe and fAf are isomorphic. 11B. Let A be a Rickart *-ring [§ 3, Def. 21 and suppose e, f' are projections in A that are algebraically equivalent (in the sense of Exercise 6). Then the *-subrings eAe and fAf have isomorphic projection lattices. 12A. Let A be a *-ring, e a projection in A, x ~ e A e ,and suppose x is invertible in eAe; say y ~ e A e x, y = y x = e . Then y ~ { x , x * ) "(the bicommutant of the set {x,x*} [I( 3, Def. 51). 13A. Let (A,),,, be a family of *-rings and let A =

nA,

be their complete

r tI

direct product (i. e., A is the Cartesian product of the A,, endowed with the coordinatewise *-ring operations). Then (i) A has a unity element if and only if every A, has one; (ii) an element x=(x,),,, of A is self-adjoint (idempotent, partially isometric, unitary, a projection, etc.) if and only if every x, is self-adjoint are projections in A, then (idempotent, etc.); (iii) if e=(e,),,, and f =(f;),,, e - f iff e,- f, for all L E I . 14B. Let A be a complex *-algebra and let M be a *-subset of A (that is, X E M implies x*EM). The following conditions on M are equivalent: (a) M is maximal among commutative *-subsets of A; (b)M is maximal among commutative *-subalgebras of A; (c) M ' = M; (d) M is maximal among commutative subsets of A. (Here M' denotes the commutant of M in A [$3, Def. 51.)Such an M is called a masa ('maximal abelian self-adjoint' subalgebra). Every commutative *-subset of A can be enlarged to a masa; in particular, if X E A is normal (i.e., x* x=xx*), then x belongs to some masa. 15A. If e d h, where h is a central projection, then e 5 h. 16A. If (A,),,, is a family of *-rings [*-algebras over the same involutive field K ] , we define their P*-sum A as follows: let B = A, be the complete direct 'GI

product of the A, (Exercise 13), write Bo for the *-ideal of all x=(a,),,, in B such that a,=O for all but finitely many 1 (thus, Bo is the 'weak direct product' of the A,), and define A to be the *-subring [*-subalgebra] of B generated by B, and the set of all projections in B. Thus, if P is the subring [subalgebra] of B generated by the projections of B, then A= Bo + P.

Chapter 1. Rickart *-Rings, Baer *-Rings,A W*-Algebras

10

17A. Let A bc the *-ring of all 2 x 2 matrices over the field of three elements, with transpose as involution. The set of all projections in A is {0,1, e, 1 -e, f , 1-J'), where

The only equivalences (other than the trivial equivalences g .f -1- f .

-

g) are e

-

I- e and

18A. The projections e,j' of Exercise 17 are algebraically equivalent, but not equivalent. 19A. With notation as in Exercise 17, eAe and ,fAf are *-isomorphic, although e and f are not equivalent.

"

20A. Let A be a *-ring with unity and let A, be the *-ring of all 2 x 2 matrices over A (with *-transpose as involution). If w is a partial isometry in A, say w* w= e, w w* =,f, then the matrix =

(,

is a unitary element of A, (that is, u* u=uu* = 1, the identity matrix). 21A. Does the Schroder-Bernstein theorem (i.e., the conclusion of Theorem 1) hold in every *-ring?

tj 2. *-Rings with Proper involution If A is a *-ring, the 'inner product' ( x ,y) = xy* ( x ,y~ A) has properties reminiscent of a Hermitian bilinear form: it is additive in x and y, and it is Hermitian in the sense that (y, x ) = ( x ,y)*. Nondegeneracy is a special event: Definition 1. Thc involution of a *-ring is said to be proper if x* x

=O

implies x=O. Proposition 1. In a *-ring with proper involution, xy=O if' and only

i f x*xy=O. Proof If x*xy=O, then y*x*xy=O, (xy)*(xy)=O,xy=O.

I

The theory of equivalence of projections is slightly simplified in a ring with proper involution: Proposition 2. In a *-ring with proper involution, w is a partial isometry i f and only if w* w is a projection.

Proof. If w* w = e, e a projection, straightforward computation yields (we- w)*(we- w)=O, hence we= w; thus w is a partial isometry [$ 1, Def. 61. 1

6 2. *-Rings with Proper Involution

11

This is a good moment to introduce a famous example:

Definition 2. A C*-alyebra is a (complcx) Banach *-algebra whose norm satisfies the identity Ilx*xll= 11x112. Remarks and Examples. 1. The involution of a C*-algebra is obviously proper. 2. If 2' is a Hilbert space then the algebra 9 ( 2 ) of all bounded linear operators in &?, with the usual operations and norm (and with the adjoint operation as involution), is a C*-algebra; so is any closed *-subalgebra of 9 ( 2 ) , and this example is universal: 3. If A is any C*-algebra, then there exists a Hilbert space .8such that A is isometrically *-isomorphic to a closed *-subalgebra of 9 ( , 8 ) (Gel'fand-Naimark theorem; cf. [75, Th. 4.8.111, [24, Th. 2.6.11). 4. If A is a C*-algebra without unity, and A , is the usual algebra unitification of A [§ 5, Def. 31, then A , can be normed to be a C*-algebra [cf. 75, Lemma 4.1.131. 5. If T is a locally compact (Hausdorff) space and C,(T) is the *-algebra of continuous, complex-valued functions on T that 'vanish at infinity', then C,,(T) is a commutative C*-algebra; in order that A have a unity element, it is necessary and sufficient that T be compact (in which case we write simply C(T)). Conversely, if A is a commutative C*-algebra and A! is the character space of A (i.e., the suitably topologized space of modular maximal ideals of A), then the Gel'fand transform maps A isometrically and *-isomorphically onto C,(A') (commutative Gel'fand-Naimark theorem [cf. 75, Th. 4.2.21). Exercises 1A. In a *-ring with proper involution, if e is a normal idempotent (that is,

e* e=ee* and e2=e) then e is a projection.

2A. A partial converse to Proposition 1: If A is a *-ring in which xy=Oiff x* x y = 0, and if x ~ Axfor every x (e. g., if A has a unity element, or if A is regular in the sense of von Neumann [5;51, Def. I]), then the involution of A is proper.

3A. In a *-ring A with proper involution, x* xAy = O implies xx* Ay = 0 4A. The complete direct product of a family of *-rings [$I, Exer. 131 has a proper involution if and only if every factor does.

5A. If R is a commutative ring # (0) and if A is the ring of all 2 x 2 matrices over R, then thc correspondence

defines an improper involution on A

6B. The involution is proper in a *-ring satisfying the (VWEP)-axiom [$7, Def. 31.

12

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

5 3.

Rickart *-Rings

To motivate the next definitions, suppose A is a *-ring with unity, and let w be a partial isometry in A. If e = w* w, it results from w= ww* w that wy=O iff ey=O iff (I -e)y=y iff y ~ (-e)A, l thus the elements that right-annihilate w form a principal right ideal generated by a projection. The idea of a Rickart *-ring (defined below) is that such a projection exists for every element w (not just the partial isometries). It is useful first to discuss some generalities on annihilators in a ring (always associative): Definition 1. If A is a ring and S is a nonempty subset of A, we write R(S)= {xEA:sx=O for all SES) and call R(S) the right-annihilator of S. Similarly, L(S)={x€A: xs=O for all SES) denotes the left-annihilator of S. Proposition 1. Let S, T and S, ( L E I be ) nonempty subsets of a ring A. Then: (1) S c L(R(S)), S c R(L(S)); (2) S c T implies R(S) I> R(T) and L(S) 3 L(T); (3) R(S)= R(L(R(S))), L(S)= L(R(L(S))); ( 5 ) R(S) is a right ideal of A, L(S) a leji ideal. (6) If J is a left ideal of A, then L(J) is an ideal of A, in other words, the left-annihilator of a left ideal is a two-sided ideal. Similarly, the rightannihilator of a right ideal is an ideal. (7) If A is an algebra, then R(S) and L(S) are linear subspaces (hence are subalgebras) . (8) If A is a *-ring then L(S) = (R(S*))*, where S* = {s*: S E S). Similarly, R(S) = (L(S*))*.

Proof. There is nothing deeper here than the associative law for multiplication. I Definition 2. A Rickart *-ring is a *-ring A such that, for each XEA, R((x})=gA with g a projection (note that such a projection is unique [§ 1, Prop. I]). It follows that L({x))= (R({x*)))*= (hA)*= Ah for a suitable projection h. The example that motivates the terminology:

# 3. Rickart *-Rings

13

Definition 3. A C*-algebra that is a Rickart *-ring will be called a Rickart C*-algebra. {These are the 'BZ-algebras', first studied by C. E. Rickart [74].) Proposition 2. I f A is a Rickart *-ring, then A has a unity element and the involution of A is proper. Proof. Write R ( ( 0 ) )= gA, g a projection. Since R ( ( 0 ) )= A , we have A=gA, thus g is a left unity for A ; since A is a *-ring, g is a (twosided) unity element. Suppose xx* = 0. Write R ( { x ) )= hA, h a projection. By assumption, x * E R ( { x ) ) ,thus hx*=x*, x h = x ; then h ~ R ( { x )yields ) O=xh=x. I Proposition 3. Let A be a Rickurt *-ring, X E A. There exists a unique projection e such that (1) x e = x , and (2) x y = 0 iff e y = 0. Similarly, there exists a unique projection f such that (3) f x = x , and (4) y x = 0 iff y f =0. Explicitly, R ( { x ) )=(1- e)A and L ( { x ) )= A ( l - f ). The projections e and f are minimal in the properties (1) and (3), respectively. Proof. Let g be the projection with R ( ( x ) ) = g A , and set e= 1 - g ; clearly e has the properties (1) and (2). I f h is any projection such that x h = x , then x(1-h)=O, e(1-h)=O, e l h . I Definition 4. W i t h notation as in Proposition 3, we write e=RP(x), f = L P ( x ) , called the right projection and the left projection o f x. Proposition 4. In a Rickart *-ring, (i) LP(x)=RP(x*), (ii) xy=O iff RP(x)LP(y)=O, (iii) if w is a partial isometry, then w* w = RP(w) and w w* = LP(w). ProoJ: (i) is obvious, (ii) is immediate from Proposition 3, and (iii) follows from the discussion at the beginning o f the section. I The following example is too important t o be omitted from the mainstream o f propositions (see also [$4, Prop. 31): Proposition 5. If % is a Hilbert space, then 9(%) is a Rickart C*algebra. Explicitly, i f T E ~ ( % )then L P ( T ) is the projection on the closure of the range of T, and I - R P ( T ) is the projection on the null space of T. Proof: Let F be the projection on T ( 2 ) . For an operator S, the following conditions are equivalent: S T =0, S=O on T ( 2 ) ,S=O on T(%), SF=O, S(1-F)=S, S E ~ ( % ) ( I - F ) .Thus L ( { T } )= L?(
24

Chapter 1. Rickart *-Rings, Baer *-Rings. A W*-Algebras

This shows that Y ( # ) is a Rickart *-ring (dualize Definition 2). It the projection on T*(S/&), therefore follows that RP(T)=LP(T*) is-I - R P ( T ) is the projection on (T*(,X))'= (T*(,X))', which is the null space of T. I -

In the preceding example, the projections happen to form a complete lattice (because the closed subspaces do); this is not true in every Rickart *-ring (or even in every Rickart C*-algebra-see Example 2 below), but when a particular supremum does exist, it has annihilation properties: Proposition 6. Let A he a Rickart *-riny and suppose (e,) is a family of projections that has a suprenzurn e. If x~ A , then x e =O iff xe, = O for all 1.

Proqf. In view of Proposition 3, the following conditions are equivalent: xe=O,RP(x)e=O,e
for every pair ofprojections e, f. Prooj'. Write x = e ( l - f ) = e-e,f and set q=RP(x). Since xj'=O, one has y f = 0, thus f + g is a projection. Write h =f + g; it is to be shown that h serves as sup(e, j ) . Obviously j < h. Also, e < h; for, O=x g - x = (e-ef') y -(e-e,f') thus e = e f + e g = e h . =eg-e(fg)-e+ef=eg-e+eJ; On the other hand, suppose e 5 k and f 5 k; it is to be shown that h < k. Clearly x k = x, thus y < k; combined with f < k, this yields f + g 5 k. Thus h = e u J: It follows at once that e n f exists and is equal to 1 - (1 - f') u (1 - e), and that

If A is a Rickart *-ring and B is a *-subring of A, then B need not be a Rickart *-ring: an obvious example is when B has no unity element, but adjoining a unity element may not be a remedy (see, e.g., Example 1 below). There is, nevertheless, a useful positive result:

Proposition 8. Let A he a Rickart *-ring and let B be a *-subring suclz that (1) B has a unity element, and (2) X E B implies R P ( x ) t B . Then B is also a Rickart *-ring; moreover, if XEB, then the right projection of x is the same whether calculated in A or in B. Proof. Write e for the unity element of B (we do not require that e = l ) . Let X E B andlet g=RP(x). By assumption, g t B ; since x e = x , we have y l e . If ~ E then B xy=O iff gy=O iff (e-y)y=y, thus the right-annihilator of x in B is ( e- g) B. I A useful application: Corollary. If A is a Rickurt *-ring and e is a projection in A, then eAe is also a Rickart *-ring (with unamhiguous RP's and LP's). We interrupt the general theory for two instructive examples. Proposition 7 suggests the question: If the projections of a *-ring form a lattice, is it a Rickart *-ring? An obvious counterexample is given in Exercise 2; a subtler example is the following:

Example 1. There exists a C*-algebra with unity, whose projection., form a lattice, hut in which, for certain elements x,there exists no smallest projection e such that x e = x. Proqf. The set of all compact operators a on an infinite-dimensional Hilbert space .f forms a C*-algebra A ; let A , be its unitification: where 1 is the identity operator on ,%.' The set of projections in A is the set of all projections of finite rank; this is a lattice. It is easy to see that the projections of A , are the projections e, I-e, where e is a projection in A (cf. [ $ 5 , Lemma 21); it follows that the projections of A , form a lattice too. (Sample calculation: if e, f are projections of finite rank, then e u (I - f ) = I-(I - e) nf is the complement of a projection of finite rank.) Fix a compact operator a with infinite-dimensional range and infinite-dimensional null space, and let y be the projection on the null space of a. Both g and I- y have infinite rank (in other words, g$ A,). Suppose h is any projection in A , with a h = a ; we show that there exists a smaller projection k in A , with a k = a (thereby thwarting the bid of 11 to be the 'right projection of a in A,'). From a(1- h)=O we have I- h s g, 1 - y I h , therefore h has infinite rank; since l z A, ~ it follows that I -h has finite rank. Choose a projection e of finite rank such that I - h < e < g and I - h f e , andset k = l - e . I

Example 2. There exists a Rickart C*-algebra whose projection lattice is not complete.

16

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

Proof. The motivating idea is that the closed linear span of a family of separable subspaces may be inseparable. Let &?' be an inseparable Hilbert space, B = 2 ( % ' ) the algebra of all bounded operators on H , and let A be the set of all operators with separable range: A= {aeB: a(%') has a countable dense subset} Obviously A, is a C*-algebra; we show that it is a Rickart *-ring with incomplete projection lattice. Regard B as a Rickart *-ring (Proposition 5); thus, if X E B then 1 -RP(x) is the projection on the null space of x. To prove that A, is a Rickart *-ring, it will suffice, by Proposition 8, to show that XEA, implies RP(x)EA,. Suppose XEA,, say x = a + l l . If A=O then RP(x) =RP(a) = LP(a*)€ A (because the closure of a separable linear subspace is separable). If A# 0, write RP(x)= 1 -8, where g is the projection on the null space of x; then (a+,? l)g=O, g = a ( - A - l g ) ~ A (because A is an ideal in B), thus I- g E A,. Finally, the projection lattice of A, is incomplete; for example, if ~ E B is any projection such that both e and I -e have inseparable range, then the separable subprojections of e can have no supremum in A,. Incidentally, the projections of A, and therefore of A,, do form a o-lattice (that is, countable sups and infs exist). {We remark that if dim%> K , then the game can be played with ideals of B other than A.) I Returning to the general theory, we assemble some elementary facts about commutants in a ring (always assumed associative), with a view to applying them to obtain new Rickart *-rings from old.

Definition 5. If A is a ring and S is a nonempty subset of A , the cornmutant of S in A, denoted Sf,is the set of elements of A that commute with every element of S: S'= {xEA:x s = s x for all SES).

We write S" = (S')', called the bicommutant of S; and S"' = (S")' (but see the following proposition).

Proposition 9. Let A be a ring, S and T nonempty subsets of' A. Then: (1) S' is a suhring of A ; (2) if S c T then S'x T'; (3) scs"; (4) S'=S"'; (5) S is commutative iff's" is commutative. (6) If A has a unity element, then 1ES' and S' contains inverses.

jj 3. Rickart *-Rings

17

(7) If A is an algebra then S' is a subalgebra of A. (8) If A is a *-ring and S is a *-subset of A, then S' is a *-subring of A. Proof. (1)-(4) and (7) are routine. ( 5 ) If S is commutative, that is, if S c S', then S " c S' = ( S ) ' by (2) and (4),thus S is commutative. The converse follows trivially from (3). (6) Assuming A has a unity element and ~ G S is ' invertible in A, say x y = y x = I, it is to be shown that Y E S ' . For all S G S , we have x s = s x ; left- and right-multiplication by y yields s y = y s . (8) The assumption is that S E S implies s*eS; the easily inferred conclusion is that xeS' implies x*eS'. I

A very useful application: Proposition 10. Let A be a Rickart *-ring, and let B be a *-subring of A such that B = B (equivalently, B = S' ,for some *-subset S of A). Then (i) X G B implies R P ( x ) eB, and (ii) B is also a Rickart *-ring (with unambiguous RP's and LP's) . Proof. For a *-subring B, the equivalence of the condition B=B" with the parenthetical condition follows from (4)and (8)of Proposition 9. By Proposition 8 we need only prove (i). Let x e B , e=RP(x), y~ B'; since B=(B')' it will suffice to show that e y = ye. Since x y = y x and x e = x , we have xCy-ye)=xy-xye=xy-yxe=xy-yx=0,

therefore eCy -ye) =0, e y = e ye. Replacing y by y* (which is also in B') we have ey*=ey*e, that is, y e = e y e ; thus ye=eye=ey. I

Corollary 1. If A is a Rickart *-ring, then the center of A is also a Rickart *-ring (with unambiguous RP's). Explicitly, if x y = y x ,for all y e A, then RP(x) is a central projection. Proof. Take B = A' in Proposition 10.

1

Corollary 2. If A is a Rickart *-ring and x* = x e A, then R P ( x ) e{x)". Proof: S= {x)' is a *-ring; take B= S' in Proposition 10.

1

Propositions 1 and 9 show an analogy between annihilators and commutants; the commutant analogue of Proposition 6 is as follows:

Proposition 11. Let A be a Rickart *-ring, B a *-subring of A such that B = B . I f (el)is a family ofprojections in B that possesses a supremum e in A, then e e B . Proof. Assuming yeB', it is to be shown that ey=ye. Since e,eB for all 1, we have

1X

Chapter 1. Rickart *-Rings, Baer *-Rings, AW*-Algebras

therefore e(y-ye)=0 Proposition 10. 1

by Proposition 6. The proof continues as in Exercises

1A. In a Rickart *-ring, every idempotent is similar to a projection; in particular, every central idempotent is a projection. , exists a smallest 2A. If A is a *-ring such that, for each element ~ E A there projection e with xe=x, it does not follow that A is a Rickart *-ring. (Consider, for example, A= C(T), where T is a connected compact space with more than one point.)

3C. Let A, be the algebra constructed in Example 1. For any XEA,, write RP(x) for the right projection of x as calculated in the algebra of all bounded operators on 2'.(i) If x = a + l l with UEA and l#O, then RP(x)€A, (RieszSchauder theory). (ii) If x = a c A and a has finite rank, then RP(X)EA,. (iii) If x = a ~ Aand a has finite-dimensional null space, then RP(x)€A,. (iv) Thus, the pathology exhibited in the proof of Example 1 is the only kind possible. 4A. Let A be a Rickart *-ring, I an ideal in A. In order that I be a direct summand of A, it is necessary and sufficient that there exist a central projection h such that I = hA. (An ideal I in a ring A is said to be a direct summand of A if there exists an ideal J such that A= I 8 J, in the sense that A = I + J and I n J = {O).) 5A. Let A be a Rickart *-ring, and suppose (h,) is a family of central projections such that (i) each h,A is a commutative ring, and (ii) h=sup h, exists. Then hA is also commutative. , exists an element YEA 6A. A ring A is called regular if, for each ~ E A there such that x y x = x ; a *-ring A is called *-regular if it is a regular ring with proper involution. For a *-ring A with unity, the following conditions are equivalent: (a) A is *-regular; (b) for each XEA there exists a projection e such that Ax=Ae; (c) A is regular and is a Rickart *-ring. 7A. An associative ring A is called a Rickart ring if, for each X E A, there exist idempotents e, f' such that R({x))= eA, L({x))= Af. (i) Every Rickart ring has a unity element. (ii) If A is a symmetric *-ring [§ 1, Exer. 71 and if A is a Rickart ring, then A is a Rickart *-ring. 8A. If A is a Rickart *-ring, B is a *-subring satisfying B = B , S is a *-subset of B. and C is the commutant of S in B, then C is also a Kickart *-ring, with unambiguous RP's and LP's; indeed, C = C". 9A. A set U of nonzero projections in a *-ring is said to be ubiquitous if, for each nonzero projection e, there exists f E U with f < e. If U is a ubiquitous set of projections in a Rickart *-ring, then, given any nonzero projection e, there exists an orthogonal family (f,) with f , U~ and e = sup f;. 10A. Suppose A is a *-ring such that, for each nonzero x in A, there exists YEA with xy=g, g a nonzero projection. Let U be a ubiquitous set of projections in A (see Exercise 9). If e is any nonzero projection in A, then there exists an orthogonal family (f,) with , f ,U~and e = sup f,. 11A. Let A be a Rickart *-ring and suppose e, f a r e projections in A that are algebraically equivalent (see [tj 1, Exer. 61). Then thc projection lattices of eAe and

6 3.

Rickart *-Rings

19

fAf are isomorphic. Explicitly, if x ~ , f A eand y ~ e A fsatisfy y x = e , x y = l ; then the formula cp(g)=RP(xgy) defines an order-preserving bijection cp of the projection lattice of eAe onto the projection lattice of fAj: Moreover, if g, I g, e, then the projections g, -g, and cp(g2)- cp(g,) are algebraically equivalent. 12A. If (A,),,, is a family of *-rings and A is their complete direct product [$I, Exer. 131, then A is a Rickart *-ring iff every A, is a Rickart *-ring. 13A. If A is a Rickart *-ring and B is a *-subring that contains every projection of A, then B is also a Rickart *-ring (with unambiguous RP's). 14C. A Boolean ring is a ring in which every element x is idempotent (x2=x). A Boolean ring with unity is called a Boolean algebra.

(i) In a Boolean ring, x +x =0 for all elements x. (ii) A Boolean ring is commutative. (iii) Every Boolean ring with unity is aRickart *-ring, with the identity involution x* = x ; explicitly, RP(x) = x for all elements x. (iv) If T is a separated (i.e., Hausdorff) topological space, then the set of all compact open subsets P, Q, ... is a Boolean ring with respect to the operations P n Q (intersection) and P A Q = (P - Q) u (Q - P) (symmetric difference) as product and sum. (v) If A is any Boolean ring, there exists a topological spacc T such that A is isomorphic to the Boolean ring of all compact open subsets of T. One can suppose, in addition, that T is locally compact and that the compact open sets are basic for the topology; then T is unique up to homeomorphism (it is called thc Stone representation space of A; it is compact iff A has a unity element). (vi) If A is a Boolean ring, then A is a lattice with respect to the operations x n y = x y and x u y = x + y +xy. (In this connection, the term complete Boolean algebra is self-explanatory.) A Boolean algebra is complete iff its Stone representation space is Stonian (i.e., the closure of every open set is open). (vii) Let B be any *-ring all of whose projections commute [cf. 8 15, Exer. 21. Let A be the set of all projections in B and define products and sums in A by the formulas ef and e + f -e$ Then A is a Boolean ring. 15.4. Let S c B c A, where A is a ring and B is a subring of A. Writc ,Y- for the commutant of S in B; thus st=B n S', where S' is the commutant of S in A. Then s~~IB~s". 16A. If u is an idempotent in a Rickart *-ring, such that LP(u) commutes with RP(u), then u is a projection. 17A. If (A,),,, is a family of Rickart *-rings, then the P*-sum of the A, [$I, Excr. 161 is also a Rickart *-ring. 18A. If A is a *-ring, we denote by A" the ring generated by the projections of A, and call it the reducedring of A; explicitly, A" is the set of all finite sums of elements of the form +el e2 ... en, where the ei are projections in A. We write e 2 f if there exists a partial isometry w in A" such that w* w = e, w w* =j (that is, e f relative to A"). (ij If A is a Rickart *-ring, then A" is also a Rickart *-ring (with unambiguous RP's). (ii) If A is a Rickart *-ring and e,f are orthogonal projections in A such that e - f , then e X f. (iii) If A is a Rickart *-ring in which 2 is invertible, and if e,,f are projections such that ueu= f with u a symmetry (= self-adjoint unitary), then e 2 f:

-

20

C:hapter 1. Rickart *-Rings, Baer *-Kings, A W"-Algcbras

(iv) If A is the *-ring of 2 x 2 matrices over the field of threc clcmcnts, then A" = A. Explicitly, if e, f are the projections described in [9 1, Exer. 171, and if

then w=2e-,fe.

5 4.

Baer *-Rings

The definition of a Rickart *-ring [ $ 3 , Def. 21 involves the annihilators of singletons; for a Baer *-ring, one considers arbitrary subsets: Definition I. A B a n *-ring is a a-ring A such that, for every nonempty subset S of A, R(S)=yA for a suitable pro.jcction g. (It follows that L(S)= (R(S*))*= (hA)*=Ah for a suitable projection 11.) The relation between Kickart *-rings and Baer *-rings is the relation between lattices and complete lattices: Proposition 1. Tlze following conditions on u *-ring A are equizlalent: (a) A is a Baer *-ring; (b) A is a Rickart *-ring whose projec,tion.sfbrm a comnplete lattice; (c) A is a Rickart *-ring in which c1c.erj>orthogona1,fumilyof'projec/ions has a szdprrmurn. P r o 4 (a) implies (b): Suppose A is a Baer *-ring and S is any nonempty set of projections in A. Write R(S)=(I e ) A, e a projection; we show that e is a supremum for S. Since 1- e E R(S), we have j'(1 - e) = 0 for all f i S, that is, f 5 e for all ,f€S. On the other hand, if g is a projection such that f ' sy for all ~ E S then , I-g€R(S)=(l-e)A, thus 1 - 9 5 1 -e, e l g . This shows that s u p s exists and is equal to e. It follows trivially that infS exists and is equal to I -sup(l - f : j ' ~ S ) . (b) implies (c) trivially. (c) implies (a): Lct S be a nonempty subset of A. Note that x€R(S) iff LP(x)cR(S) [$3, Prop. 4, (ii)]. It follows that if the only projection in R(S) is O then R(S) = (0) = OA. Otherwise, let (e,) be a maximal orthogonal family of nonzero projections in R(S). By hypothesis, e = supe, exists; we conclude the proor by showing that R(S)=eA. At any rate, e~ R(S) 153, Prop. 61, therefore eA c R(S). Conversely, assuming x€R(S), we assert that x €eA. Setting y = x - ex, it is to be shown that y=O. Note that Y E R(S), therefore LP(y)eR(S). Also, for all I, we have e,y= e,x-e,ex=e,x-e,x=O, therefore e, L P(y)=O. It follows from maximality that L P k ) =0, thus y = 0. I

# 4. Baer *-Rings

21

Completeness of the projection lattice permits a useful characterization of annihilators in a Baer *-ring: Proposition 2. If' A is a Baer *-ring and S is a nonempty subset qf' A, then R(S)= (1 -g) A, where g = sup{RP(s): ss S).

Proqf. The point is that XER(S) iff gx=O, and this is immediate from [$3, Prop. 61. I The concept of Baer *-ring is due to I. Kaplansky (see 1521, 1541). It arose from his study of two special cases: the 'complete *-regular rings' (this is the class of rings described in Exercise 4, (ii)) and the class of C*algebras described in the following definition: Definition 2. An AW*-algebra is a C*-algebra that is a Baer *-ring. {A term consistent with 153, Def. 31 would be 'Baer C*-algebra'; but, with most of the literature of such algebras probably already written, it is too late to change.}

AW*-algebras were proposed by Kaplansky (1471, 1481, [49]) as an appropriate setting for certain parts of the algebraic theory of von Neumann algebras (defined below). There are two elemental examples, of which most other examples are refinements or derivatives: (I) the algebras C(T), where T is a Stonian space (these are discussed in Section 7), and (2) the following: Proposition 3. If X is a Hilbert space, then Y(Af) is an AW*-algebrn.

Proof. We know already that T(.X') is a Rickart C*-algebra [$3, Prop. 51; in view of Proposition 1, it suffices to check that its projection lattice is complete. Indeed, the correspondence E HE ( . X ) is an isomorphism of the projection lattice of Y ( X ) onto the lattice of closed linear subspaces of Z, and the latter is obviously complete. {Morc explicitly, if Y is a nonempty subset of 9 ( , X ) and if F is the projection on the closed linear span of the ranges of the S E ~ it, is easy to see directly that L(.Y)=Y ( X ) ( l - F).) I A capsule resum6 of the rest of the section is 'subrings of Baer *-rings'. Other 'catogorical' matters requiring attention are direct products (see, e. g., Exercises 7 and 8, and Section lo), quotient rings (see Chapter 7) and matrix rings (see Chapter 9); this is not the place to go into detail, but it ought to be reported here that the general results obtained are shallow, and deeper results are obtained only at the cost of restrictive hypotheses. In the category of Baer *-rings [cf. $23, Exer. 61, what is the appropriate notion of 'subobject"? To be precise, suppose A is a Baer *-ring: what is the appropriate notion of 'sub Baer *-ring'? An obvious choice would be to consider *-subrings B that are themselves Baer *-rings. The

22

Chapter 1. Rickart *-Rings, Baer *-Ring,, A W*-Algebras

disadvantage of this choice is that there is ambiguity in the lattice operations: a set S of projections in B has two competing supremasup S as calculated in the projection lattice of A, and sup,S as calculated in the projection lattice of B-and they may be different [$7, Exer. 41. The difficulty is avoided by choosing the right definition:

Definition 3. Let A be a Baer *-ring, B a *-subring of A. We say that B is a Baer *-subring of A provided that (i) X E B implies RP(x)E B, and (ii) if S is any nonempty set of projections in B, then s u p S ~ B . It follows that B is itself a Baer *-ring, with unambiguous lattice constructs:

Proposition 4. If' A is a Baer *-ring and B is a Baer- *-.subring of A, tlzm B is also a Baer *-ring, and RP's, LP's, sups and infs in B are unambiguous. Proof. Let e = s u p ( R P ( x ) : x ~ B ) ; by hypothesis eEB, and it is clear that e is a unity element for B. It follows from [53, Prop. X I that B is a Rickart *-ring, with unambiguous RP's and LP's. In addition, since the projection lattice of A is complete (Proposition I), it follows from Definition 3 that the projection lattice of B is also complete, therefore B is a Baer *-ring (Proposition 1) with unambiguous sups and infs. I Right-annihilators in a Baer *-subring are 'unambiguous' too:

Corollary. Let A be a Baer *-ring, B a Baer *-subring of A, e the unity element of B, and let T be a nonempty subset o j B. Write R ( T )= (1 - g) A, g a projection. Then the right-annihilator of T in B is (e - g) B. Proof. From Proposition 2, we know that g = sup(RP(t):t r T ) ; since RP's and sups in B are unambiguous, it follows from Proposition 2 (applied to B) that the right-annihilator of T in B is (e-g)B. I A convenience of Definition 3 is that it is obviously stable under intersection :

Proposition 5. 1 j A is a Baer *-ring and (B,) i~ any farnily oj Baer *-subrings of A, then B, is also a Baer *-subring. {lt follows in the obvious way that if S is any subset of a Bacr *-ring A, there exists a unique smallest Baer *-subring of A that contains S, called the Baer *-subring generated by S. However, the latter concept will play no direct role in the present text; the Baer *-subrings that do occur are more conveniently described in other ways.) The next two propositions yield the major examples of Baer *-subring:

Proposition 6. If A is a Baer *-ring and e is a projection in A, then r Ae is a Baer *-subring o j A.

4. Raer *-Rings

23

Proof. Since the pro-jections of eAe are precisely the projections of A I that are 5 e, the criteria of Definition 3 are trivially verified.

Proposition 7. If A is a Baer *-ring and B is a *-subring such that B = B" (equivalently, B= S' for some *-subset S of A ) , then B is a Baer *-subring of A. Proof. The criteria (i) and (ii) of Definition 3 are fulfilled, by [$3, Prop. 101 and 193, Prop. 111, respectively. I

Corollary 1. Let A be a Baer *-ring, B a *-subring such that B= B". If T is a *-subset of B and if C is the commutant of T in B, then C is u Baer *-subring of A ; indeed, C = C". ProoJ: By definition, C = B n T ' = (B' u T)', where B' u T is a *-subset of A ; quote Proposition 7. 1

Corollary 2. If A is a Baer *-ring, then the center Z of A is a Baer *-subring of A. More generally, if B is a *-subring of A such that B = B rind iJ C is the center of B, then C is u Baer *-.subring of A ; indeed, C = C . Proof. Put T = B in Corollary I;explicitly, C = B n B' =(B1u B)'.

I

The foregoing results apply, in particular, to A W*-algebras, modulo the following definition:

Definition 4. Let A bc an A W*-algcbra, B a *-subalgcbra of A. Wc say that B is an A W*-subalgebra of A provided that (1) B is a norm-closed subset of A (hence B is a C*-algebra), and (2) B is a Baer *-subring of A in the sense of Definition 3. Proposition 8. Let A be an AW*-algebra. (i) If B is an AW*-subalgebra of A, then B is also an A W*-ulgehra,and RP's, LP's, sups and infi in B are ~tzatnhiguou~s; morpovcr, { f T is a nonempty subset of B and i f R(T)=(I - g ) A , g a projection, then the r.icghtannihilator of' T in B is ( e - g) B, where e is the unity element of B. (ii) If (B,) is any jatnily qf AW*-.subalgehra.s of' A, then r) B, is also an A W*-~uhalgebra. (iii) I f e is a projection in A, then eAe is an AW*-subalgebra oj' A. (iv) IJ'B is a *-subalgebra of A such that B = B" (equivalently, B = S' ,for some *-subset S of A ) , then B is an AW*-subalgebra of A. (v) The center r!f' A is an AW*-subalgebra of A ; more generullj~, (vi) if' B is a *-subalgebra qf A such that B = B", if T is a a-.subset of' B, and zf' C is the commutant of' T in B, then C is an AW*-.subalgebra of A ; indeed, C = C . Proof: (i) Since B is a C*-algebra, this is immediate from Proposition 4 and its corollary.

24

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

(ii) All that needs to be added to Proposition 5 is the remark that the intersection of a family of closed sets in closed. (iii) This is immediate from Proposition 6, and the observation that eAe is a closed subset of A . (iv)-(vi) are immediate from Proposition 7 and its corollaries, plus the observation that cornmutants in A are closed subsets. I The single most important application of all of this: Definition 5. If ,X is a Hilbert spacc, a uon Neumann algebra on Y/t' is a *-subalgebra .d of 9(,%?) such that d = .d". Proposition 9. Every vvon Neumann algebra is an A W*-alcgebru. Puooj: Immediate from Proposition 3, and Part (iv) of Proposition 8. I Remarks. The converse of Proposition 9 was shown to be falsc by J. Dixmier 1201, who showed that there exist commutative AW*-algebras that cannot be represented (*-isomorphically) as von Neumann algebras on any Hilbert space (see [$7,Exer. 2,3]). At the other end of the commutativity spectrum, it has recently been announced by J. Dyer [25] that there exist AW*-algebras with one-dimensional center that are not *-isomorphic to any von Neumann algebra. In between, it is known that certain types of A W*-algebras are von Neumann-representable provided that their centers are (cf. [$IS, Exer. 7, 81). Historical note. If .d is a von Neumann algebra on a Hilbert space

-8, it is elementary that .d is closed in 9(2) for the weak operator topology. Von Neumann showed, conversely, that if .d is a weakly

closed *-subalgebra of 9 ( X ' ) containing the identity operator, then .d=dl'[cf. 23, Ch. 1, $3, No. 41. These algebras have also been called 'W*-algebras' (von Neumann called them 'rings of operators'), the W referring to closure in the weak operator topology; thus 'A W*-algebras' are proposed as an abstract (or algebraic) generalization of W*-algebras (see I. Kaplansky 1471). Exercises 1A. If S is a nonempty set of projections in a Baer *-ring A, then

(inf S)A = r ) eA , rtS

2A. (i) A *-ring with unity and without divisors of 0 (that is, xy=O implies x = O or y = 0) is trivially a Bacr *-ring. Examples: any division ring with involution ; any integral domain (in particular, any field) with the identity involution. (ii) A Baer *-ring need not be semisimple.

$ 4 . Baer +-Kings

25

3A. (i) The following conditions on a Rickart *-ring A are equivalent: (a) 0,1 are the only idempotents in A; (b) 0,1 are the only projcctions in A; (c) A has no divisors of 0. Such a ring is trivially a Baer *-ring, in which y x = I implies x y = 1; if, in addition, A has finitely many elements, thcn A is a (commutative) field. (ii) A nonzero projection e in a Rickart *-ring A is minimal if and only if eAe has no divisors of 0. (Anonzcro projection cJ in a *-ring is said to be nzinimal if 0 and e are its only subprojections.) 4A. (i) If A is an associative ring, any two of the following conditions imply the third: (a) if S is any subset of A, thcn R(S)=eA for a suitable idempotent e; (b) if S is any subset of A, thcn L(S)=A j for a suitable idempotent j; (c) A has a unity element. A ring with these properties is called a Baer ring. (ii) The following conditions on a *-ring A are equivalent: (a) A is a regular Baer *-ring; (b) A is a *-regular Baer ring. (See [53, Excr. 61 for the definitions of regular and *-regular.) Such rings are also called complete *-regular rings.

5A. If A is a symmetric *-ring [$I, Exer. 71 and if A is a Bacr ring (see Exercise 4), then A is a Baer *-ring. In particular, the A W*-algebras may be described as the C*-algebras that are Baer rings. 6A. If A is a Baer *-ring and B is any *-subring of A that contains all projections of A, then B is a Baer *-subring of A. In particular, the reduced ring A' of A is a Bacr *-subring [$3, Excr. 181.

7A. If (A,),,, is a family of Baer *-rings and A =

n

A, is their completc dircct

'tl

product, then A is also a Baer *-ring. The supremum of a nonempty set of projections in A may be calculated coordinatewise. 8A. If (A,),,, is a family of Baer *-rings, then the P*-sum of thc A , [§ 1, Exer. 161 is also a Baer *-ring.

9A. Let A be a Baer *-ring, B a *-subring such that B = B . Lct T bc a nonempty subset of B, and write R(T)=rA, e a projection. Prove that ~ E from B first principles. 10A. If A is a Baer*-ring and J is a right ideal of A , then R(J) is a direct summand of A [cf. 5 3, Exer. 41. 11C. If ./if is a Hilbert space and .d is an A W*-subalgcbra of Y(:/f) containing the identity operator, then .d is a von Ncumann algcbra on .Y.(See also [g 18, Exer. 81.) 12A. Let A be a *-ring, XEA, (, a projcclion in A. In analogy with opcralors on Hilbert space, the condition e x e = x e may be interpreted as "e is invariant under x", and e x = x e as "e reduces x". (i) If e x e = x e then (I - e)x*(l - e)= x*(l - e ) . So to speak, if e is invariant under x then its orthogonal complemcnt is invariant under x*. (If A has no unity element, then the use of 1 is formal: x*(l -e) stands for x*-x*e.) , let (e,),,, be an orthogonal family of (ii) Let A be a Baer *-ring, ~ E A and projections such that e,xe,=xe, for all L E I . For any subset J of I, define e,=sup{e,: I E J ) . Then e J x e J = x q . (iii) Suppose, in addition to the hypothesis of (ii), that supr,= I. Thcn e,x*e,=x*e, for all 1 , hence xe,=e,x for all 1 . 13A. Assume (i) C is a *-ring with unity and proper involution, (ii) A is a *-subring of C, (iii) all projections of C are in A, and (iv) for cvery ~ E C1,+x*x

26

Chapter I. Rickart *-Rings. Baer *-Rings, A W*-Algebras

is invertible and (1 +x*x)-'EA. Then A is a Rickart *-ring [Baer *-ring] iff C is a Rickart *-ring [Baer *-ring]. 14A. A real AW*-algebra is a Banach *-algebra A over the field of real numbers, such that I(x*xl(=llxl12 for all XEA, and such that A is a Baer *-ring. Every AW*-algebra is trivially a real AW*-algebra (by restriction of scalars). A nontrivial example is the algebra of bounded sequences that are 'real at infinity', defined as follows. Let A be the set of all sequences x=(l,) of complex numbers, such that IA,l is bounded and Im l, +0. (i) With the coordinatewise operations, A is a commutative algebra over the field of real numbers. (ii) Setting x*=(l,*), A is a *-ring with proper involution. (iii) The projections in A are the sequences of 0's and 1's. (iv) Every orthogonal family of nonzero projections in A is countable. (v) A is a Baer *-ring. (vi) Setting llxl(= sup ll,l, A is a real Banach algebra, such that ((x*xll=Ilxl12. Thus A is a real AW*-algebra. (vii) A is symmetric [$ 1, Exer. 71. (viii) Write x 2 0 in case 1,2 0 for all n (equivalently, x = y * y for some y t A). If x 2 0 then there exists a unique y 2 0 in A such that x = y 2 . It follows that A satisfies that (UPSR)-axiom [jj 13, Def. 101 (note that S'=A for every nonempty subset S of A). (ix) If x e A , x#O, there exists YEA such that x y = e , e a nonzero projection. It follows that A satisfies the (EP)-axiom [$ 7, Def. I]. 15A. The algebra of all bounded operators on a real Hilbert space is a real A W*-algebra (see Exercise 14). 16A. (i) Every complete Boolean algebra is a commutative Baer *-ring [cf. 5 3, Exer. 141. (ii) Let B be a Rickart *-ring all of whose projections commute, and let A be the set of all projections in B. Endow A with the Boolean algebra structure described in [$ 3, Exer. 14, (vii)] and the identity involution. Then B is a Baer *-ring iff A is a complete Boolean algebra iff A is a Baer *-ring. 17A. If A is a Baer*-ring and S is a nonempty subset of the center of A, then R ( S ) is a direct summand of A . 18A. A Rickart *-ring with finitely many elements is a Baer *-ring. 19A. The ring A of all 2 x 2 matrices over the field of three elements. with transpose as involution, is a Baer *-ring [cf. 5 1, Exer. 171. 20C. Let D be a division ring with involution l-A*, let n be a positive integer, and let D,, be the *-ring of all n x n matrices over D (with *-transpose as involution). The following conditions are equivalent: (a) D, is a Baer *-ring; (b) D, is a Rickart *-ring; (c) if ILieD(i= I, ..., n) and 1:A, +...+3,,*/1,=0, then A,=...=1,=0 . (See also [$ 56, Exer. I].) 21B. An AW*-algebra B is said to be AW*-embedded in an AW*-algebra A if (1) B is a closed *-subalgebra of A, and (2) if (e,) is any orthogonal family of projections in B, then supe, (the supremum as calculated in A) is also in B. {It follows that supe, is the same whether calculated in A or in B.) (i) If B is an AW*-subalgebra of A, then B is AW*-embedded in A. B RP(x)€B (RP as cal(ii) If B is AW*-embedded in A, then ~ E implies culated in A).

$ 5 . Weakly Rickart *-Rings

27

In fact, the concepts "AW*-embedded and "AW*-subalgebra" are equivalent (Exercise 27). 22C. Let Z' be a Hilbert space and let d be a norm-closed *-subalgebra of Y (A?) such that (i) d is an AW*-algebra, (ii) I E . ~ and , (iii) d is AW*-embedded in 2 ( . P ) . Assume, in addition, that (iv) =dis *-isomorphic to a von Neumann algebra. Then d = d " in Y ( Z ) , that is, d is already a von Neumann algebra on 2. {Better yet, the assumption (iv) can be omitted-see Exercise 24.) 23C. Let A be an AW*-algebra. A linear form q on A is said to be positive if q(x*x) 2 0 for all XEA. {It follows that if (e,),,, is any orthogonal family of projections in A, then q(e,)=O for all but countably many L E I , and the sum cp(e,)- that is, the supremum of all finite subsums-is finite and I cp(sup e,).}

1

rol

A positive linear form cp on A is said to be completely additive on projections (CAP) if cp(supe,) = q(e,) for every orthogonal family (e,) of projections in A. Let .VP be the set of all such cp (conceivably, .Y=(0)). The following conditions on A are equivalent: (a) A is *-isomorphic to a von Neumann algebra; (b) there exists a Hilbert space .%?and a *-monomorphism 0: A- Y ( X ) such that @ ( A )is an AW*-subalgebra of 9 ( . f ) ; (c) there exists a Hilbert space 2 and a *-monomorphism 0: A+ 2(.%?) such that the A W*-algebra @(A)is AW*-embedded in 9 ( 2 ) ; (d) for each nonzero XEA, there exists ~ € 9 ' with ~ ( x#) 0; (e) for each nonzero X E A, there exists ~ € with 9 q(x* x) > 0. 24C. Let 2' be a Hilbert space and let . d be a norm-closed *-subalgebra of Y ( 2 )such that (i) d is an AW*-algebra, and (ii) I€.&. The following conditions on d are equivalent: (a) .d is an AW*-subalgebra of Y(Af'); (b) d is AW*-embedded in 2 ( . f l ) ; (c) .d is a von Neumann algebra on A? (i. c., .d=d"in Y ( . f ) ) .

1

25C. Let ,%? be a Hilbert space and let d be a norm-closed *-subalgebra of 2 ( 2 ) such that (i) I E . ~ and , (ii) if (E,) is any orthogonal family of projections in d , then sup E, (as calculated in 2 ( 2 ) ) is also in d. The following conditions are equivalcnt: (a) . d = d " in 9 ( 2 ) , that is, d is a von Neumann algebra on (b) d is *-isomorphic to a von Neumann algebra; (c) .d is an AW*-algebra; (d) d is an Rickart C*-algebra; (e) T E . ~implics R P ( T ) € d (RP as calculated in Y ( 2 ) ) . 26C. Let .%?be a IIilbert space and let d be a norm-closed *-subalgebra of 2 ( 2 ) with I E ~ . The following conditions on d are equivalent: (a) d = i d "in 9(:#); (b) if .Y is any set of positive elements of d that is increasingly directed (i. e., S, T E Y implies there exists UE,Y with S I U and T I U) and bounded in norm, then sup .CP (as calculated in the partially ordered set of self-adjoint elements of Y ( 2 ) ) is also in d . 27A. Let A be a Baer *-ring and let B be a *-subring of A. Thc following conditions on B are equivalent: (a) B is a Baer *-subring of A; (b) X E B implies RP(x)EB (RP as calculated in A), and if (e,) is any orthogonal family of projections in B, then sup e, (as calculated in A) is also in B.

5 5.

Weakly Rickart *-Rings

What about the unitless case? The answer, in principle, is to modify the arguments or attempt to adjoin a unity element, and in practice it is

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

28

congenial and instructive to do a little of both. First, let us review the unit case : Proposition 1 . A *-ring A is a Rickurt *-ring if and only if (i) A hus a unity element, and (ii) for each x~ A, there exists a projection e such that R({xI)= R(teS).

Proof. If A is a Rickart *-ring, then A has a unity element [ $ 3 , Prop. 21, and e=RP(x) meets the requirement of (ii) [$3, Prop. 31. Conversely, if A satisfies (i) and (ii), then the relations R({x))= R({e)) = ( I -e)A show that A is a Rickart *-ring. I Condition (ii) alone is of no help (Exercise 1). The appropriate extension to the unitless case is as follows: Definition 1. Let A be a *-ring, x~ A, e a projection in A. We say that e is an annihilating right projection (briefly, ARP) of x in case (i) x e = x , and (ii) x y = O implies e y =O. The notion of annihilating lcji projection (ALP) is defined dually. (Thus e is an ARP of x iff it is an ALP of x*.) We say that A is a weakly Rickart *-ring if every x in A has an AK P.

If x and e are related as in Definition 1 then e is uniquely determined by x ; that is, if e and f' are ARP's for x then e = f: Indeed, x(e-f) = x e - x f = x - x = 0, therefore e(e-f) = 0 and j'(e-f) = 0, thus e = e f =f ' = f. Definition 2. If x and e are related as in Definition 1, we write e = R P(.\-) This is consistent with the usagc for Rickart *-rings, in view of the following: = LP(x*).

Proposition 2. The following conditions on u *-ring A are rquiralent: (a) A is u Rickart *-ring; (b) A is a weakly Rickart *-ring with unity.

Proof. Immediate from Proposition 1 and Definition 1.

I

A characterization of weakly Rickart *-rings: Proposition 3. The following conditions on a *-ring A are equivalent: (a) A is a weakly Rickurt *-ring; (b) A has proper involution, and. for each x t A , theup exists u projection e such that R({x})= R({e]).

Proof. (a) implies (b): Suppose xx* =O and let e=RP(x). On the one hand, xx*=O implies ex* =0, thus xe=O; on the other hand, x e = x , thus x=O. (b) implies (a): Assuming R((.u))= R({e)), e a projection, it will suffice to show that x e = x. For all Y E A, e(y - e y ) = 0, hence xO, - ey) = O

9: 5 . Weakly Rickart *-Rings

29

by hypothesis. Putting y = x*, we have x ex* = xx* ; a straightforward computation yields (x-xe)(x-xe)* =0, therefore x -xe=O. I The obvious question posed by Proposition 2: Can every weakly Rickart *-ring be embedded in a Rickart *-ring (e.g., by some process of 'adjoining a unity')? In the last part of the section we show that the answer is often yes, but in general the problem is open; thus there is some point in developing the basic properties of weakly Rickart *-rings directly from the definition, as we do in the next several propositions.

Proposition 4. In a weakly Rickart *-ring, (i) xy=O ifand only {f R P ( x ) L P b ) = O ; (ii) e =RP(x) is the smalle.st projection such that x e = x Proof: (i) is immediate from the definitions. (ii) I f f is any projection with x f = x , then x(f-e)=x f-xe=x-x=O, therefore e(f - e) = 0, e = eJ; thus e 5 f . I

Proposition 5. Let A be a weakly Rickart *-ring, und let B be a *-.subring of A such that L3 = W (equivalently, B = S' for some *-subset S of A ) . Then (i) x E B implies R P(x)EB, (ii) B is also a weakly Rickart *-ring (with unambiguous RP's and LP's). Proof. Same as 143, Prop. 101.

1

Corollary 1. If A is a weakly Rickart *-ring, then the center of A is also a weakly Rickart *-ring. Explicitly, if x is in the center oj A, then RP(x) is a centralprojeclion. Corollary 2. If A is a weakly Rickart *-ring and x * = x ~ A , then RP(X)E (x)". A useful technique for reducing to the Rickart *-ring case:

Proposition 6. If A is a weakly Rickart *-ring and e is a projection in A, then eAe is a Rickurt *-ring, with unambiguous RP's and LP's. Proof. If x ~ e A ethen RP(x)€eAe by Proposition 4, (ii), thus A is a weakly Rickart *-ring with unambiguous RP's and LIJ's; quote Proposition 2. 1

Proposition 7. The projections of a weakly Rickart *-ring form a lattice. Explicitly, e u f =f + R P [ e - e f ] , JOT

e n f =e-LP[e-ef]

every paw of projections e, f.

Proof. The first formula is proved exactly as in [$ 3, Prop. 71; whereupon, dropping down to (e u f ) A(e u f ) , the whole matter may be referred back to the case of a Rickart *-ring (Proposition 6). 1

30

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

Definition 3. Let A be a a-ring. We define a unitification A , of A, provided there exists an auxiliary ring K , called the ring of Lsculur.s (denoted 2, p, .. .), such that (1) K is an integral domain with involution (necessarily proper), that is, K is a commutative *-ring with unity and without divisors of zero (the identity involution is permitted); ( 2 ) A is a *-algebra over K (that is, A is a left K-module such that, identically, I a = a, 12(ah)= (A a) h = u(A h), and (2 a)* =A* a*); and (3) A is a torsion-free K-module (that is, Au=O implies A = O or a =0). In two special situations, we agree on the following choices of auxiliary ring: (I) If A is a *-algebra (in the usual sense) over a field F with involution, then we take K = F (note that (3) is satisfied automatically, since A is a vector space over F). (11) If A is not a *-algebra over a field with involution, but the additive group of A is torsion-free, then we take K = Z (thc integers. equipped with the identity involution). Returning to the general case, we define A, = A @K

(the additive group direct sum) ;

thus (a, A) = (h, p) means, by definition, that a = h and /2 = p, and addition in A , is defined by the formula Defining

(0, A)+(b,p)=(a+b, A t p ) . (a, A)(b, p ) = ( a b + p u + A h , Ap), p(a, 4 = (pa, p 4, (a, 2)* =(a*, I"*),

evidently A , is also a *-algebra over K, has unity element (0, I), and is torsion-free; moreover, A is a a-ideal in A , . We write a+?, 1 in place of (a, A). The following lemmas are aimed at the proof that if A is a weakly Rickart *-ring possessing a unitification A , in the sense of Dcfinition 3, then A , is a Rickart *-ring. The proofs of the first two are straightforward.

Lemma 1. Notation as in Definition 3. If the involution of A i\ propc~, then so is the in~.olutionof A , . Lemma 2. Notation ah in Definition 3. Tlze projections of A , arc flze projections e, 1 -e, where e is a projection in A . Lemma 3. Notation as in Definition 3. Let a € A, e a projection irz A. Tlzen e is an ARP of a in A if and on1.v if it is an ARP of a in A , .

31

$ 5 . Weakly Rickart *-Rings

Proqj: The "if' part is trivial. Conversely, assume e is an ARP of a in A, and suppose ay=O, where y = h + p l ~ A , . Then where e h + p e A, ~ therefore e(eb+pe)=O, thus e y =O.

I

It is in the following key lemma that the torsion-free hypothesis (condition (3) of Definition 3) is used:

Lemma 4. Notation NS in Defitzitio~3 . Assutne, in uddition, that A is a weakly Rickart *-ring. If a * = a ~ Aand ?,E K, l # 0, then there exists a largest projection g in A such that a g =A+ Proof. Let e=RP(a)=LP(a); then a - i e ~ e A ' . Define h=RP(u-Ae); thus 11 < e and h is the smallest projection in A such that (a - A c)h = u - ~e (Proposition 4). Setting g = e - h, we have

O=(a-ie)g=ag-leg=ag-i,g,

thus ag=ag. On the other hand, suppose k is a projection in A such that u k = i k. Since ea=cr, we have e ( l k) = A k, A(k - ek) = O ; since A is a torsionfree K-module, we have k - e k = 0, thus k < e. Then

therefore hk=O. thus k < e-h=g. be dropped (see Theorem 1 below).)

{The condition that a * = a can I

Lemma 5. Let B he a *-ring with proper inaolution, let XEB, and let e he a projection in B. Then e is an ARP of x if and onl-y if it is an AR P of x*x. Proof. Since R({x)) = R({x*x}) [$2, Prop. I],the essential point is that x*xe=x*.x implies (xe-x)*(xe-x)=O. I

Theorem 1. I f A is a weakly Rickart *-ring, and if A lzas a unitificution A, in the sense of Definition 3, then A, is u Rickurt *-ring. Proof. Since the involution of A is proper (Proposition 3) so is the hence it will suffice to show that every involution of A, (Lemma I), self-adjoint element of A , has an ARP (see Lemma 5 and Proposition 2). Let x * = x ~ A , .Say x = a + l l ; thus a * = u and i*=A. If i = 0 , that is, if .x = a € A, and if e is the ARP of a in A, then e is also an ARP of u in A , (Lemma 3). Assirme A#O. By Lemma 4, thcrc exists a largest projection g in A such that ag=(-A)g, that is, x y =O. Set e = l -y; we show that e

32

Chaptcr 1. Rickart *-Rings, Bacr *-Rings, A W*-Algebras

is an ARP of x in A , . Of course x e =x. Assuming xy=O, where y = h + p l ~ A ~it, is to be shown that ey=O. We have therefore 0=Ap (hence p=0 since K has no divisors of zero) and O=ab+pa+Ab=ab+Ah, thus

+

Let f = L P(b)in A. Writing (a) as (a A f ) b = 0, we infer that (a + 2 f ) f = 0, thus a f =(-A) f ; it follows that f < g , hence f ( l -g)=O, that is, fe=O. Then

Exercises

1A. Let A be a *-ring with A'= (0). Then 0 is the only projection in A, and R({x))=R({O))= A for every element x, but xO # x when x # 0. 2A. Let A be a weakly Rickart *-ring, and let (B,),,, bc a family of *-subrings of A such that, for each L , X E B ,implies RP(x)eB,. Then B = n B, is also a r t l

*-subring such that x € B implies RP(x)€B, hence B is a weakly Rickart *-ring with unambiguous RP's and LP's.

3A. Let A be a weakly Rickart *-ring, B a *-subring of A such that B = B . If (e,) is a family of projections in B that has a supremum e in A , then ~ E B . 4A. Let A be a weakly Rickart *-ring, and suppose (e,) is a family of projections in A that has a supremum e. If X E A and xe,=O for all 1, then xe=O. 5A. If A is a weakly Rickart *-ring such that every orthogonal family of projections in A has a supremum, then A is a Baer *-ring.

6A. A weakly Rickart *-ring with finitely many elements is a Baer *-ring 7A. Let A be a weakly Rickart *-ring, let x, y , z ~ A ,and let e=RP(x), j'=RP(y), g=RP(z). If x*x+y*y=O, then e=j'. If x*x+y*y+z*z=O, then e uf = e u g = f u g . 8A. Let A be a weakly Rickart *-ring such that x* x +y* y = 0 implies x =y = 0. (i) If e=RP(a) and f'=RP(b), then e u j'=RP(a*a+b*b). (ii) In particular, e u f = RP(e+ j') for every pair of projections e, ,f in A . (iii) If e, ,f are projections in A such that f - e = a * a for some a € A, then elf'.

9C. If A is a weakly Rickart C*-algebra, then A, (with suitably defined norm) is a Rickart C*-algebra. 10A. If A is a weakly Rickart *-ring and a t A, there exists a largest projection g in A such that x g = g . 11A. Let A be the set of all complex sequences x=(l,) such that 3,,=0 ultimately, with the usual *-algebra structure. Thus A, is the set of all 'ultimately constant' sequences. Then A is a weakly Rickart *-ring whose projection lattice

5 6.

Central <:over

33

is conditionally complete, but A, is not a Baer *-ring. (See also the ring A described in [$ 3, Example 21.) 12A. Let A be a wcakly Rickart *-ring without unity, and suppose (e,),,, is a maximal orthogonal family of nonzero projections in A . Then (i) I is infinite; (ii) if A has a unitification A, in the sense of Definition 3, then the family (e,),,, has supremum 1 in A,. 13A. Let A be a weakly Rickart *-ring without unity, and suppose that the projection lattice of A is countable (see, e. g., Exercise 11). If A admits a unitification A, in the sense of Definition 3, then A, cannot be a Baer *-ring. 14A. Let B be a Boolean algebra [$3, Exer. 141 containing more than two elements. In order that there exist a Boolean ring A without unity, such that A, =B, it is necessary and sufficient that B be infinite. 15A. Does there exist a weakly Rickart *-ring A # (0) without unity, possessing a unitification A, in thc sense of Definition 3, such that A, is a Baer *-ring?

16A. If A is a *-ring in which

x* xi= 0 implies x, = .. .= x, = 0 (for any n),

then the additive group of A is torsion-free 17A. Let A be a *-ring, let K be a commutative *-ring with unity, and suppose that A is an algebra over K, that is, A is a left K-module satisfying l a = a and A(ab)=(Ia)b=a(Ab) for all a , b ~ Ai,e K . Write &=&(A, +) for the endomorphism ring of the additive group of A; each UEA determines an element La of 6 via L,x=ax; and each ~ E Kdetermines an element AI of & via (A I)x=Ax. Let A, = A@K with the *-algebra operations described in Definition 3. Each (a, A) in A, determines an element La + i l of 6, and the mapping (a, I ) rt La+ i l is a ring homomorphism of A , onto a subring .dl of 8, namely, the subring of d generated by the left-multiplications Lo and the homotheties I I. Defining p(L, + I to be the ring product (PI) (L,+i I), d,becomes an algebra over K, and the mapping @,A)rt Lo+ A I an algebra homomorphism of A, onto dl. Let N be the kernel of this mapping, and write A , = A,/N for the quotient algebra. Denote the coset (u,A) + N by [a, A] ; thus [a,I] is the equivalence class of (u,A) under the equivalence relation @,A)- (b, p) defined by a x + I x = b x + p x for all ~ E A . (i) Write Z=[a,O] for UEA; the mapping a - 5 i s an algekra homomorphism of A into A^,. Writing u = [0,1] for the unity element of A,, we have [u,A]=z+Au. (ii) If L(A)= {0) (e. g., if the involution of A is proper), then the mapping a u Z is injective, and we may regard A as 'embedded' in A,. (iii) If the involution of A is proper, then [a, I] = 0 implies [a*,A*] = 0, a?d the formula [u,A]*= [a*,i*] defines unambiguously a proper involution in A,. (iv) Lf A is a weakly Rickart *-ring, a e A , and e=RP(a), then F is an ARP of Z in A , . 18D. Problem: Can every weakly Rickart *-ring be embedded in a Rickart *-ring? with preservation of RP's?

fj6. Central Cover The concept of central cover is most useful in Baer *-rings, but it can be formulated in any *-ring:

34

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

Definition 1. Let A be a *-ring, XEA. We say that x possesses a central cover if there exists a smallest central projection h such that h x = x ; that is, (i) h is a central projection, (ii) lzx=x, and (iii) if k is a central projection with k x = x then h k. If such a projection h exists, it is clearly unique; we call it the central cover of x, denoted h = C(x); when it is necessary to emphasize the ring A, we write h = C,(x).

Central cover works best when the lattice of central projections is complete, but a few remarks can be made in a more general setting: Proposition 1. Let A he a *-ring in wlzich c'tjery projection r hus a central corjer C(e). (i) If'e is a projection in A, then C(e) is the smallest cc3ntral projection h such tlzat e 5 h. ,f; then C(e) I C(f ) . (ii) If'e andfare projections such that 6. I (iii) C(he)= h C(e) ij'h is a c ~ n t r uprojection. l (iv) e=supe, implies C(e)=supC(e,). (v) If' tlze involution of' A is proper, then e - f inzplirs C(e)=C(f), and e 5 f implies C(e) 5 C(,f'). Suppose, in addition, that A is a Rickart *-ring, with center Z. Then: (vi) Ez~ery element x of' A has a central cover C(x); cxplicit1.y. C(X)= C(LP(x))= C(RP(x)). (vii) C(xx*)= C(x* x) fbr all x~A. (viii) C(h x) = h C(x) !f' h is a central projection. A Z E Z ,tlzen zx=O fi zC(x)=O. (ix) If' ~ E and

Proof. (i) is obvious, and (ii) follows immediately from it. (iii) Let k = C(h e); it is to be shown that k = h C(r). By (ii) we have k 5 h and k < C(e), thus k < h C(e). On the other hand, k(h e) = h e, (h-kh)e=O, (h-klz)C(e)e=O; then C(e)-(11-klz)C(e) is a central projection such that

therefore C(e) 5 C(e) - (h - k h) C(e). It follows that C(e)[(h - k h) C(e)] = 0, thus hC(e)=khC(e)=k. (iv) The hypothesis is that (e,) is a family of projectioils in A that possesses a supremum e; it is asserted that the family (C(e,))admits C(e) as supremum. By (ii), C(e,)5 C(e) for all I. Assuming lz is a central projection such that C(e,) < h for all 1, it remains to shou~that C(e) 5 h; 5 C(e,)5 h, therefore r 5 h, thus C(e) I h indeed, for all r we have by 6). (v) In view of (ii), it suffices to consider the case that r - f : Say x*x=e, xx*=,f. It is to be shown that if h is a central projection, then Ize=e iff h f =,f. Since the involution is proper, it is routine to check

5 6. Central Cover

35

that h x * x = x * x iff ( x h x ) * ( x h x ) = O iff x - h x = 0 iff (x-hx)(x-hx)*=O iff hxx*=xx*. (vi) Assume, in addition, that A is a Rickart *-ring, let xgA, and let f=LP(x). If h is a central projection, then, citing [$ 3, Prop. 31, h x = x iff h 2 f iff hf =f ; by hypothesis, there exists a smallest such h (namely, C(j')), thus x has a central cover, and C(x)= C(f ) . Similarly C(x)= C(RP(x)). (vii) Since the involution in a Rickart *-ring is proper [$3, Prop. 21, the calculation in (v) shows that C(x*x)= C(xx*) for every ,YE A . (viii) Let x~ A, let h be a central projection, and write f = LP(x). Clearly h x and h f have the same left-annihilator, therefore LP(1z.x) = LP(hf'); citing (vi) and (iii), we have C(hx) = C(lzf ) = lz C(f ) = h C(.x). (ix) Let xcA, ZEZ. If zC(x)=O then zx=zC(x)x=O. Conversely, suppose z x = 0. Writing h = RP(z), we have h x = 0. Since h is a central projection [$ 3, Cor. 1 of Prop. 101, it follows from (viii) that O = C(hx)= hC(x), hence zC(x)=O. I The key to the theory of central cover in Baer *-rings is the following result:

Proposition 2. If A is a Buer *-ring and J is a ricght ideal in A, t h ~ n R(J)= h A, where h is a central projection. Proof: Write R(J)=hA, Iz a projection. Since J is a right ideal, R(J) is an ideal (two-sided) [$3, Prop. 11; in particular, A(lz A) c h A, therefore A h c h A . Thus h(xh)=xh forall ~ E Athenalso ; hx*h=x*h, thus h x = h x h = x h for all x e A . I

Proposition 3. In a Baer *-ring A, every element x ha., a central cover. Explicitly, if e =RP(x) and f = LP(x), then C(x)= C(e)= C( f ) and

Proof. The existence of central covers is immediate from [$ 3, Prop. 61 and the fact that the central projections form a complete lattice 154, Cor. 2 of Prop. 71. We give here an alternate proof that exhibits the desired formulas. Write R(xA)=(l - h)A, h a central projection (Proposition 2). In particular, I -h right-annihilates x, thus x h = x . On the other hand, if k is a central projection such that k x = x , then x(1- k)= 0, xA(1- k) = x ( l -k)A=O, I -kc(l-h)A, I- k < l -h, h < k. This proves that C(x) exists and is equal to h (see Definition 1). A similar proof shows that if L(Ax)=(I - k) A , then k= C(x). Thus

36

Chaptcr 1. Rickart *-Rings, Baer *-Iiings, A W*-Algchras

Since C(x)= C(e)= C(f ' ) by Proposition 1, the remaining formulas follow at once. I

Corollary 1. If A is a Burr *-ring arzd x, y~ A, then .l-A!,= 0 if and only if C(x)C(y)= 0. Proof. Let h=C(x), k=C(y). If hk=O then xAy=(xlz)A(ky) =xA(h k)y =O. Conversely, if xAy = 0 then X E L(Ay)= (1 - 12) A, therehk=O. I f o r ~( 1 - k ) x = x , h=C(x)11-k, The phenomenon in Corollary 1 occurs often enough to merit a name:

Definition 2. Projections r, f in a Baer *-ring are called vecv ortliogonal if C(P)C(f ) = O (equivalently, in view of Corollary 1, eAf= 0). Corollary 2. Let A hc a B u o *-ring wlzos~only cerztralproje~.tion.sare O and 1. I f x, y t A then xAy=O if andonly if x = O or y=O. Proof. If x f 0 and y#O then C(x)=CCy)=l (0 and 1 are the only available values), thus C(x)C(y) # 0; citing Corollary 1, we have xAy#O. I The condition on A in Corollary 2 is an import recurring theme:

Definition 3. If A is a *-ring with unity, whose only central projections are O and 1, we say that A is fuctorial, or that A is a fuctor. A factorial Baer *-ring is called a Baer *Tfilcto~.; a factorial A W*-algebra is called an AW*-,furtor. Remarks and Examples. 1 . In a factorial *-ring, every element has a central cover in the sense of Definition 1. 2. If A is a *-algebra with unity, over an involutive field K, and if the center of A is one-dimensional over K, then A is factorial. (Proof: In any *-algebra, orthogonal nonzero projections are linearly independent.) 3. A factorial Baer *-algebra need not have one-dimensional center. {Consider, for example, the complex polynomial ring A=C[t], t indeterminate, with the identity involution.] 4. If .F is a Hilbert space, then 9 ( X ) is a factor. (Suppose E is a nonzero projection such that T E = E T for every TtK'(,Yf), and let . /i be the range of E; it is to be shown that ,a = .#. Fix a unit vector x in ,it. Given any y t P,consider the operator T z = (zlx)y ( ~ € 2 In )particu. lar, Tx=(xlx)y= y, thus y t T(A2). By hypothesis, .A?' is invariant under T (indeed, .Atreduces every operator in Y(.%')), thus y t T ( . M ) c& ,).' 5. If A is a Baer *-ring and B is a *-subring such that B = B", and if x t B, then C,(x) l C,(.r) (because B contains every central projection of A), but in general equality does not hold. ( F o r example, let d be

4 6 . Central Cover

37

a commutative von Neumann algebra on a Hilbert space .F,thus

.df3.d=.d"in 9 ( X ) ; if E is a projection in .d different from O and I,

then its central cover in .dis E, but its central cover in 2'(.X) is I (by Example 4 above).) Such *-subrings B are Baer *-subrings of A in the sense of [54, Def. 31; for Baer *-cubrings of the form eAe, the situation is clearer:

Proposition 4. Let A he a Baer *-ring, e a projection in A, and ,f a projection in eAe (tlzat is, j' 5 e). Tlze following conditions on f are equivalent: (a) J' is a central projection in eAe; (b) .f=eC(J'); (c) f = eh for some central projection h ofA. If A is a juctor, then so is eAe. Proqf. Obviously (b) implies (c), and (c) implies (a). (a) implies (b): For all a E A, we have j'(eae) = (eae)J ; that is, ,/up = euf; it follows that fhe(1 - f ) = O for all UEA, thus fAe(1 -,/')=O. Citing Proposition 3, we have -.f)€R(fA)

=

(1 - C ( f ) ) A ,

therefore e(1- f')C(f)=O, e C ( f ) = e f C ( f ) = fC(f')= f . It follows that if O and 1 are the only central projections in A, then O and e are the only central projections in PAP. I

Corollary 1. Let A he a Baer *-ring, with center Z, and let e be a projection in A . (i) The renter of' eAe contains c%e, that is, e Z C ( ~ An~( ~ ) AP)'. (ii) e Z and (eAe) n(eAe)' contain the same projections. (iii) If' x ~ e A ethen e C(x) is the central cover of' x in eAe. Proof: (i) is obvious. (ii) If ,f' is a projection in (eAe) n (CAP)', then, by Proposition 4, there exists a projection ~ E with Z f = eh. (iii) Let ,f=C,,,(x), h = C(x); it is to bc shown that j'= eh. Since eh is central in eAe, and ( e h ) x = c ( h x ) = e x = x , we have j's 'h. On the other hand, since f is central in eAe, we have f = e C ( f ) by Proposition 4; I then x C ( f ) = x e C ( f ) = x f = x , therefore h 5 C ( f ) , h e < e C ( f ) = f . When A is an A W*-algebra, the inclusion in Corollary 1, (i) is an equality; the proof borrows a result from Section 8: Corollary 2. If A is an AW*-algjebru, with center Z , and if e is a projection in A, then e Z is the center of' eAe, that is, e Z = (cAe) n(eAe)'. Proof. It is immediate from [$ 4, Prop. 81 that e Z and (eAe) n (eAe)' are AW*-algebras, and by Corollary 1 they contain the same projec-

Chapter 1. Kickart *-Rings, Baer *-Rings, A W*-Algebras

38

tions; the desired conclusion follows from the fact that an A W*-algebra is the closed linear span of its projections [$ 8, Prop. I]. I Definition 4. A projection f in a Baer *-ring A is said to be fuithf'ul (in A) if C(J')= 1. Corollary 3. Let f he a juithjul projection in a Baer *-ring A . (i) If' e is any projection in A such that j' 5 e, then f is ,faitlzful in eAe. (ii) If' h is any central projection in A, then hf is faithful in hA. Proc?f. (i) Citing (iii) of Corollary 1, we have C,,,( f ) = e C ( f ' ) = e . (ii) Similarly, C,,(h f ) = hC(hf)=h(hC( f ) ) = h . I The final result of the section is for application later on (cf. [$20, Prop. 51, [$ 29, Lemma 11, [$ 30, Lemma I]); the gist of it is that, in a certain sense, the consideration of finitely many projectioiis can be rcduced to the case of faithful proicctions: Proposition 5. Let A he a *-ring with unity, in which eaeryprojection has a central cover, and let e, (i= 1, . . .,n) hr projections in A . Then tlzerr rxis t ortlzogonal central projections k, (v = I,. . .,r), with k, + . .. + h, = I, such that, for each pair of' indices i, r, either k, r, = O or C(k, e,)= k,. Proof. Let k=sup{C(e,):i=l, ..., n) and note that ( I k ) e , = O for all i; dropping down to kA (cf. Exercise 4), we can suppose that sup C(e,)= I. Disjointify the C(e,), that is, let k,, ... ,h, be orthogonal central projections with k, +...+k,= 1, such that each C(e,) m the sum of certain of the k,. {For example, consider the projections where c, = + 1 or - 1, with the understanding that C(e,)+' = C(e,) and C ( e , ) ' = 1 - C(e,).) For any palr of ~ n d ~ c i,e sv we have either k, C(r,)= O or k, < C(e,) (because C(e,) is the sum of those k, that it contains, and IS therefore orthogonal to the rest). It follows that ~f k,C(e,) # 0 then k, 5 C(e,), hence, citing Proposition 1, C(k, r,)= k , C(e,)= h, On the other hand, ~f k, C(e,)= 0 then k, e, = 0. I Exercises 1A. A projection e in a *-ring A is central iff e A(l

- e)=O

2A. Let A be a Baer *-ring, and suppose that S and J are nonempty subsets A such that J S c J. Write R ( J ) = h A , h a projection. Prove: (i)SR(J) c R(J); h s h = s h for all SES; (iii) if S is a *-subset of A , then h e r . Proposition 2 is

a special case.

3A. Proposition 1 holds with "weakly Rickart *-ringn in place of "Rickart *-ringn.

# 6 . Central Cover

39

4A. Let h be a central projection in the *-ring A. Thc central projections of hA are the central projections k in A such that k 5 h; in other words, a projection in hA is central in hA if and only if it is central in A. 5A. Let A bc a *-ring in which every projection has a central cover, and let h be a central projection in A. If e is a projection in hA, then C,,(e)=C(e). 6C. Let A be an AW*-algebra, e a projection in A . Then C(e)=sup{ueu*: u unitary). 7A. Let A be a Baer *-ring, let e be a projection in A, and define It is easy to see that e 5 h 5 k 5 C(e), and that u h u* = h for every unitary U E A. Write U = { ~ E Au:unitary), P= {SEA: g a projection). Prove that h= k = C'(e) under either of the following three hypotheses: (i) Every projection in U' is central. (ii) Every projection in P' is central, and 2 x = 0 implies x=O. (iii) A is an A W*-algebra. 8A. If A is a Kickart *-ring with PC [tj 14, Def. 31, then a projection In A 1s central iff it commutes with every projection of A (thus a projection is central in A iff it is central in the reduced ring A' [$ 3, Exer. 181). 9A. If A is a *-ring with the property that A and A" have the same central projections (cf. Exercise 8), then every direct summand has this property. 10A. In a Baer *-ring with PC [$ 14, Def. 31, the Collowing conditions on a pair of projections e , j' are equivalent: (a) C:(e) 5 C(/'); (b) e=supe, with ( r , ) an orthogonal family of projections such that e , s f for all 1; (c) e=supe, with (e,) a family of projections such that e, 5 f for all 1. 11C. In a Rickart *-ring with orthogonal GC [$ 14, Def. 41, a projection is central if and only if it has a unique complement. {Projections e, f are said to be complementary if e v j'= l and c nf =O; either is called a complement of the other. Every projection e has at least I -e as a complement.) 12A. If e is a minimal projection 144, Exer. 31 in a Baer *-ring A , then the direct summand C(e)A is a factor. 13A. Let B be a Baer *-ring, A a *-subring such that A = A". (Model: B thc algebra of all bounded operators on a Hilbert space 2, A any von Neumann algebra on =iY' [94, Dcf. 51.) Let D = (A n A')' =(Af u A)"; thus D is the smallest *-subring of B such that A c D, A' c D and D = D" (in this sense, it is 'generated' by A and A'). Then A, A' and D havc common center D' = (A' u A)"' = (A' u A)' = A n A ' ; hence if UEA then C,(a)= C,(a), and if U'E A' then C,.(u1)= C,(ul).

14A. Let B be a Baer *-ring and adopt the notation of Exercise 13. If UEA and u'EA', the following conditions are equivalent: (a) there exists a prqject~onc in the center of A such that ac=O and ca'=af; (b) there exists an element c in the center of A such that uc=O and cur=a'; (c) aDa'=O; (d) C,(u)C,(u')=O; (e) ~ ~ ( a ) c , , ( a ' ) = O(f) ; aa'=O. 15A. Let B be a Baer *-ring, with notation as in Exercise 13. Let e be a projection in A. Define a mapping cp: A'+ eA'e (= eA'= A'e) by the formula cp(a')=ale. Thus cp is a *-epimorphism, with kernel

40

Chapter 1. Rickart *-Rings, Haer *-Rings. A W*-Algcbras

Let h = CA(e)(= C,(e)). Prove: ker cp = (I h) A'. In particular, cp is injcctivc if and only if h = l , that is, e is faithful in A. -

16A. Let B be a Baer *-ring, with notation as in Exercise 13. Write Z to indicatc center; thus Z,, = ZAP = Z, = A n A'. Let e be a projection in A. Then (i) Z,,, 3 e ZA; (ii) Z,,, and eZ, contain the same projections. (iii) If, moreover, B is an AW*algebra, then Z,,,= e Z,, in other words (calculating commutants in B), (CAP)n (eAe)' = e(A nA'). 17A. Let B be a Baer *-ring, with notation as in Exercise 13. Let e be a projection in A. Then (eBe)n(eAe)' 3 eA1e(=eA'). If f is a projection in B, then , f ~ ( e B en(eAe)' ) iff f 2 e and JA(e- J)=O. 18A. Let A be a ring with unity, B= A, thc ring of n x n matrices over A (i) The center of B is the sct of all 'scalar' matrices (fiijz)= diag(z, ..., z ) with z in the center of A. In particular, the central idempotcnts of A and B may be identified. (ii) Suppose, in addition, that A is a *-ring, and B is given the natural involution (*-transposition). If every UEA has a central cover C(a) (in the sense of Definition 1) then every x=(aij)€B has a central cover, namely, C(x)= sup{C(aij):i,j= 1, ..., n). (This is a reason for considering central covcr for clcments other than projections.) (iii) If A is a Baer *-ring and x, y e B, then x By = 0 iff C(x) CCy) = 0. 19A. If A is a factorial Rickart *-ring, then thc ccntcr of A is an integral domain.

20A. If h is a central projection in a weakly Rickart *-ring, then RP(hx) = hRP(x) for every element x.

5 7.

Commutative A W*-Algebras

The commutative C*-algebras A with unity are the algebras A = C(T), T compact [92, Example 51. What properties of 7' are needed to make A an AW*-algebra'? The following answer is the main rcsult of the section: Theorem 1. Let A be a commutative C*-ulgehra with unity and write A=C(T), T compact. In order that A be an AW*-algebra, it is necessary and sujficient that T be a Stoniun space. A compact space is said to be Stonian (or extremally discorznecte~ if the closure of every open set is open (hence closed and open 'clopen'). For clarity, we separate the proof of sufficiency (Proposition 1) and necessity (Proposition 2). The key to the analysis is the observation that the projections of A = C ( T ) correspond to the clopcn scts in T: if ecA, then e* = e = e 2 iff e(t)=O or 1 for all t~ T iff e2= e iff e is the characteristic function of a clopen set P in T (namely, P = i t : c(t)= I)). Thus, the projection lattice of A = C ( T ) [$ 1, Prop. 31 is isomorphic to

5 7 . Commutative A W*-Algebras

41

the lattice of clopen subsets P of T (ordered by inclusion), via the correspondence P + + / , , (the characteristic function of P). The first lemma is valid in any topological space:

Lemma. Let (P,) be a jumily qf'clopen sets, k t U = U P,, and suppose tlzut is clopen. Then the fumily (6)has U as supremunz in tlze cluss of ull clopen sets (ordered by inclusion). PI-oof. Let P = 0.On the one hand, P, c P for all I . On the other Q for all I , then U c Q, therefore hand, if Q is clopen and P = ~ J C ~ = Q .I

cc

Proposition 1. If T is Stonian, then A = C(T) is ~znAW*-a1~gehr.u. Proof: Since T is Stonian, it is clear from the lemma that the lattice of clopen sets is complcte, in othcr words, the projection lattice of A is complete. To show that A is a Bacr a-ring, it will therefore suffice to show that it is a Rickart *-ring [1$ 4, Prop. I]. Let .YEA; we seek a projcction P such that R({x))=(l -P) A . Set U = { t :x(t) # 01, P = U . Since U is open, P is clopen; let e = 1,. If y E A, then x y = O iff y =O on U iff y=O on U = P iff ey=O. Thus R({x))=R({c~))=(l-r)A. I We approach the converse through a pair of lemmas. If A = C ( T ) is an AW*-algebra, we know that the clopcn sets of Tform a complcte lattice [$ 4, Prop. I ] ; what remains to be vcrificd is the explicit formula for supremum indicated in the above lemma.

Lemma 1. If A = C(T) is a Rickurt *-ring, then the clopen set3 of T separate the points of T . Proof. Suppose .s, t~ 7: s # t ; we seek a clopen set P such that J EP and t~ T - P. Choose ncighborhoods U , V of s, t with U n V = and choose x, YEA such that x(s)# 0, x=O on T - U , and y(t) # 0, y=O on T- V. Obviously xy=O; thus, setting e=RP(x), we have ey=O. Write P=/,, P clopen. Since x e = x and x(s) # 0, we have e(s)= I, S E P ; on thc other hand, r y -0 and y(t) # 0, therefore r(t)=O, ~ET-P. I

a,

In essence, thc next lemma is a result about thc Stone representation space of a Boolean algebra:

Lemma 2. I f A = C ( T ) is a Rickurt *-ring, tllm the cloprn Jets in T are basic for the. topology. Prooj. Let U be an open set, t~ U ; we seek a clopen set P such that t c P c U. For each J E T - U , choose a clopen set P, such that S E P, and t E 7 - P, (Lemma 1). S~nce T - U 1s compact,

42

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

for suitable s,,. ..,s,; the clopen set P = ( T - P S I )n... n( T -P,,,) has the required properties. I Proposition 2. I f A = C ( T ) is an A W*-algebra, then T is Stonian.

Proof. Let U be an open set in T; it is to be shown that U is open. By Lemma 2, there exists a family (PI)of clopen sets such that U = UP,. As remarked following Proposition 1, the clopen sets form a complete lattice; let P be the supremum of the family (P,). Since P, c P for all 1 , we have U c P, therefore U c P ; it will suffice to show that U = P. Assume to the contrary that the opcn set P - U is nonempty. Let Q be a nonempty clopen set such that Q c P - U (Lemma 2). Since Q nPI= (ZI for all 1 , that is, P, c T - Q for all 1 , it follows that P c T - Q (because T-Q is clopen and P is the supremum of the P,). Thus P n Q = (ZI; but this is contrary to Q c P, Q # @. I In many situations, Proposition 2 passes for 'spectral theory' in an AW*-algebra. Form example: Proposition 3. Let A he any A W*-algebra, X E A, x 2 0, x f 0. Given any E > 0, there exists y e {x)", y 2 0, such that (i) x y = e, e a nonzero projection, and (ii) Ilx -xeJJ < c .

Proof. The condition x 2 0 means that x* = x and the spectrum of x (as an element of A) is nonnegative. Since. {x)" is a commutative A W*-algebra [fj 4, Prop. 81, we have {x}" = C ( T ) , T a Stonian space (Proposition 2). The spectrum of x as an element of {x)" is the same as its A-spectrum [fj 3, Prop. 9, (6)], therefore x 2 0 as a function on T. We can suppose that 0 < c < llxll. Define

u = jt:x(t)>

f);

U is a nonempty open set, P = U is clopcn, and the characteristic function of P is a nonzero projection e. Since x(t) < t.12 on T - U , and therefore on T - U , we have

-

thus Ilx(1 -e)ll < c / 2 On the other hand, since x(t)> c/2 on U , we have x(t) 2 c/2 on U = P, therefore the function t l/,~(t)(t E P) is continuous; it follows that the function y on T defined by 1 f o r t s f , y(t)=O for ~ E T - P

5 7. Commutative A W*-Algebras

43

is continuous, that is, y ~ l x ) " . Since x(t)y(t)= I on P, and = O on I T-P, we have x y = e .

Corollary. Let A he any A W*-algebra, X E A , x # 0. Given any e > 0, there exists y~ {x*x)", y 2 0, such tlzat (i) (x*x)y2 = e, e a nonzero projection, and (ii) Ilx - xell < E . Prooj'. One knows from C*-algebra theory that x* x 2 0. By Proposition 3, there exists Z E {x*x}", z 2 0, such that (x* x) z = e, e a nonzero projection, and 11x*x - (x* x) ell < E~ ; set y = z; . Since e commutes w i t h x * ~we , have Ilx-xe1/2 =Il(x-xe)*(x-xe)ll =Ilx*x-x*xell < c2. I The conclusion of the corollary is a mixture of algebra and analysis: the ideal generated by x contains nonzero projections e, and x can be approximated from the ideal generated by such projections. The purely algebraic part can be formulated in any *-ring:

Definition 1. A a-ring A is said to satisfy the existence of' projections axiom (briefly, the (EP)-axiom) if for every xeA, x f 0 , there exists yr;{x*x)" such that y* = y and (x*x)v2=e, e a nonzero projection. As long as we're on the subject, this is a convenient place to record two weaker axioms that sometimes suffice:

Definition 2. A *-ring A is said to satisfy the weak (EP)-axiom (briefly, the (WEP)-axiom) if for every X E A, x # 0, there exists Y E {x*x)" (y is necessarily normal, but not necessarily self-adjoint) such that (x* x) (y*y) = e, e a nonzero projection. Definition 3. A *-ring A is said to satisfy the very weak (EP)-axiom (briefly, the (VWEP)-axiom) if for every ~ E A x#O, , there exists y e {x*x)' such that (x* x) (y*y) = e, e a nonzero projection. Obviously (EP) * (WEP) * (VWEP) => the involution is proper. Exercises 1C. A C*-algebra A is an AW*-algebra if and only if (A) in the partially ordcred set of projections of A, every nonempty set of orthogonal projections has a supremum, and (B) every masa [#I, Exer. 141 in A is the closed linear span of its projections.

2C. Let A be a commutative AW*-algebra. In order that A be *-isomorphic to some (commutative) von Neumann algebra, it is necessary and suficient that there exist a family 9 of linear forms on A having the following three properties: (i) each fi9 is positive, that is, f'(x*x)2 0 for all X E A ; (ii) each ~ E isYcompletely additive on projections, that is, f'(supe,)=x f(e,) for cvcry orthogonal family of projections (e,);(iii) 9 is total, that is, if xt-A is nonzero then f(x*x)>O for some f €9. 3C. Let T be a compact space. In order that C ( T )be *-isomorphic to some (commutative) von Neumann algebra, it is necessary and sufficient that T bc

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algcbras

44

hyperstonian. {A Stonian space is said to be hypecc.tonian if the supports of its normal measures have dense union (a measure is normal iff it vanishes on evcry closed set with empty interior).)

4C. There exists a Hilbert space .# and a commutative *-algcbra .dof operators on such that (i) the identity operator belongs to ,d, (ii) d is an AW*algebra, and of (iii) =QZis not a von Neumann algebra. It follows that there exists a family (EL) projections in .d such that sup EL, as calculated in the projection lattice of ..A, is distinct from the projection on the closed linear span of the ranges of the E,. 5A. Let A bc a *-ring, B a *-subring of A such that B=B". If A satisfies the (EP)-axiom [(WEP)-axiom] then so does B. 6A. Let A be a *-ring, a projection in A. If A satisfies the (EP)-axiom [(WEP)axiom] then so does eAe. 7A. Let A be a Baer *-ring satisfying the (WEP)-axiom, and let XEA,x #O. Lct be a maximal orthogonal family of nonzero projections such that, for each 1 , there exists y , {x* ~ x)" with (x* x)(y, *y,)= e,. Then sup e, = RP(x). (p,)

8A. Let A be a Baer *-ring, and let B be a *-subring of A such that (1)if S is any nonempty set of orthogonal projections in B, then s u p S ~ B and , (2) B satisfies the (WEP)-axiom. Thcn thc following conditions arc equivalent: (a) X E B implies RP(x)EB; (b) B is a Baer *-ring; (c) B is a Bacr.*-subring of A . 9A. Let A bc an AW*-algebra, and let B bc a closcd *-subalgebra of A such that sup S E B whenever S is a nonempty set of orthogonal projections in B. Then thc following conditions are equivalent: (a) X E B implies RP(x)EB; (b) B is an AW*algebra; (c) B is an A W*-subalgebra of A.

10A. A compact space is Stonian if and only if (i) the clopen sets are basic for the topology, and (ii) the set of all clopen sets, partially ordered by inclusion, is a complete lattice. 11C. A commutative AW*-algebra A is 'algebraically closed' in the following sense: If p(t)= tn+ a , 1"- ' +...+a,_, 1 +a, is a monic polynomial with coefficients a , , ..., a, in A, then p(a)=O for some UEA.

5 8.

Commutative Rickart C*-Algebras

If T is a compact space, when is C ( T ) a Rickart C*-algebra?; precisely when the clopen sets are basic and form a o-lattice:

Theorem 1. Let A be a commutatiue C*-algebru with unily, and wrilr A = C(T), T compact. The following conditions are necessary and sufficient fir A to be a Rickavt C*-algebra. (1) the clopen sets in T a r ~ huszc for the topology, and (2) if P,, is any sequence of clopen sets, cmd

U P,,, ' I )

if U =

then

U

is clopen.

n= 1

The proofs of necessity (Proposition 1) and sufficiency (Proposition 2) are separated for greater clarity. We begin with three general lemmas :

5 8.

Commutative Rickart C*-Algebras

45

Lemma 1. I f , in a weakly Rickart *-ring, every orthogonal sequence of projections has a supremum, then every sequence of projections has a supremum. Proof. If en is any sequence of projections, consider the orthogonal sequence f , defined by ,fl=el and J n = (el v . . . u e , ) - ( e l u . . . u e ,,-,) for n > l .

I

Lemma 2. If B is a weakly Rickart Banach *-algebra (real or complex scalars), then every sequence of projections in B lzas a supremum. Proof. By Lemma 1 , it suffices to show that any orthogonal sequence of nonzero projections en has a supremum. Define

, + O as m, n-t co, since 2-kllekl11ekhas norm 2 - k , it follows that ~ l x -xnll m

thus we may define x = lim x,. (Formally, x = 2 " Ilenll ' en.) Let 1 e = R P ( x ) ; we show that e=sup en. Iff is any projection such that en I f for all n, then x, f =xn for all n ; passing to the limit, we have x j = x , therefore e < f . It remains to show that en5 e for all n. Fix an index m. By orthogonality, -

e,xn=2-mllemll-1 em for all n 2 m , therefore e,x = 2-* llemll- ' em, that is, em= 2, lle,ll emX . Since x e = x , it follows that e,ne = em, thus em5 e. I In particular: Lemma 3. In u weakly Rickart C*-ul<jebra, every sequence of projections has a supremum. Proposition 1. Let T be a compact space such that A = C ( T ) is u Rickart C*-algebru. Then: ( 1 ) The clopen sets in T are basirfor the topology. m

(2) If Pn is a sequence of clopen sets and i f U = UP,,, then 1

U

is clopen.

(3) A is the closed lineur spun of its projections. (4) If X E A and U = ( t : x ( t )# 0}, then U is the union of u sequence of clopen sets, 0 is clopen, and the churucteristic function of 0 is RP(x). Proof. (1) See [$7, Lemma 21.

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

46

(2) By Lemma 3, there exists a clopen set P which is a supremum for the P,. Since P, c P for all n, we have U c P, therefore c P; the proof that U = P proceeds as in [47, Prop. 21. (3) Let B be the linear span of the projections of A. Evidently B is a *-subalgebra of A, with 1E B; moreover, it is clear from (1)that B separates the points of T, therefore B is dense in A by the Weierstrass-Stone theorem. (4) Note first that if C c U , with C compact (i.e., closed) and U open, then there exists a clopen set P such that C c P c U (this follows from (1) and an obvious open covering argument). Let x € A andlet U = { t : x ( t ) # O ) . For n=1,2,3, ... set

u

m

then C, is compact and U = such that C , c P, c U ; then

U C,.

For each n choose a clopen set P,

-

thus U = UP,, therefore 0 is clopen by (2).

n;

1

Let e = R P ( x ) and let f be the characteristic function of it is to be shown that e= f. Since x e = x, clearly e(t)= I for t e U , hence for t~ U , thus e 2 f . On the other hand, since f = I on hencc on U , wc have x f = x , therefore e l f . I

u,

In the reverse direction:

Proposition 2. If' T is a compact space satisfying conditions (1) and (2) of'Proposition 1 , then A = C ( T ) is a Rickart C*-algebra.

n

Proo$ If X E A and U = { t : x ( t )# 0 ) , then is clopen by the argument in the proof of (4) above; writing e for the characteristic function of U, we have R ( { x ) ) = R ( { e ) ) = ( l e ) Aas in the proof of [47, Prop. I]. I This completes the proof of Theorem 1. Another characterization of these algebras is as follows:

Proposition 3. Let A he a commutative C*-algebra with unity. Then A is a Rickart C*-algebra if and only if (i) A is the clo.ted linear span of its projections, and (ii) eaery orthogonal sequence of projection.^ in A has a supremum. Proof Write A = C ( T ) , T compact.

5 8.

Commutative Rickart C*-Algebras

47

If A is a Rickart C*-algebra, then (i) and (ii) hold by Proposition 1 and Lemma 3, respectively. Conversely, suppose A satisfies (i) and (ii). Let B be the linear span of the projections of A ; by hypothesis (i), B is dense in A. Since A separates the points of T, so does B ; it follows that if s and t are distinct points of T, then there exists a projection e such that e(s)# e(t). In other words, the clopen sets in T are separating, and the argument in 137, Lemma 21 shows that they are basic for the topology. To complete the proof that A is a Rickart C*-algebra, it will suffice, by Proposition 2, to show that if --

U

=

UP,,

where P, is asequence ofclopen sets, then

is clopen. Replacing

1

P,,, for n > 1, by the clopen set

we can suppose without loss of generality that the P, are mutually disjoint. By hypothesis (ii), there exists a clopen set P which is a supremum for the P,; then 0 = P by the argument in 137, Prop. 21, thus 0 is clopen. I The foregoing results are the basis for 'spectral theory' in Rickart C*-algebras. For example : Proposition 4. Let A he any Rickart C*-algebra, x~ A, x 2 0, x # 0. Gi~ienany s > 0, there exists y e { x ) " , y 2 0, such that (i) xy = e, e u nonzero projection, and (ii) Ilx -x ell < c. Proof. Since {x)" is a commutative Rickart C*-algebra [# 3, Prop. 101, we have {x)" = C ( T ), where T is a compact space with the properties (I), (2) of Proposition 1. As argued in [#7, Prop. 31, x 2 0 as a function on T. We can suppose 0 < c < llxll. Define

since llxll> 42, the open set U is nonempty. Writing z = x -(~/2)1 + Ix- (42)11, we have z~ A and

therefore is clopen by (4) of Proposition 1 ; let e be the characteristic function of a. The proof continues as in [37, Prop. 31. 1

48

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

Corollary. Let A be any Rickart C*-algebra, XEA,x f 0. Given uny e > 0, there exists y E {x*x)", y 2 0, such that (i) (x*X) y2 = e, c a nonzero projection, and (ii) Ilx - x ell < E . Proof: Same as [97, Cor. of Prop. 31.

1

In particular: Every Rickart C*-algebra satisfies the (EP)-axiom 197, Def. I]. Exercise

1A. If A is a Rickart C*-algebra, in which every orthogonal family of nonzero projections is countable (e.g., if A can be represented faithfully as operators on a separable Hilbert space), then A is an AW*-algebra.

3 9.

Commutative Weakly Rickart C*-Algebras

In this section the results of the preceding section are generalized to the unitless case. The unitless commutative C*-algebras A are the algebras C,(T), where T is a noncompact, locally compact space, and C,,(T) denotes the algebra of all continuous, complex-valued functions .x on T that 'vanish at infinity' in the sense that is compact for every e > 0; clearly the projections e t A are the characwhere P is compact and open. We seek conditions teristic functions e = x,, on T necessary and sufficient for A = C,(T) to be a weakly Rickart C*-algebra. A natural strategy is to adjoin a unity element [95, Exer. 91; the effect of this on the character space is to adjoin a 'point at infinity' to T (the 'one-point compactification'), and the rclation between A and 7' can be studied by applying the results of the preceding scction to their enlargements A , and T u {a). It is eclually easy-and illore instructive-to work out the unitless case directly, as we now do. The central result is as follows: Theorem 1. Let A be a cotn~zututiv~ C*-ulqebru witl~outunity, and write A = C,(T), w h ~ r eT is locally compact and noncornpact. 7'hc following conditions ure necessary and sufficient for A to he a ~:ecrklj~ Rickurt C*-ulgebru: (1) the conzpnct-open sets in T are basic for the topology, and (2) if P,, is any sequence qf'compact-open sets, and then U is c,otnpuct-open.

iJ' U =

u cx,

P,,,

1

The proofs of necessity (Proposition 1) and sufficiency (Proposition 2) are separated for greater clarity; throughout these results, we assume

$ 9 . Commutative Weakly Rickart C*-Algebras

49

that A = C,(T), where T is a noncompact, locally compact space, with extra assumptions on A or T as needed. Lemma 1 . If A = C , ( T ) is a weakly Rickart C*-crlgebra, tlzen any two points qf' T may be sepa~atedby disjoint compact-open sets.

Proof'. Assuming s, t~ T, s f t , we seek compact-open sets P and Q such that .YEP, ~ E Qand P n Q = (a. Let U , V be neighborhoods of s, t with U n V = @ . Choose x,Y E A so that x ( s ) # 0, x=O on T - U , and y ( t ) # 0 , y=O on T - V. Evidently x y = O ; writing e = R P ( x ) , ,f'=RP(y), we have eJ'=O. Let P and Q be the compact-open sets such that e=x,, j'=zp. Thc orthogonality of e and f means that P and (2 are disjoint. Since x e = x and x ( s )#O, we have cis) = I,s t P ; similarly ttQ. I Lemma 2. I f any two points qf T can he separated by disjoint coi?zpactopen ..ret,s, then A = C,(T) is the closed linear span of its projections.

Proof. Let B be the linear span of the projections of A ; B is a *-subalgebra of A. If s, ~ E Ts,# t , by hypothesis there exist projections e, f such that e f = O and e(s)= f ( t )= 1 ; it follows that B separates the points of T , and no point of T is annihilated by every funct~onin B, thercfore B is dcnsc in A by thc Weierstrass-Stone theorem. I Lemma 3. I f A = C , ( T ) is tlze closed linear span of its projections, then the compact-open sets of T are basic for tlze topology.

Proof. Let U be an open set, .YE U ; we seek a compact-open set P such that S E P c U . Choosc X E A with xis)= I and x=O on T- U. By hypothesis, there exists an element .YEA, y a linear combination of projections, such that Ilx-yli < 112. Say .v=/Z, e , +...+Anen, where the en are projections; we can suppose that the P , are orthogonal, and that e, # 0, A,# 0. Say e, = x,,~, P, compact-open. Since

necessarily y(s)#O, thus there exists an index j such that ej(.s)#0, that is, s t P j The proof will be concluded by showing that Pj c U. If t ~ then q y ( t )= Li by the assumed orthogonality, thus

i*?

+ > lx(t)-y(t)l

=

Ix(t)-/Z,l ;

in particular, )..(

;>

I X ( ~ ~ ) - A ,= I

11

-1~~1.

50

Chapter 1. Rickart *-Rings. Baer *-Rings, A W*-Algebras

It follows from (*) and (**) that t €5 implies Ix(t)- 11 c x(t)#O, hence ~ E U . I

+ i, therefore

Proposition 1. Let T be a noncompact, locallj~compact space such that A = C, ( T ) is a weakly Rickart C*-alqebra. Then: (1) The compact-open sets in T are basic ,for the topology. m

(2) If P, is a sequence of compact-open .sets und i f U = UP,, then U is compact-open. 1 ( 3 ) A is the closed linear spun of its projections. (4) If x~ A and U = ( t :x ( t ) # O ) , then U is the union of' a sequence of compact-open sets, 0 is compact-open, and the characteristic function of' iS is R P ( x ) . Proof. (1) and (3) are covered by Lemmas 1-3. (2) Since every sequence of projections in A has a supremum [$8, Lemma 31, there exists a compact-open set P which is a supremum for the P,. Since P n c P for all n, we have U c P ; it will suffice to is show that P. Assume to the contrary that the open set Pnonempty. By (I), choose a nonempty compact-open set Q with Q C P - 0 ;thus Q n P = Q # (ZI and QnP,,=(ZI for all n. It follows that if en, e and f' are the characteristic functions of P,, P and Q, then fe = ,f # 0 and fen = 0 for all n. Thus - f' is a projection, and (e - f)e,=ee,- fen =en-0 shows that e,, 5 e f' for all n, therefore e 5 e -f ; it follows that f = 0, a contradiction. (4) The argument in [$ 8, Prop. 1, (4)] may be used verbatim, ProI vided 'clopen' is replaced by 'compact-open'.

o=

-

Proposition 2. If ?' is a noncompact, loculll: compuct Jpuce satisfyinq conditions (1) und (2) of Propo~itionI , then A = C,(7') is (i w e ~ k l ~ v Rickart C*-ulgebm.

Proof. If x s A and U = ( t :x ( t ) # 01, then C/ is compact-open by thc proof of (4) above. If r is the characteristic function of U , then x e = x and R ( { x ) ) = R ( j c ) )as in the proof of [ $ 7 , Prop. I ] , thus e isanARPofx[$5,Def.1]. I Another characterization : Proposition 3. Let A be u commutative C*-algebra. Then A is a weakly Rickart C*-algebra if and only i f (i) A is the closed linear span of its projections, and (ii) every ovtlzogonal sequence of projections in A has a supremum.

Prooj'. We can assume A is unitless (the unity case is covered by

[9 8, Prop. 31).

$ 9 . Commutative Weakly Rickart C*-Algebras

51

If A is a weakly Rickart C*-algebra, then (i) holds by Proposition 1, and (ii) holds by [§ 8, Lemma 31. Conversely, suppose (i) and (ii) hold. Write A=C,(T), T locally compact. By Lemma 3, the compact-open sets in 7' are basic, thus condition (1) of Proposition 2 holds; to complete the proof that A is a weakly Rickart C*-algebra, it will suffice to verify condition (2). Let m

U = UP,, where P, is a sequence of compact-open sets; as argued in 1

[§ 8, Prop. 31, we can suppose the P, to be mutually disjoint, and hypothesis (ii) yields a compact-open set P which is a supremum for the P,,. The proof that = P proceeds as in the proof of (2) in Proposition 1. I

In a compact space, 'clopen' means the same as 'compact-open', and every continuous function 'vanishes at infinity'. Since the term 'weakly Rickart' does not rule out the presence of a unity element, it follows that the results of this and the preceding section can be stated in unified form; the details are left to the reader. An application to 'spectral theory': Proposition4. Let A be any weakly Rickart C*-algebra, ~ E A , x 2 0, x # 0. Given any c > 0, there exists y e {x)", y 2 0, such tlzut (i) x y = e, e a nonzero projection, and (ii) Ilx -xell < c. Proof. Let g =RP(x), drop down to the Rickart C*-algebra gAg, and apply [§ 8, Prop. 41; a minor technical point-that y~{x)"-is settled by the elementary observation that the bicommutant of x relative to gAg is contained in (x)". I

Corollary. Let A be any weakly Rickart C*-algebra, ~ E A x, # 0. Given any c > 0, there exists (x* x)", y 2 0, sucli that (i) (x* X) y2 = p , e a nonzero projection, and (ii) Ilx-xell < c. Proof. Same as [§ 7, Cor. of Prop. 31.

1

In particular: Every weakly Rickart C*-algebra satisjies the (EP)axiom [§ 7, Def. I]. Exercises

1A. If A is a weakly Kickart C*-algebra in which every orthogonal family of nonzero projections is countable, then A is an AW*-algebra (in particular, A has a unity element). 2A. Let A be a commutative C*-algebra that is the closed linear span of its projections. If X E A , x # 0, and if c > 0, then there exists a nonzero projection e such that (i) e = x y for some Y E A , and (ii) Ilx-xell< 8:.

3A. Let A be a C*-algebra in which every masa [$I, Exer. 141 is the closed linear span of its projections. Suppose that (e,)is a family of projections in A that possesses

52

Chapter 1. Rickarl *-Rings, Baer *-Rings, A W*-Algebras

a supremum e. Let then xcl=ex.

.YEA.

(i) If x e , = O for all

I,

then x e =O. (ii) If xc,= ~ . for w all

1,

4A. Let A be a weakly Rickart C*-algebra and let R be a closed *-subalgebra of A such that if (r,) is any orthogonal sequence of projections in B, then sup e , (as calculated in A) is also in B. The following conditions are cquivalcnt: (a) X E B implies RP(X)EB(RP as calculated in A ) ; (b)B is a weakly Rickarl C*-algcbra. In this case, R P's and countable sups in B are unambiguous-i.c.. thcy are the same whether calculated in B or in A.

If (A,),,, is a family of Baer *-rings and A =

n I€

r

A, is their complete

direct product [$ 1, Excr. 131, it is easy to see that A is also a Baer *-ring. However, if (A,),,, is a family of A W*-algebras, and A is their complete direct product (as *-algebras), it may not be possible to norm A so as to make it an AW*-algebra (Exercise 1); in other words, for AW*-algebras, the complete direct product is the wrong notion of 'direct product'. The right notion is the C*-sum: Definition 1. If (A,),,, is a family of C*-algebras, the C*-s~inrof the family is the C*-algebra B defined as follows. Let B be the set of all families .x= (a,),,, with a,cA, and llu,(1 bounded; equip B with the coordinatewise *-algebra operations, and the norm Ilxll =sup llu,ll. (It is routine to check that B is a C*-algebra.) Notation: B =@A,. !€I

Proposition 1. I f (A,),,, is a furnily of weukly Rickart C*-a1grhr.a.s [Rickurt C*-algehvus, A W*-ul~jehras], then their C*-sum B =@A,, is l i

1

also a weakly Rickart C*-algebra [Rickurt C*-algebra, A W*-algebra]. Proof: Let A =

n LEI

A, be the complete direct product of thc A , ,

equipped with the coordinatewise *-algebra operations [cf. $ 1, Exer. 131. Since the projections of A are the families e=(e,), with e, a projection in A, for each L E I , and since the projections in a C*-algebra have norm 0 or 1, it is clear that B contains every projection of A. Suppose cvcry A, is a weakly Rickart C*-algebra. It 1s routine to A check that A is a weakly Rickart *-ring; explicitly. if x= ( ~ , ) E and if, for each 1 , e,=RP(a,), then the prqjection e=(e,) is an AKP of .x in A. Since B contains all projections of A . it follows that B is a weakly Rickart C*-algebra. If, in addition, every A, has a unity element, then so does B; this proves the assertion concerniilg Rickart C*-algebras [cf. 5 3, Exer. 12, 131.

Finally, suppose every A, is an AW*-algebra; it is to be shown that B is a Bacr *-ring. Sincc B contains every projection in A, it is sufficient lo show that A is a Baer *-ring [cf. 4 4, Exer. 6, 71. Let S be a nonempty subset of A ; we seek a projection t l A~ such that R(S)= eA. Write n,: A +A, for the canonical projection, and let S, = n,(S). Clearly R(S)= (xEA: TC,(X)ER(S,) for all L E I ) .Write R(S,)=e, A,, el a projection in A,, and set e=(c,); evidently x € R ( S ) iff e,n,(x)=n,(x) for all L E I iff c x = x , thus R ( S ) = r A . I Proposition 1 is a result about 'external' direct sums; let 11s now look at 'internal' ones. If (A,),,, is a family of C*-algebras with unity, and if, for each x ~ l h,=(6,,1) , is the element of B = @ A , with 1 in 1 ~ 1

the xth place and 0's elsewhere, it is clear that the / I , are orthogonal central projections in B, and that sup h, exists and is equal to 1. Conversely, under favorable conditions, a central partition of unity in an algebra induces a representation as a C*-sum; the next two propositions are important examples.

Proposition 2. Let A he an A W*-ul
qf A onto

@ h, A. LEI

Proof: Since the h,A are AW*-algebras [$ 4, Prop. 8, (iii)], their C*-sum B =@ h, A is also an A W*-algebra (Proposition 1). re1

If u r A then Illz,ull 5 \lull shows that the family (h,rr),,, is in B, and, defining rp(rr)=(h,r~),we have Ilrp(a)ll 5 llall. Clearly rp: A -.B is a *-homomorphism, and Ilrp(a)ll< llall shows that rp is continuous. Moreover, rp is injective (if h,u = O for all 1 then u=O [$ 3, Prop. 61). It follows that l\cp(u)ll=llall for all U E A (scc Exercise 3). Sincc A is complete in norm, so is its isometric image cp(A), therefore cp(A) is a closed *-subalgebra of B. It remains to show that (p(A)=B; since B is the closed linear span of its projections [cf. 4 8, Prop. I], it will sufficc to show that y(A) contains every projection of B. Suppose e € B is a projection, say e = (e,), where e, is a projection in h, A, that is, e, < h,. Define f ' = s u p ~ , ; it is elementary that h , , f ' = e , for all 1, thus p()=e.

I

The above argument requires that the projection lattice of A be complete (via the definition of f ) ; this hypothesis can be weakened provided that we retreat from families to sequences:

Chapter 1. Rickart *-Rings, Baer *-Rings, A W*-Algebras

54

Proposition 3. Let A be a Rickart C*-algebra and suppose (h,) is an orthogonul sequence of central projections in A with sup h,= I.Tlzen A i.5 isometrically *-isomorphic to the C*-sum of t l z ~h,A, viu the mupping OU

cp : A

+@

hnA defined b,v cp ( a )= (h,a).

n= 1

Proof: In the proof of Proposition 2, replace the index set I by the set of positive integers, and read "Rickart C*-algebra" in place of "A W*-algebra"; no other change is necessary, except to note that every sequence of projections in A has a supremum [$S, Lemma 31. 1 Exercises

1C. (i) If a Banach algebra is a regular ring [$3, Exer. 61, ~t must be finitedimensional. (ii) If, for n = I,2, 3, ..., A, is the algebra of all 2 x 2 complex matrices, and if

n U)

A=

A, is the complete direct product, then A cannot be normed to be a Banach

n= 1

algebra.

2A. If (A,),,, is a family of C*-algebras, the C*(m)-.sum of the family is the closed *-subalgebra of @A, consisting of those x=(a,),,, such that, for every E > 0, 'tl

Ila,ll< E for all but finitely many indices. (This amounts to putting the discrete topology on I and requiring that Ila,ll + O at a,in the sense of the one-point compactification of I.)

3A. (i) In a C*-algebra, if x = z i i e i , where the ei are orthogonal, nonzero 1 projections, then llxll= max /Ail. (ii) If (p: A + B is a *-homomorphism, whcrc A is a Banach *-algebra with continuous involution and B is a C*-algebra, then (p is continuous. (iii) If A and B are C*-algebras, and if (p: A + B is a *-monomorphism, then Il(p(x)ll= llxll for all X E A. When A and B are weakly Rickart C*-algebras, a simplc proof can be based on (i) and (ii). 4A. If (A,),,, is a family of C*-algebras, then their P*-sum [$I, Exer. 161 is a subalgebra of their C*-sum. 5A. Let (T,),,, be a family of connected, compact spaces, let A,=C(T,), and let A be the P*-sum of the A,. If x=(a,),,, is in A, then, for all but finitcly many 1, a, is a scalar multiple of the identity of A,; moreover, only finitely many scalars can occur as coordinates of x.

Chapter 2

Comparability of Projections

5 11.

Orthogonal Additivity of Equivalence

Let A be a Baer *-ring, let (e,),,, and (f,),,, be orthogonal families of projcctions indexed by the same set I , let e=supe,, f =sup f,, and suppose that el- f, for all L E I . Does it follow that e - f? I don't know (see Exercise 3). If the index set I is finite, the question is answered affirmatively by trivial algebra [$ 1, Prop. 81. The present section settles the question affirmatively under the added rcstriction that e f = 0; this restriction is removed in Section 20, but only under an extra hypothesis on A . Some terminology helps to simplify the statements of these results:

Definition 1. Let A bc a Baer *-ring (or, more generally, a *-ring in which the suprema in question are assumed to exist). If the answer to the question in the first paragraph is always affirmative, we say that equivalence in A is additive (or 'completely additive'); if it is affirmative whenever card 1 1 N, we say that equivalence in A is N-udditive; if it is affirmative whenever ef = 0, we say that equivalence in A is orthogonally additive (see Theorem 1). The term orthogonally N-additive is selfexplanatory. Suppose, more precisely, that the equivalences e l - f , in question are implemented by partial isometries w, ( 1 E I). We say that partial isometries in A are addable if e f via a partial isometry w such that we, = w, = f,w for all L E I . The terms N-addable, orthogonally addable, and orthogonally N-addable are self-explanatory. The main result of the section:

-

Theorem 1. In any Buer *-ring, partial isometries are orthogonullq, addable; in particular, equivalence is orthoyonally additive. Four lemmas prepare the way for the proof of Theorem 1.

Lemma 1. In a weakly Rickart *-ring, suppose (lz,),,, is an orthogonal family of projections, and (e,),,, is a (necessarily ortlzogonal) family S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

Chapter 2. C:oniparability of I'rojections

56

of projections such that el _< h, for all r and such tlzat Then e, = h,e for all 1 .

rj = sup r,

cxi,rts.

Prood. Fix r and set x = h , e - e l ; obviously x e = x . If x f r then xe, = h,e, - e,e,= 0 by the assumed orthogonality; also xe,= h , e , c ,= 0 ; therefore x e = 0 [$ 5, Exer. 41, that IS, .x=O. I

-

Lemma 2. If' A is a Rickart *-ring containing u projec.tion e .\uclz that e I - c., then 2 = 1 + 1 is invertible in A . Proof. Let w be a partial isometry such that w* w = e , w w * = 1 e , and write R ( { ew*))= f A , f a projcction; wc show that x= fe + wf 11. satisfies 2x = 1. From ( e - w*)j = 0, wc have f w = / e. Since t v ~ ( 1 -e) Ac, it follows that ( e - w*) ( e + MI)= 0, therefore f ( P + n,)= r + w ; notliig that f w = f P , this yields -

Right-multiplying (*) by IV*, we have joints, we obtain Addition of (*) and (**) yields 1 = 2x.

MI* = 2

fw* - ( 1 - r ) ; taking ad-

I

Lemma 3. Let A be a ~ e a k l yRiclcurt *-ring in which e o e v ortlzogonr~l ,family of' projections of' cardinality < X hc~sLI supremum. Let (h,),,, be an orthogonal firmilv of' central projections, ~ ~ i t h card I I K , and suppose tlzat, fbr ecrclz 1, e, and f ; are ortlzogonal projc~ctions with el+ f ; = h,, c . , - , / ; . Let e=sup e,, J'=sup f;. Tlzen e /: More prrcisc~ly, jf' the equinalences r , f ; ure inzpkenzented by purtic~l isometries w,, then there c.xist.s u partial isomc7trj3 implemcntin~g e -- J; such tlzat wr, = w , = ,J'w ,fbr all I .

-

-

Procf. Since h, A is a Rickart *-ring [$ 5, Prop. 61 and w , t h , A , we have e l - f ; in h , A . By Lemma 2, 2h, is invertible in lz,A ; say a , ~ l zA, with lz,=(2/z,)a,=2n1. Since 2 h , is self-adjoint and central in h,A, so is a,. Write u,= M ~ , + w T ; clearly u, is a symmetry ( = self-adjoint unitary) in Iz,A . that is, uT =u,, uf =h,. Defining (informally, y, = (112)(Ill+ w,+w?)), it is easy to check that I/, is a projection in h, A . Define g = sup q,. Citing Lemma I,we have Iz,g = g,, Iz,e = r , , h, f = f ; . Finally, define w = 2.fje;

11. Orthogonal Additivity of E q ~ ~ i v a l e ~ l c e

57

the proof will be concluded by showing that M? is a partial isometry having the desired properties. Note that j ; u , = w,= u,e,; it follows easily that J ; u , P , = a,w,,

therefore 2 ,f',g,e,= 2 a, w, = lz, MI,= w,. Then 17, kt' = W , ;

(1)

for, lz,w= h , ( 2 , f g e )=2(/z,.f')( h , ~ l(h,e)=2,/',q1c~, ) = M),. It follows that (2)

w er = w = r / ‘. l W ;

for example, w c , = ~ ~ ~ ( / ~ , e , ) = ( l z , w ) e , = w ~ , c , = w , . It remains to show that w* w = c and WJ w* = J'. Let h = sup 11,. Since e,
(w*w - t ~ ) l l=, (12, cv)*

(12, w)- 12, c =

w: w ,- c, = 0 ,

therefore (w* 1.1: c )h = O [$ 5, Exer. 41, thus w* w - e = 0. Similarly ww*f=O. I -

Lemma 4. If' A is u weakly Rickart *-ring in whiclz every or/hogonul ,fumily of projection.^ oj' cardinality 5 W has a .suprmzum, then partial isometries in A are orthogonally W-crdduhle. Proof: Let (w,),,, be a family of partial isometrics, card l < W , with orthogonal initial projections e,, orthogonal final projections Ji, and such that, setting e = sup e l , f = sup ,I;we , have cf'= 0. We seek a partial isometry w such that w* w = e , w w * = f and we,=w,=,f;w for all 1 . Write h,= el j;, S = (h,:1 E 11, and let B = S'. According to [$ 5, Prop. 51, B is also a wcakly Rickart *-ring, with unambiguous KP's. Since the h, are orthogonal, S is a commutative set; it follows that B 3 B', and that B has center B n B' = B' = S". In particular, the h, are central projections in B. Each w, belongs to B ; for, 11,w,= w, h, ( = w,)and 17, w,= MI,h, ( = 0) whenever x # 1 , thus ~ , E S=' B. In particular, el ,/; in B. Finally, it is clear that e , ,f E B (cf. the proof of L$ 3, Prop. I l l ) , thus the desired w exists by Lemma 3. 1

+

-

Proof of' Theorem I. When A is a Baer *-ring, thc hypothesis of Lemma 4 is verilied for every W. I

We now take up several other useful consequences of Lemma 4. Proposition 1. L6.t A he u weakly Rlckort *-ring in whiclz every ortlzogonul funzily of projections of cc~vciinality 5 W has a supremum.

rhapter 2. Comparability of Projections

58

Let (e,),,, be an infinite family of mutually equivalent, orthogonal projections, with card 15 K, and let J c 1 with card J = card I. Define e = s u p { e , : ~ ~ ~ }f ,= s u p ( e , : ~ ~ J )

Then e - f . Proof. Dropping down to eAe, we can suppose that A is a Rickart *-ring [tj 5, Prop. 61. Write J = J' u J " , where J' nJ" = ja and card J' = card J" = card J ( = card I). Define

Since the e, are orthogonal, clearly f ' g =O, thus sup -( f ' , g ) but J' u [J" u (I - J ) ] = I , therefore

= ,f' + g ;

by the associativity of suprema. Similarly,

(2) f =ff+.f". Since J' and J" u ( I - J ) have the same cardinality, and since f 'g=O, we have (3)

f'-s

by Lemma 4. Similarly,

(4) Adding (3) and (4), we have f ' f-e. I

f"

-

+ f"

f".

-

g + ,f", that is, citing (1) and (2),

Proposition 2. Let A be a Buer *-ring, e and f projections in A with (h,) an orthogonal ,fhmily of central projections, und h = sup ha. If hue5 h, f jbr all a, then h e 5 hf: More precisely, if, for each cc, w, is a partial isometry such that w,* w, = h,e and w,w,* = ,fa 5 haf , then there exists a partial isometry w such that w* w = h e and w(h,e)= w, = f,w fir all a.

ef

= 0,

Proof. Since (sup h,e) (sup f,) = h e sup f ,5 h ef = 0, and since the f , are also mutually orthogonal (because the ha are), Lemma 4 is applicable to the partial isomctries w,. I In a weakly Rickart C*-algebra, countable suprema are available [$ 8, Lemma 31, thus Lemma 4 holds with N = KO,implying the obvious sequential forms of Propositions 1 and 2.

6 12. A General Schroder-Bernstein Theorem Exercises

59

-

1A. Let e, f be orthogonal projections in a Rickart *-ring. In order that e J; it is necessary and sufficient that there exist a projection g such that 2ege=e, 2fg.f=f; 2geg=2gfg=g. 2A. Let C be a Baer *-ring possessing a projection e such that a-I -e (for example, the *-ring of all 2 x 2 complex matrices). Let B be the complete direct

n 00

product of EC, copies of C, that is, B =

A, with A,= C for all n [cf. 1, Exer. 131.

Let B, be the weak direct product of the A,, that is, the ideal of all x=(u,) in B such that a,=O for all but finitely many n. Write f = I -e, F=(e, e, e, .. .), S=(,f,,f,,f; . _ ) = I F , and let S be the *-subring of B generated by F and 1; thus, S = {mF+n f ' : m, n integers). Define A = B , + S ; thus, A is the *-subring of B geIf the additive group of C is torsion-free, then A is a Kickart nerated by B,, 4 and l. *-ring. 2, 3, ... write em= (6,, e), ,f, = (S,,J'). Then, relative to the *-ring A, For m = I, we have em- f, for all m, sup e,=F, sup J,= f , Ff = 0, but F is not equivalent to Thus Theorem 1 does not generalize to Rickart *-rings.

7.

3D. Let C be a Baer *-ring, let C" be the reduced ring of C [$3, Exer. 181, and suppose there exists an equivalence e - f in C which cannot be implemented by any partial isometry in C" (that is, e J' but not e 2 f). Let A be the P*-sum of No copies of C [cf. $4, Exer. 81. For m= I,2, 3, ... let emand f,be the projections in A defined by the sequences e, = (6,, e), j = (6,,J'). Then em-I, for all m, but sup emis not equivalent to SUP./,. Problem: Does therc cxist such a Bacr *-ring C? (Cf. [$17, Exer. 201.).

-

4D. Problem: Is equivalence No-additive (i.e., 'countably additive') in a Kickart C*-algebra'? 5A. If A is a Baer *-ring such that the ring A , of all 2 x 2 matrices over A is also a Baer *-ring (with *-transpose as involution), then partial isometrics in A are addable. 6A. In the notation of Definition 1, if there exists an element x such that xe,=w,=f;x for all 1 , then the partial isometries w, are addable (the desired partial isometry is w=xe).

5 12. A General Schriider-Bernstein Theorem

-

We say that the Schrodev-Bernstein theorem holds in a *-ring if the relations e 5 f and f 5 e imply e ,f. In Section 1, it was shown that the Schroder-Bernstein theorem holds in any *-ring whose set of projections is conditionally complete [$ 1, Th. 11-in particular, it holds in any Baer *-ring [cf. 4 4, Prop. I]. The fixed-point theorem employed there requires lattice completeness; by reverting to the format of the classical set-theoretic proof, one can get along with countable lattice operations :

60

Chapter 2. Comparability of Projections

Proposition 1. If A is a weakly Kickart *-ring in which etery countable family o f orthogonal projections has a supremum, then the SchroderBernstein theorem holds in A. It is convenient to separate out an elementary lemma:

(.,

Lemma. If is u decr~xzsinqsequence of projections in such a *-ring. that is, if el 2 r , 2 e , > .. . , then inf e, exists. Explicitly, inf en = el g , ,... ) .

where g=sup{e,-e,+,:n=1,2,3

-

Pro<$ of' Proposition 1. Assuming e f" 5 f ' and f'- e' 5 cJ, it is to be shown that e j'. Let w be a partial isometry such that w* ltl= f , w w* =el. Sctting v=M'f", we have v* r;= f " ; thus 2. is a partial isometry, and, writing e" = 11 2 : * , we have j" c" 5 e' .

-

-

Combining this with e

- f',

we have the following situation

(1

c" 5 e'

<e

and e"

-

P.

- -

On the basis of (I), it will be shown that e'-e (the observation f e' e then ends the proof); no further reference to f is necessary. there exists a partial isometry u such that By (I), u*u = (." ,

uu*

= 6'.

Since g ++ cp(g)=u*gu is an order-preserving bijection of LO, c] onto 10, e"] (scc [$ 1, Prop. 9]), we may define a sequence r,, r,, e,, . . .,c,,,. . . of subprojections of r as follows:

Define another sequence c,,c,,c,, . . ., r , , ,, . .. of subprojections of by the same technique, starting with e':

$12. A General Schroder-Bernstein Theorem

Observe that (Indeed, (1) may be written e,, 2 P , 2 r , ; application of q yields e, 2 e, > e4; etc.) We now look at the 'gaps' in the decreasing sequence (2). Since, by definition, u*e,u=e,,+, ( n = 0 , 1 , 2 , 3,... ), we have ~ * ( c , , - ( ~ , + , ) u -en+ - en+, thus (3)

~ ~ - e , -en+,-en+, ,+~

(n=0,1,2,3 ,...)

(the equivalence (3) is implcmcnted by the partial isometry u*(cn- en+ ,)). By the lemma, we may define Obviously any truncation of the sequence en has the same infimum, in particular,

Consider the following two sequences of orthogonal projections:

(the second sequence merely omits the second term of the first sequence). In view of (4), it follows from the lemma that thus, by the associativy of suprema, the sequence (*) has supremum em,+(e, - e x ) = e, = e. It follows similarly from (5) that the sequence (**) has supremum e , +(el - e ), = e, = e'. The desired equivalence e-e' is obtained by putting together the pieces in (*) and (**) in another way. We define y = s ~ p ( ~ ~ - e , , e , - e , , e ~ - ~...,, , ,I q' = sup (P, - e,, cJ4 - e5, e6- e7, . . .}, h=e,,+sup{e, -e,, e,-e,, e,-e,, ...).

By the associativity of suprema, q + h coincides with the supremum of the scquence (*), thus similarly, y' + h is the supremum of the scquencc (**), thus

62

Chapter 2. Comparability of Projections

-

It follows from (3), and the definitions of g and g', that g Prop. 11; in view of (6) and (7), this implies e e'. I

-

g' [$11,

The principal applications:

Corollary. The SchrBder-Bernstein theorem holds (i) in any Baer *-ring, and (ii) in any weakly Rickart C*-algebra. Proof. Of course (i) is also covered by 151, Th. 11;(ii) follows from the fact that every sequence of projections in a weakly Rickart C*-algebra has a supremum [$8, Lemma 31. 1 Exercises 1A. Let A be a weakly Rickart *-ring in which every countable family of orthogonal projections has a supremum. If e is any projection, write [el for the equivalence class of e with respect to -, that is, [el = ( / : / e}. Define [el I [ f ] iff ed f . This is a partial ordering of the set of equivalence classes.

-

-

2A. The Schroder-Bernstein theorem holds trivially in any finite *-ring. (A *-ring with unity is said to be finite [9:15, Def. 31 if e 1 implies e = I.) 3A. Let .%? be a separable, infinite-dimensional Hilbert space, with orthonormal basis e,,e,,e ,,.... Let T be the operator such that Te,=e,+, for all n ; thus T* T= I, T T* = E, where E is the projection with range re,, e,, e5, ...1. Let F be the projection with range [e,, e3, e,, ...I. Finally, let .d be The *-ring generated by T and F. F and F < I The relations T* T= I, T T * = E 5 F and F I I show that relative to the *-ring d.Is F I relative to .&?

-

Is

5 13. The Parallelogram Law (P) and Related Matters The law in question is reminiscent of the 'second isomorphism theorem' of abstract algebra:

Definition 1. A *-ring whose projections form a latticc is said to satisfy the parallelogram law if for every pair of projections e,J: The projections of every weakly Rickart *-ring form a lattice [$5, Prop. 71, but even a Baer *-ring may fail to satisfy the parallelogram law (Exercise 1). Occasionally, the following variant of (P) is more convenient:

Proposition 1. Let A he a *-ring with unity, whose projections fbrm a lattice. The following conditions are equivalent: (a) A satisfies the paralleloyram law ( P ) ; (b) e - e n ( 1 -j') f - (1 - e) n f for every pair ofprojections e,f .

-

5 13. The Parallelogram Law (P) and Related Matters

63

Proof. Replacement o f f by 1 -f in the relation (P) yields e-en(1-j')-[eu(I-

f')]-(1-,f)=,f-[I

- e u ( I - f')] I

=f-(I-e)nf.

Proposition 1 may be interpreted as saying that, in the presence of (P), certain subprojections of e,f (indicated in (b)) are guaranteed to be equivalent; this conclusion reduces to the triviality 0 0 precisely when e = e n (I - f ) , that is, when ef = 0. The projections that occur in (P) are familiar from 155, Prop. 71:

-

-

Proposition 2. IfA is a weakly Rickart *-ring such that LP(x) RP(x) jbr all XEA, then A sati.sfie.~the parullelogram law (P). Proof. Apply the hypothesis to the element Prop. 71. 1

x= e-eS

[$5,

An important application : Corollary. Every von Neumann algebra sati.sfies the purallelogram law (P). Proof. Let .dbe a von Neumann algebra of operators on a Hilbert space 2 [$4, Def. 51. If T is any operator on X , the 'canonical factorization' T = WR is uniquely characterized by the following three properties: (i) R 2 0, (ii) W is a partial isometry, and (iii) W* W is the projection on the closure of the range of R, that is, W* W=LP(R) as calculated in 9(%). It follows that W* W=RP(T), W W*=LP(T), thus LP(T) -RP(T) in 9 ( , X ) .The proof is concluded by observing that if T e d then WE& (therefore LP(T) RP(T) in .d). Suppose T E ~ . If U ~ . d 'is unitary, then T = U T U* = (U W U*) (UR U*); since the properties (i), (ii), (iii) are satisfied by the positive operator UR U* and the partial isometry U W U*, it follows from uniqueness that U W U* = W, thus W commutes with U. Since d' is the linear span of its unitaries (as is any C*-algebra with unity [cf. 23, Ch. I, Ej 1, No. 3, Prop. 3]), it follows that WE(&')' =.d. I

-

Later in the section it will be shown, more generally, that every A W*-algebra-indeed, any weakly Rickart C*-algebra-satisfies the parallelogram law (P). Thc proof will avoid the use of LP R P (known to hold in any A W*-algebra [Ej20, Cor. of Th. 31, but of unknown status in Rickart C*-algebras). The general strategy is to reduce the consideration of arbitrary pairs of projections e,f to pairs of projections in 'special position'; the following concept is central to such considerations:

-

64

Chapter 2. Comparability or 1'1-ojcctions

Definition 2. Let A be a *-ring with unity, whose projections form a lattice (for example, A any Rickart *-ring). Projections e, f in A arc said to be in position p' in case

{Equivalenty, e n (I - J') = 0 and e u (I - f ) = I ; that is, the projections e, 1 - f are comp1ementary.j The condition is obvio~~sly symmetric in e andf. In Rickart *-rings, the concept has a useful reformulation: Proposition 3. In a Rickurt *-ring, the ,fi,llowiny conditions on a pair

of' projections e,j'imply one another:

(a) e, f are in position p'; (b) LP(ef')=e and RP(ef)=,f. Proof: Let .x= e,f= [I -(I LP(x)=e-en(1-

f'),

-

j ) ] . Citing [$3, Prop. 71, we have RP(x)=eu(I -f)-(I-

thus, the conditions (b) are equivalent eu(1-f)=I. I

to e n (I

f); -

f ) =0

and

In a Rickart *-ring, the parallelogram law can be reformulated in terms of position p': Proposition 4. The jollowing conditionn on a Rickart *-ring A are equivalent: (a) A satisfies the parallelogram law (P); (b) if e,f are projections in position p', thrn e f .

-

Proof. (a) implies (b): If e n (I J') = (I - e) n f'= 0, then, in the presence of (P), e -- J' by Proposition 1. (b) implies (a): Let e,f be any pair of projections, and set e'= LP(e.1'). j" =RP(ej'). Since e j'= er(e,f'Xl"= e'f", it follows from Proposition 3 that e1,f' are in position p'; therefore, by hypothesis, el-j", that is, -

Since e,f are arbitrary, it follows from Proposition 1 that A satisfies (PI I The proof of Proposition 4 yields a highly ~ ~ s e f udecompositioii l theorem: Proposition 5. Let A he a Rickart *-ring suti.s/ying t/?e parallelogranz law (P). If' e, f i.7 any pair qf'prqjections in A, there c~.~ist orthogonal decompositions e=el+e", j ' = fl+J'"

$ 1 3 The Parallelogram Law (P) and Related M'ltters

with e', f ' in position p' (hencc~e'

-f '

by Proposition 4) and e V f =e f"

65 = 0.

Proof. Let e' = L P(ef ) , f ' = R P(ef ); as noted in the proof of Proposition 4, e', f ' are in position p'. Set e" = r -el, f ' " =f -f " ; obviously e"(e,f)=(ef)f"=O, thus e " f = e f U = 0 . I The rest of the section is concerned with developing sufficient conditions for ( P )to hold. With an eye on Proposition 4, we seek conditions ensuring that projections in position p' are equivalent. For the most part, victory hinges on being able to analyze position p' considerations in terms of the following more stringent relation: Definition 3. Let A be a *-ring with unity, whose projections form a lattice. Projections e, f in A are said to bc in position p in case {Equivalently, each of the pairs e,f and e, I - f is in position p'.) The condition is obviously symmetric in e and f.

Tf e,f are in position p, then so is any pair g, h, whcrc g = e or I - e, and lz= f or I - f . In Proposition 3, position p' is characterized in terms of the elemcnt e , f ; the characterization of position p involvcs both e,f and its adjoint: Proposition 6. In a Rickart *-riny, theJollowm~gcondition., on a puuof projections e, f imply one another. (a) e, f are in position p ; (b) RP(ef - fe)= I . Proof. (b) implies (a): Set x = ef f e. Since R P ( x )= 1, the relations e n f= 0 and e u f = I are implied by the obvious computations -

But e(1 - f ) - (I - f ) e = x also has right projection 1, therefore e n ( 1 - f ) = O and ~ u ( 1 f-) = l . Thus e n , f = ( l - e ) n ( I - f ) =en(I -f)=(l -e)nf=O. ( a ) implies (b): Let x = e f - fe, g = R P ( x ) ; assuming e,f are in position p, it is to be shown that g= 1. (For an insight on the success of the following strategem, compute ( a h- ha)' for a pair of 2 x 2 matrices a, h over a commutative ring.} Set z = X* x = - x 2 ; by direct computation,

Chapter 2. Comparability of Projections

66

From the last two formulas, it is clear that e and f commute with z. On the other hand, [$3, Cor. 2 of Prop. 101, therefore g commutes with e and with 1. Set h = I - g. Since g = RP(x) and since h commutes with e and f ; we have Prop. 31, we have thus eh, f h are commuting projections; citing [$I, thus (1)

(eh)(.fh)=(eh)n(.fh)~en.f=O,

(e,f)h = 0 .

Since e(1- f )-(I -f ) e = - x also has right projection g, and since e n (I -f ) = 0 by hypothesis, the same reasoning yields (2)

[e(l- f)]h=O.

Adding (1) and (2), we have eh = 0. Similarly f h = 0. Thus e I 1 -h = g and f lg ; since e u f'= I, we conclude that g = 1. I An obvious way to fulfill condition (b) of Proposition 6 is to assume outright that ef-,fe is invertible; in the next proposition, it is shown that the invertibility of ef -fe implies e f, provided one also assumes a condition on the existence of 'square roots'. Historically, the first condition of this type, considered by I. Kaplansky ([52], [54]), was the following:

-

Definition 4. A *-ring is said to satisfy the square-root axiom (briefly, the (SR)-axiom) in case, for each element x, there exists r ~ ( x * x ) "such that r * = r and x*x=r2. Occasionally, the following weaker axiom suffices (later in the section, stronger axioms will be employed):

Definition 5. A *-ring is said to satisfy the weak square-root axiom (briefly, the (WSR)-axiom)in case, for each element x, there exists r e {x* x]" (necessarily normal, but not necessarily self-adjoint) such that x* x = r* r ( = r r*). A sample of the wholesome effect of square roots:

Lemma. If A is a *-ring satisfying the (WSR)-axiom, and if the projections e, f are algebraically equivalent in the sense that y x = e and xy = f for suitable elements x, y~ A , then L. f .

-

5 13. The Parallelogram Law (P) and Related Matters

67

Proof Replacing x and y by , f x e and e y f , we can suppose x r f A e , y ~ e f:A We seek a partial isometry w such that w* w = c , w lu* = ,f'. Choose r ~ {y*y)" with y* y = r* r = rr*, and set w = rx. Then On the other hand, ww*=rxxYr*; to proceed further, we show that r commutes with x x * . Since r ~ { y * y ) " it, suffices to note that x x * ~ { y * y ) ' ;indeed, xx* and y*y are self-adjoint elements whose product ( x x * ) ( y * y )= x ( y x ) * y = x e y = x y = j is also self-adjoint. Thus

YE

(x* x)', and

W W * = Y X X * Y=*x x * r r * = ( x x * ) ( y * y ) = , f .

I

Armed with square roots, a considerable dent can be made on the parallelogram law problem: Proposition 7. I f A is a *-ring with unity s~ltisfyingthe (WSR)-~~xiorn, and if e, f are projections in A such that ef - f e is invertible, then -,f. e - f -1-e-1 Proqf. Since the invertibility hypothesis for the pair e , ,f' clearly holds also for the pairs e, I- f and 1 -e, ,f, it is sufficient to show that e f. Let z=(ej - je)* (ef - je) = - (ef - je)' and write B = { z J r . As noted in the proof of Proposition 6, e, f E B. Since ( z ) c { z ) ' , we have B = { z ) '2 (z)"= B', thus B has center B nB' = B' = ( z ) " . In part~cular, z is central in B. We assert that efe is invertible in eBe. The proof begins by noting that s = z P 1 is also central in B ; then z s = s z = l implies ( e z e )(ese) =(ese)(eze)=e, thus e z e = e z is invertible in eBe. From one of the formulas for z in the proof of Proposition 6, we have

-

thus the invertibility of e z in e Be implies that of efe. Let t ~Bee with t(efe)=(efe)t=e, that is,

(*I

t,fe = yft

=e

(explicitly, t =s(e - e.fe)). By the lemma, it will suffice to show that e and f are algebraically equivalent. To this end, define

68

Chapter 2. Comparabil~tyof Projections

Obviously X E f A e , y ~ e A f ;and .vx=(ef) (ft)=e,f't=e by (a). On the other hand, xy=(f't) (cf')= f t f ; citing (*) at the appropriate step, we have (gf -,fe)xy = (ef -,/)f'tf'= e f t j ' - f'eftf = (e,f't)J' f (ef t ) /' = ef' j'e J' = (ef - f ' r ) /; -

-

therefore x y = f by the invertibility of ef

-

fe.

I

The technique of Proposition 7 suffices to establish the parallelogram law in the C*-algebra case: Theorem 1. Every wrakl-y Rickart C*-al~grbt-asati,sfies tlze purallelogram law (P). Prooj'. If A is a weakly Rickart C*-algebra, then the projections of A form a lattice [$ 5, Prop. 71. Let e. f be any pair of projections in A. To verify that e, f' satisfy the relation (P), it suffices to work in the Rickart C*-algebra ( e u j') A(e u J') [$ 5, Prop. 61; dropping down, wc can suppose without loss of generality that A has a unity clement. Set z = (ef f e)* (ef fe) = ( ef 'fe)2 and consider the Rickart C*-algebra ( z ) ' [$ 3, Prop. 101. As noted in the proof of Proposition 7, e, { ~ { z ) ' and {z}' has center (z]". Dropping down to {z)', we can suppose that z is in the center Z of A . (This will yield the sharper conclusion that the equivalcnce e - n,f e u f - ,f can be implemented by a partial isometry in {(ef - f'e)2)'.) Write Z = C ( T ) , T a compact space with the properties noted in [$ 8, Prop. I]. By C*-algebra theory, we have z 2 0 in Z (see the proof of [$ 7, Prop. 31); setting -

-

it follows that U is a clopen set whose characteristic function h is RP(z) [$8, Prop. I]. If U is empty, that is, if z = 0 , then e l ' f e = O and the desired relation (P) reduces to the triviality r -e f = ( r + f -e f ) - ,f' [$ 1, Prop. 31. Assuming U is nonempty, write U = UP,,, where P,, is a scqucncc (possibly finite) of disjoint, nonempty clopen sets (cf. the proof of [$ 8, Prop. 31). Let h, be the characteristic function of P,; thus the h, arc orthogonal central projections with suph,=h (cf. 157, Lemma to Prop. I]). Since z is bounded below on the compact-open set P,,, it follows that zh, is invertible in 11, A; but -

zh, therefore

=

-(eJ

-

f el211,

=

-

[(e A,,)( f lz,,) - ( f 12,) (e h,)I2 ,

13. The Parallelogram Law (P) and Rclated Matters

69

by Proposition 7 (note that every C*-algebra satisfies the (SR)-axiom by easy spectral theory [cf. 5 2, Example 51). By Proposition 6, ch, and fh, are in position p in h,A; in particular, therefore (1) may be rewritten as Since h, is central, the foregoing relation can, by lattice-thcorctic trivia, be rewritten as Since hA is the C*-sum of the h,A [$ 10, Prop. 31, and since every partial isometry has norm 51, it follows from the relation (1') that (e-en f)h

(2)

What happens on 1 -lz?

-( r uf - f)h.

Since h=RP(z), we have

0 = z(1- h) = (ef

-

f e)* (ef

-

j'e) (1 - h) ,

therefore (ef - f e ) (I - h) = 0, that is, e(1 h) and f(1 - h) commute. Write e' = e(1- h), f ' =f ( l - h); as noted earlier, the relation -

-

n f"

holds trivially, thus ( e - e n f ' ) ( l -h)

(3)

-

Adding (2) and (3), we arrive at (P).

"

J" - f '

( e u f ' - f ) ( 1 -11) I

To proceed further, it is necessary to sharpen the conclusion of Proposition 7 (the price, of course, is a sharper hypothesis). As it stands, the relations e f and 1 - e I- f obviously imply that r. and j' arc unitarily equivalent, that is, u r u * = f for a suitable unitary clement u; the sharper conclusion needed is that u can be taken to be a symmetry in the sense of the following definition:

-

-

Definition 6. In a *-ring with unity, a ,sj~nlmetr.yis a self-adjoint unitary (u* = u, u2 = 1). In a *-ring with unity, the mapping e -t u = 2 e - 1 transforms projections e into symmetries u ; if, in addition, 2 is invertible. then this mapping is onto the set of all symmetries, with inverse mapping u*($) (I +u).

Definition 7. If e, j are projections such that ueu = f for some symmetry u (hence also uf'u=e), we say that P and f are exchanged by the symmetry u.

70

Chapter 2. Comparability of Projections

It can be shown that if, in Proposition 7, one assumes the (SR)axiom, then the projections e, J' can be exchanged by a symmetry (see Exercise 5). We content ourselves with a much simpler result (it is complicated enough) based on a stronger axiom. The stronger axiom depends on a general notion of positivity available in any *-ring (and therefore generally useless), consistent with the usual notion of positivity in C*-algebras:

Definition 8. In any *-ring, an element x is called positive, written x 2 0, in casc x =yT y, + .. . +y,*y, for suitable elements y,, .. .,y,,. The following properties are elementary: (1) if x 2 0 then x* = x ; if x 2 0 then y*xy 2 0 for all y; (3) if x 2 0 and y 2 0, then - y 2 0 . (Warning: x 2 0 and -x 2 0 is possible for nonzero x; equivalently, the relations x 2 0, y 2 0 and x +y = 0 need not imply x=y=O.} In particular, elements of the form x*x are positive; thus the following is an obvious strcngthcning of the (SR)-axiom:

Definition 9. A *-ring is said to satisfy the positive square-root axiom (briefly, the (PSR)-axiom) in case, for every x 2 0, there exists y ~ j x ) " with y > O and x = y 2 . The axiom wc want is still stronger:

Definition 10. A *-ring is said to satisfy the unique positive squarcJroot axiom (briefly, the (UPSR)-axiom) in case, for every x 2 0, there exists a unique element y such that (1) y 2 0, and (2) x = y 2 ; we assume, in addition, that (3) Y E jx)" (but conditions (1) and (2) are already assumed to determine y uniquely). Every C*-algebra A satisfies the (UPSR)-axiom. {Proof: If xgA, x 2 0, there exists a unique Y E A such that y 2 0 and x = y 2 ; since x 2 0 as an element of the C*-algebra jx)", it follows from uniqueness that y~ (x}".) The kcy to the rest of the section is the following result:

Proposition 8. Let A be a *-ring with unity andproper involution, satisfying the (UPSR)-axiom. If e, f are projections such that e f - f o is insc>rtible,then e and j can he exchanged by a svmmetry. Of course the pair e, 1- f also satisfies the hypothesis of Proposition 8, as do the pairs I-e, J' and I -e, 1- f ; the statement of the conclusion is confined to the pair e, J' for simplicity. {Proposition 8 holds more generally with (UPSR) weakened to (SR), but with a considerably more complicated proof (Exercise 5).) To break up the rather long proof of Proposition 8, we separate out some of the earlier steps,

5 13.

The Parallelogram Law (P) and Related Matters

71

which are valid under a weaker hypothesis, in the form of an admittedly ugly lemma:

Lemma. Let A be a *-ring with unity andproper involution, satisfj,ing the (WSR)-axiom, and suppose e, f are projections such that ef' - f e is invertible in A. Define x = f e. Then

(1)

x* x

= ef

e

is invertible in eAe .

Let u be the inverse of e f e in eAe; thus, (2)

u ~ c A e , a*=a,

a ( e f e ) = ( e f e ) a = r ( t h a t i ~ ~, z f e = e f a = e ) .

Choose r E {x*x)" with x* x = r* r. Then (4)

r is invertible in eAe, with inverse. ur* = r* a,

Define v=xar*. Then

vv* = l'. Proof. ( 1 ) See the proof of Proposition 7. (2) The self-adjointness of u follows from that or e J e . (3) By the (WSR)-axiom, we may choose r E { x * x ) "= {efe)" such that e fe= r* r= rr*. Since etz je f P ) ' , it follows that re = e r ; a straightforward calculation then yields (re - r)* (re- r)= 0 , therefore re - r = 0 (the involution is assumed proper). Thus r = r e = e r, r E eAe. (4),( 5 ) Since r* r=rr* = e J e is invertible in eAe, so is r ; explicitly, the calculations e = ( e f e ) u= (rr*)u = r(r*u),

(8)

show that the inverse of r in eAe is r* a = ur*. Taking adjoints in the last equation, we havc ur= r a . (6) Setting v = x a r Y , we have c * ~ : = r u x * x a r * = r u [ ( e f e ) u ] r * =ruer*=(ra)r*=(ar)r*=a(efe)=e by ( 5 ) and (2). (7) v r = ( x a r * ) r = x [ a ( v * r ) ] = x [ a ( e f e ) ] = x e = x . (8) Writing g=vv*, it remains to show that q= f: At any rate, q is * (f'e)ar*E f'A), a projection [ij 2, Prop. 21 and f y =g (because ~ $ = x a r= thus g 5 . f . To show that f -g=O, it will suffice, by the invcrtibility of e f - f ' e , t o show that

Chapter 2. Comparability of Projections

72

in fact, it will be shown that e f ( f - y)= f e ( f -(1) =O. A straightforward computation yields g = f af, therefore by (2); thus e ( f -y)=0. On the one hand, this implies f e ( f - g ) = 0 ; on the other hand, since f -q < j ' we have also f ( f - y) =e( f ' - y) = 0. I Proof oJ' Propositiotl 8. With notation as in the lemma, we assume, in addition, that r 2 0. Similarly, let y = - (1 - f ) (I - e) (the minus sign is intentional) and consider y* y = (1 - e) (1 - f )(1 - e). Since

(.

(1-e)(1-,/')-(I-

f ) ( 1 - e ) = ~ , f- f e

is invertible, the lemma is again applicable, as follows. (1')

y*y

= (1 e

) (I f ) 1 e ) is invertible in (I -e)A(l - e ) .

If h is the inverse of (I - e) (I - f ) (I - e) in (I - e)A ( l

-

e), then

j " s 2 0 and y* =s2, we have Choosing s ~ ( ~ * y with s is invertible in (I - e) A(l - e), with inverse hs = s h (4') (recall that s* = s ; thus (5') is redundant). Defining w = y hs, we have (the minus sign in the definition of y gives no trouble)

(6') (7')

w*w= I - e ,

)' = WS,

Define u = v + w. Obviously u is unitary and ueu* = f ; thc proof will be concluded by showing that u is self-adjoint. Set t = r + s . From (4) and (47, it is clear that t is invertible in A (with tp' = u r + h s ) . Since ut = ~ l r + c s + w r + w s= x+O+O+y, and since x + y = f'e-(I - f ) ( l - e ) = r + j -1, we have u t = e + j -1. Thus, setting z = e + f - I ,

we have

(*) z = ut, where u and t are invertible and z* = z. Since t = r +s. where r 2 0 and s 2 0, we have t 2 0. Since, in addition, (*) yields

6 13

The Parallelogram I.aw (P) and Related Mattcrs

73

it follows from the (UPSR)-axiom that t is the unique positive square root of z2, and in particular t c (z2}"; but z~ (z2)', therefore tz = z t, that is, z t p l = t p l z . Citing (*), we see that u = z t p ' = t-' z is the product of commuting self-adjoints, therefore u* = u . I In a Rickart *-ring, a condition weaker than the invertibility of ef - f e is RP(ef - f e)= I, that is, position p (Proposition 6); still weaker is position p'. To arrive at the parallelogram law (P), we must show that projections in position p' are equivalent (Proposition 4); it would suffice to show that they can be exchanged by a symmetry. Thus, to establish the parallelogram law, it would suffice to prove the conclusion of Proposition 8 under the weaker hypothesis that e, J' are in position p'; this is done in the next group of results (but the proofs require added axioms on A). It is convenient to separate out the intermediate case of position p as a lemma: Lemma. Let A he u Buer *-ring suti.yfying the (EP)-ax ion^ und /he (UPSR)-uxiom. If e, f are prejections in position p, then c and f ' can he exchanged by a symmetry (in particulur, e f 1 - e 1 - f ) . Proof. We show that e and ,j' can be exchanged by a symmetry; it is then automatic that 1 - e -1 J; and the parenthetical assertion of the lemma follows from the observation that e, 1 - f are also in position p. Let x = e f - , f c ~ , z = x * x = -(qf -,f'e)', and write B= ( z ) ' . As noted in the proof of Proposition 7, B has center R'= jz)", and B contains and J' (hence also x). By hypothesis, RP(z) =RP(x) = 1 (Proposition 6); we shall reduce matters to the situation of Proposition 8 by constructing a central partition of 1 in B such that z is invertible in each direct summand. Lct (12,) be a maximal orthogonal family of nonzero projections in (z)" such that, for each 1 , zh, is invertible in h,B (the Zorn's lemma argument is launched by an application of the (EP)-axiom). We assert that sup h,= I (recall that suprema in B are unambiguous [jj 4, Prop. 71). Writing h=suph,, it is to be shown that I- h = O ; since RP(z)=1, it will suffice to show that z(1 - 12)=0, equivalently, x(1 l z ) = O . Assume to the contrary. Then, by the (EP)-axiom, there exists an element Y E ((1 - h)x* s(l - h))" = (z(1 - h))" c { z } " such that z(1- h ) y = k , k a nonzero projection. Obviously k c { z J " , k I - h, and z k is invertible in k B, contradicting maximality of the family (h,). We propose to apply Proposition 8 in each h,B; to this end, we note that the (UPSR)-axiom is satisfied by B (Exercise 2) and therefore by h, B = h, Bh, (Exercise 3). Since ieh,) ( f h , )- (f'h,) (phi) = xhl

- - -

-

(.

74

Chapter 2. Coinparability of Projections

is invertible in h, B (because (xlz,)( xh,)* = x x * h, = ( xh,)* ( xh,)= z h, is invertible in h, B), it follows from Proposition 8 that there exists a symmetry u, in h,B such that (*)

u,(eh,)u, = f h,.

It remains to join the u, into a symmetry u exchanging e and f'. {If A were an AW*-algebra, the C*-sum technique would do the trick; in a Baer *-ring, we must be more deft.) The strategy is to express the symmetry u, in terms of a projection g, of h,A (see the remarks following Definition 6), take the supremum g of the g,, and define u =2g - 1. Part of the conclusion of Proposition 8 is e h, h, - e h,; therefore 2 h, has an inverse a, in h, B [fj 11, Lemma 21, thus g, = a,(h,+ u,) is a projection in h,B , such that u, = 2g, - lz,. Define g = sup g,, u = 2g - 1. Since gh,=g, for all 1 [$ 11, Lemma I],it follows that

-

u h , = 2cqh,-h,

= 2g,-h, = u , ;

thus (*) yields ( u e u - f)h,=O for all suph,=I. I

1,

and u e u - f = 0

results from

The above proof actually yields information for an arbitrary pair of projections: Theorem 2. Let A be a Baer *-ring satisfying the (EP)-axiomand the (UPSR)-axiom.Ife, f is anypair ofprojections in A, there exists upvojection h, central in the subring B = { - (ef -f ~ ) ~ )such ' , that (1)e h and f lz are in position p in Bh (hence may be exchanged by a symmetry in B h ) , and (2) e(1 - h) and f ( I - h) commute. Explicitly, h = R P ( e f - j e ) .

Proof. With notation as in the proof of the lemma (but with the hypothesis R P ( x )= 1 suppressed), set h = sup h,; the argument given there shows that h=RP(x). On the one hand, x(1 -h)=O shows that e(1- h) and J'(1- h) commute. On the other hand, (eh)(f h ) - ( f h ) ( eh) =xlz=x has right projection h. therefore e h and f h are in position p in BIT (Proposition 6). 1 We now advance to position p':

Lemma. Notation as in Theorem 2. 11; in addition, e,f are in position p', then e(1 - h ) = f (I - h). Proof. Write k = 1 - h and set e" = e k, f " =f k ; we know from Theorem 2 that e" and f " commute. By hypothesis, e n ( l -j') = ( 1 - e ) n f = 0 ;

since k is central in B, it follows that

er'n(k-,f"') = ( k - e l ' ) n f " = 0 ,

5 13.

The Parallelogram Law (P) and Related Matters

75

that is, in view o f the commutativity o f e" and ,f", e U ( k - f " )= (k-e") f " ' = 0 . Thus e " = e U f " =f " .

I

Theorem 3. Let A be a Baer *-ring satisfying the (El')-axiom and the (UPSR)-axiom. If e,j' are projections in position p', then e and f can he exchanged by a symmetry 29 - 1, g a projection. Proof. W i t h notation as in the proof o f Theorem 2, set el=ph, eU=e(l-h), f l = f h , f " = f ( l -h); thus e=ef+e",

f=ff+

f".

By Theorem 2, e' and f' are in positionp in B h, and there exists a symmetry u' in B h such that u1e'u'=f ' ; by the lemma, en=f'". Then u=u' +(1-h) is a symmetry in B (hence in A) and it is straightforward t o check that ueu= f. A second look at the proof o f Theorem 2 (rather, its lemma) shows that u1=2g'-h for a suitable projection g', thus u=2g-1, where g=gt+(l -h). I Combining Theorem 3 with Proposition 4, we arrive at the climax o f the section (see also Exercise 7 ) :

Theorem 4. The parallelogram law ( P ) holds in any Baer *-ring satisfjing the (EP)-axiomand the (UPSR)-axiom. Theorems 3 and 4, combined with Proposition 5, yield an important decomposition theorem (see also Exercise 8):

Theorem 5. Let A be a Baer *-ring sutisfjiing the (EP)-axiom and the (UPSR)-axiom.If e,f i s any pair ofprojections in A, there exist orthogonal decompositions e=ef+e", f = f f + f "

-

such that e' j ' and e" f = e f " = 0. Explicitly, e' = LP(ef ), f' = RP(e f ), e" = e - e', f" =f -f '; e' and f ' are in position p', and can be exchungrd by a symmetry. Exercises 1A. In the Baer *-ring of all 2 x 2 matrices over the field of three elements [$I, Exes. 171, the parallelogram law (P) fails; so docs the (SR)-axiom; so does the (EP)-axiom.

2A. Let A be a *-ring, B a *-subring such that B = B . If A satisfies the (WSR)axiom [(SR)-axiom, (PSR)-axiom, (UPSR)-axiom] then so does B. 3A. Let A be a *-ring with proper involution, and let e be a projection in A . If A satisfies the (WSR)-axiom [(SR)-axiom, (PSR)-axiom, (UPSR)-axiom] then so does eAe.

76

Chapter 2. ('omparability of Projectio~ls

4A. If A is a weakly Rickart *-ring satisfying the (WSR)-axiom, and if e, f are projections such that ef- fe is invertible in (e u . f ) A(r u j'), then e-f - e u f - e - e u . f - f. SC. Let A be a *-ring with unity and proper involution, satisfying the (SR)axiom. If e,f are projections such that ef- j'e is invertible, then e and f can be exchanged by a symmetry. (This generalizes Proposition 8.) 6C. Let A be a Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom. If e, f are projections in position p', then e and,fcan be exchanged by a symmetry. (This generalizes Theorem 3.) 7C. The parallelogram law (P) holds in every Baer *-ring satisfying the (EP)axiom and the (SR)-axiom. (This generalizes Theorem 4.) 8C. Let A be a Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom. If f is any pair of projections in A, there exist orthogonal decompositions cJ= e' + c", J'=fl +f " with e', f ' in position p' and e" f = ef " = 0; in particular, e' j", indeed, e' and f ' can be exchanged by a symmetry. (This generalizes Theorem 5.)

-

c.,

9A. The following conditions on a *-ring are equivalent: (a) the involution is proper, and the relations x 2 0, y 2 0, x + y = O imply x = y =O; (b) implies x, =. ..= x,= 0 (n arb~trary).

x, *.y,=O 1

IOA. In a *-ring satisfying the conditions of Exercise 9, the (PSR)-axiom and the (UPSR)-axiom are equivalent. IIA. Let A be a *-ring with proper involution, satisfying thc following strong square-root uxiom (SSR): If x t A, x 2 0 , then there exists y ~ ( x ) "with y*=y and x=y2. (The (SR)-axiom provides such a y only for positives x of the special form x = t* t.) Assume, in addition, that (1) A has a central element i such that i2 = - 1 and i* = - i, and XI - x,= 0 (cf. Exercise

(2) 2 x =0 implies x = 0. Then 9).

1x,*x, =0

impl~es

1

12A. Let A be a *-ring with proper involution, satisfying the conditions (I), (2) of Exercise 11. In such a *-ring, the (PSR)-axiom and the (UPSR)-axiom are equivalent. 13A. If A is a *-ring satisfying the (WEP)-axiom and the (SK)-axiom, then A satisfies the (EP)-axiom.

-

14A. Let A be a Bacr *-ring, let (e,),,, and (f,),,, be equipotent families of orthogonal projections such that e, f , for all L E1, and let e = sup e,, J'=supj',. We know that if ef'=O then e-f [$11, Th. I]. If A satisfies the parallclogram law (P), then the weaker condition e n f = 0 also implies r -,f. ISA. If e,,fare projections in a *-ring A, such that e-f can be exchanged by a symmetry in (e+f') A(r+f).

and ef=O, then e and f

16A. Theorems 2-5 hold in any Rickart C*-algebra; in particular, any pair of projections in position p' can be exchanged by a symmetry. 17A. Suppose A is a Rickart *-ring in which every pair of'projections in position p' can be exchanged by a symmetry. If e, f is any pair of projections in A, there exists a symmetry u such that u(e f ) u = fe. 18A. In an arbitrary Baer *-ring, projections in positionp necd not be equivalent.

# 14. Generalized C:omparability

77

-

19C. In a von Neumann algebra A, projections r , f are in position p' (relative to A) if and only if e j' relative to the von Neumann algebra generated by e and 1.

5 14.

Generalized Comparability

Projections e,,fin a *-ring A are said to be comparable if either e 5 j or f 5 e. Rings in which any two projections are comparable are of interest in the same way that simply ordered sets are interesting examples of partially ordered sets [cf. 8 12, Exer. I]. In general, the concept of comparability is of limited use. (For example, if A contains a central projection h different from 0 and 1,and if e,f are nonzero projections such that e I h and f I 1- h, then e and f cannot be comparable.) The pertinent concept in general *-rings is as follows:

Definition 1. Projections e,f i n a *-ring A are said to be generalized comparable if there exists a central projection h such that (When A has no unity element, the use of 1 is formal and the condition need not be symmetric in e and f:) We say that A has generalized comparability (briefly, A has GC) if every pair of projections is generalized comparable. Generalized comparability may be reformulated in terms of the following concept, which generalizes, and is consistent with, an earlier definition [§ 6, Def. 21:

Definition 2. Projections e,f in a *-ring A are said to be very orthogonal if there exists a central projection lz such that h e = e and 12 f = 0. (That is, e 5 h and f I I-h, where 1 is used formally when A has no unity element--in which case, the relation need not be symmetric in e and J:) If e,fare projections in a Baer *-ring A, then the following conditions are equivalent: (a) e,f a r e very orthogonal; (b) C(e)C(j')= 0; (c) e Af = 0 [46, Cor. 1 of Prop. 31. The relevance of very orthogonality to generalized comparability is as follows:

Proposition 1. If e,j'areprojection.~in a *-ring, the J~llowingconditions are equivalent: (a) e,fare generalized comparable; (b) there exist orthogonal decompositions e= e , + e,, f ' =f',+f, with el j', and f , e, very orthogonal.

-

Chapter 2. Comparability of Projections

78

Proof. (a) implies (b): Choose h as in Definition 1, say he-f;
(1-1z)f-ey<(l-h)e.

Writing e; =he and f ; ' = ( 1 - h)J; we have ei-I;,

(*)

e;-f;'.

Obviously el e';= O and f ; f ;' = 0 ; setting el=c>',+ e ; ,

f',=f; +f;',

it follows from (*) that el - f , [$I, Prop. 81. Since el < r and f ' ,I f ; we may define e, = e - e l ,f , =f -f , ; it is routine to check that /ze, =O and h,f2=f,. (b) implies (a): Assuming there exists such a decomposition, let I1 be a central projection such that hf;=.fz and he,=O. Then he= he, h f l 5 hf [$ I , Prop. 71, thus he 5 hj', and similarly ( I - h)j' ( Ih e . I

-

If e,j' are generalized comparable, but are not very orthogonal, then Proposition 1 shows that e, {have nonzero subprojections e,,j', such that el f , ; this is a phenomenon worth formalizing:

-

Definition 3. Projections e,f in a *-ring A are said to be partially comparable if there exist nonzero subprojections e, < e, J',, < f such that eo -,A. We say that A has partial comparability (briefly, A has PC) if eAf # 0 implies e,j'are partially comparable. GC is stronger than PC:

Proposition 2. I f A is a *-ring with GC, then A has PC. Proof. Assuming e , f are projections that are not partially comparable, it is to be shown that eAf=O. Write e = r , +e,, f = f , +f; as in Proposition 1. By the hypothesis on e,J; necessarily r l = , f ,=0, thus f ; e are very orthogonal; if h is a central projection with hf =J' and he=O, then eAf=eAIzf=elzA,f=O. I PC is implied by axioms of 'existence of projections' type; for instance:

Proposition 3. If A is a *-ring sutisjying the (VWEP)-axiom, then A has PC. Proof. Suppose e,f are projections such that eAf # 0, equivalently, f A e f 0. Let x ~ f A e ,x f 0. By hypothesis, there exists an element Y E {x* x)' with b*y ) ( x * x )= e,, e, a nonzero projection 157, Def. 31, thus e, =y*(x*x)y=(xy)*(xy). Writing w = x y , we have w* w=e,; since the involution of A is proper [$2, Exer. 61, w is a partial isometry [$2, Prop. 21. Set f o = w w*. Since x ~ . f A e the , formula e, =(y*y ) ( x * x ) shows that e, < e, and ,fo = w w* = ( x y )w* shows that fb I j'. I

5 14. Generalized

Comparability

79

In Baer *-rings, generalized comparability is intimately related to additivity of equivalence [ijll, Def. I]; in fact, a Baer *-ring has GC if and only if it has PC and equivalence is additive [$20, Th. 21. The "only if" part appears to be fairly difficult--the proof we give in Section 20 involves most of the structure theory discussed in Part 2. The "if' part is easy:

Proposition 4. 1fA is a Baer *-ring with PC and if equivalence in A is additive, then A has GC. Proof. Let e,fbe any pair of projections in A. If eAf=O then e,f ' arc vcry orthogonal and the gencralizcd comparability of e and f is trivial.Assuming eAf# 0, let (e,),,,, (f,),,, be a maximal pair of orthogonal families of nonzero projections such that e, I e, f ,sf and e, J; for all 161 (an application of PC starts the Zorn's lemma argument). Set e' =supe,, f l = s u pf,, el'=e-e', f " = f -f'. On the one hand, e' -f" by the assumed additivity of equivalence. On the other hand, e" Af"' = 0 (if not, an application of PC would contradict maximality), therefore e",f" are very orthogonal. In view of Proposition 1, the decompositions e = e' + e", f = f ' +f " show that e,f are generalized comparable. I

-

It is a corollary that every von Neumann algebra A has GC; for, it is easy to see that partial isometries in A are addable (e. g., they can be summed in the strong operator topology), and the validity of the (EP)-axiom [ij 7, Cor. of Prop. 31 ensures, via Proposition 3, that A has PC. For AW*-algebras, essentially the same argument may be employed (except that the proof of addability is hardcr-see Section 20), but an alternative proof will shortly be given. Proposition 4, and the fact that equivalence is orthogonally additive in any Baer *-ring [ij 11, Th. I], naturally suggest the following definition:

Definition 4. We say that a *-ring has orthogonal GC if every pair of orthogonal projections is generalized comparable. This condition is automatically fulfilled in a Baer *-ring with PC:

Proposition 5. If A is a Burr *-ring with PC, then A has orthogonal GC. Proof. Let e, f be projections with ej'= 0. The proofs proceeds as for Proposition 4, except that el- f ' results from a theorem [jj 11, Th. I]rather than an assumption. I In the presence of the parallelogram law, GC and orthogonal G C are equivalent hypotheses:

Chapter 2. Comparability of Projections

80

Proposition 6. l f A is a Rickart *-ring with orthogonal GC, and A satisfies the parallelogram law (P), then A has GC.

iJ'

Pvoqf. Let e, f be any pair of projections in A. By the parallelogram law, write e=e1+e", f'=f"+f" with el- f' and e" f = ef " = O [$ 13, Prop. 51. Since, by hypothesis, the orthogonal projections e", j" are generalized comparable, Proposition 1 yields decompositions ?ti = e 1+e2, f"=fl+f~ with el

- J;

and e,, ,f2 very orthogonal. Then

-

where e' + e, f ' + f ; and e2, fz are very orthogonal, therefore e, f are generalized comparable by Proposition 1. I In a Baer *-ring satisfying the parallelogram law, the concepts PC, G C and orthogonal G C merge:

Proposition 7. If' A is a Baer *-ring satisfying the parallelogram law (P), then the following conditions on A are equivalent: (a) A has PC; (b) A has orthogonal GC; (c) A has GC. Proqf. (a) implies (b) by Proposition 5; in the presence of (P), (b) implies (c) by Proposition 6; and (c) implies (a) by Proposition 2. 1 Corollary 1. Every AW*-al~gebrahas GC. Proof: An AW*-algebra A satisfies the parallelogram law (P) [$ 13, Th. I]; since A satisfies the (EP)-axiom [$ 7, Cor. of Prop. 31, and therefore has PC (Proposition 3), it follows from Proposition 7 that A hasGC. I

-

Corollary 2. I f A is a Baer *-ring such that LP(x) RP(x) for all x in A , then A has GC and sat is fie.^ the parallelogram law ( P ) . Prod. Since A satisfies (P) [$ 13, Prop. 21, by Proposition 7 it suffices to show that A has PC. Suppose e , f are projections such that eAf #O, say x = e a f #0; then e,=LP(x), f,=RP(x) are nonzero subprojections of e , f such that e, f,. I

-

The parallelogram law is not the most natural of hypotheses. Some ways of achieving it were shown in Section 13; an application (see also Exercise 5):

Theorem 1. If A is a Baer *-ring satisfying the (EP)-axiom and the (UPSR)-axiom. then A has GC and satisfies the parallelogram law (P).

5 14.

Generalized Comparability

81

Proqf. A satisfies ( P ) [§ 13, Th. 41 and has PC (Proposition 3), therefore A has GC by Proposition 7 . 1 Incidentally, Theorem 1 provides a second proof of the AW* case (Corollary 1 of Proposition 7). We close the section with two items for later application. The first is for application in Section 17 [§ 17, Th. 21:

Proposition 8. Let A be a Rickart *-ring with GC, satisfying the parallelogram law (P). If e , f is any pcrir of' projections in A, there exists a centrul projection h such that

Proef. Apply GC to the pair e n ( I - f ) , (1 -e) n,f: there cxists a central projection h such that (1

h [ e n ( I - f ) ] 5 h [ ( 1- e ) n f ] , (1 - h) [(I - e) n f ]

(2)

5 (1-h) [ e n (1-.f)].

It follows from the parallelogram law (see [$ 13, Prop. I ] ) that

e-en(1- f ) - f-(1-e)n

f'

and (replacing e , f by 1 - e, 1 - f )

( I -e)-(I - e ) n f - ( I - f ) - e n ( 1

-

j'),

therefore

Adding (1) and (3) yields h e 5 hf ; while (2) and (4) yield (1 - h) ( I - e ) 5 - 1 f 1 The final proposition is for application in [$ 18, Prop. 51:

Proposition 9. Let A be a Baer *-ring with PC, and suppose (e,),,, is u family of projections in A wlth the jollowiny property. for every rzonzero central projection h, the set of indices is infinite; in other words, there exists no direct summand qf' A (other than 0) on which all but finitely many of' the el vanish. Then, given any positi~~einteger n, there exisl n dislinct indices i,, . . ., in, and nonzero projections g, I e," (v = 1,. ..,n), such that

Chapter 2. Con~parabilityof Projections

82

Proc?f'. The proof is by induction on n. The case n = l is trivial: the set { I : 1 el # 0) is infinite, and any of its members will serve as I , , with g , = e l , . Assume inductively that all is well with n - I , and consider n. By and nonzero projecassumption, there exist distinct indices I,, ..., i n tions ,f,,...,f,-, such that f ,5 el,, (v = I , . . .,n - I ) and f , ... f,._ Since C ( f ; ) #O, it is clear from the hypothesis that there exists an index I , distinct from L , , ..., I , _ , such that C ( J ; ) e l n# 0. Then C ( J ; ) C ( e l n )# 0, thus f l A elm# 0 [$ 6, Cor. 1 of Prop. 31; citing PC, there exist nonzero subprojections g , 5 j; and g, < eln such that g1 g,. For v = 2 , ...,n - I,the equivalencc f ; f ; transforms g, into ( v = 2 ,..., n - 1 ) . a subprojection g,< f , with g l - g , . Thus g , - g l - g , {The proof shows that the indices for n may be obtained by augmenting the indices for n - I ; but as n increases, the projections y, will in general shrink.] I

- -

-

,.

-

Exercises 1A. A Baer *-ring with orthogonal GC, but without PC (hence without GC): Exer. 171. the ring of all 2 x 2 matrices over the field of three elements [$I,

2B. A Baer *-ring A has G C if and only if (i) A has PC, and (ii) equivalence in A is additive. 3A. In a Baer *-ring with finitely many elements, PC and G C are equivalent.

4B. In a properly infinite Baer *-ring [§ 15, Def. 31, PC and G C are equivalent. 5C. If A is a Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom, then A has G C and satisfies the parallelogram law (P). (This gcneralizcs Theorem 1.) 6A. (i) If A is a *-ring with G C and if y is any projection in A, then gAg has GC. (ii) If A is a Baer *-ring, if g is a projection in A, and if e,f are projections in gAg that are generalized comparable in g A g , then e,f are generalized comparable in A.

7A. Ife,f are partially comparable projections in aBaer *-ring, then C(e)C ( , f )#O. 8A. If A is a Rickart *-ring satisfying the parallelogram law (P), and if e.j are projections in A such that r f # 0 , then e,f are partially comparable. 9A. Let A be a Baer *-ring satisfying the parallelogram law (P). If A satisfies any of the following conditions, then A has GC: (1) For every projection r, C(r)= sup {e' : r' e) [cf. 4 6, Exer. 71. (2) If e,J' are projections such that PA,\'# 0. then there exists a unitary u such that r u f # 0. (3) If e,f' are projections such that eAf #O, then there exists a projection g such that e(2g)j'# 0.

-

10A. The following conditions on a *-ring A are equivalent: (a) A has GC; (b) A has orthogonal G C and, for every pair of projections e,f , there exist orthogonal decompositions e = e'+ en, f ' = f l + , f " with e'- J" and e"f" = j'" en.

-

11A. Let A be a Rickart *-ring in which every sequence of orthogonal projections write [el = { J : j e} and define [ r ] 5 [ J ] has a supremum. As in [ij 12, Exer. I],

5 14. Generalized Comparability

83

iff e 5 f . If A has GC then the set of equivalence classes is a lattice with respect to this ordering. 12A. Let A be a Baer *-ring satisfying the (EP)-axiom and the (SK)-axiom (or let A be a Rickart C*-algebra with GC). If e,f is any pair of projections in A, therc exist orthogonal decompositions e = e , +e,, f = f , +j', such that c , and r , arc exchangeable by a symmetry and e,, fz are very orthogonal. 13B. If A is a Baer *-ring satisfying the (WEP)-axiom, then thc following conditions are equivalent: (a) A has GC; (b) LP(x)-RP(x) for all x t A ; (c) A satisfies the parallelogram law (P). 14B. Let A be a Baer *-ring with GC, and let e,f be any pair of projections in A. Either (1) f 5 e, or (2) there exists a central projection h with the following property: for a central projection k, k e 5 k f iff k < h. In case (2), such a projection h is unique. h21-C(e), and (1-h)f5(1 -h)e. 15A. If A is a Baer *-ring with PC, the following conditions on a pair of projections e,f'imply one another: (a) C(e) _< C ( f ) ; (h)e = sup (., with (e,)an orthogonal family of projections such that e, 5 f for all 1 ; (c) e = sup e, with (c,) a family of projections such that e, 5 j' for all 1 . 16A. Let A be a Rickart *-ring with GC, let n be a positive integcr, and suppose that the n x n matrix ring A, is a Rickart *-ring satisfying the parallelogram law (1'). Then A , has GC.

17C. Let A be a Rickart *-ring with orthogonal G C (e.g., let A be a Baer *-ring with PC) and let e be a projection in A. The following conditions on cJ are equivalent: (a) e is central in A ; (b) e commutes with every projection in A (that is, e is central in the reduced ring A"); (c) e has a unique complement. 18A. (i) If A is a Baer *-ring with PC, then a projection in A is central iff it commutes with every projection of A (thus a projection is central in A in it is central in the reduced ring A" [$3, Exer. 181). (ii) The converse of (i) is false: there exists a Bacr *-ring A such that A' = A but A does not have PC. 19A. Let A be a *-ring with unity. A partial isometry u in A is said to be c~.ut/.rmol if the projections I-u*u and 1 u u * arc very orthogonal in the sense of Definition 2. {The terminology is motivated by the fact that if A is an A w*-algebra, then the closed unit ball of A is a convex sct whose extremal points are precisely the extremal partial isometrics.) For example, if u is an isometry (u*u= 1) or a co-isometry (uu* = I)then u is an extremal partial isometry; when A is factorial, there are no others 196, Def. 31. If A has GC and if w is any partial isometry in A, then there exists an extremal partial isometry u that 'extends' w, in the sense that u(w* w)= MI. 20D. Problem: If A is a Baer *-ring with PC, does it follow that A has GC? 21D. Problem: If A is a Baer *-ring satisfying the parallelogram law (P), does it follow that A has PC? 22D. Problem: If A is a Baer *-ring with PC, does it follow that A satisfies thc parallelogram law (P)?

Part 2: Structure Theory

Chapter 3

Structure Theory of Baer *-Rings Part 1 of the book dealt with more or less general Baer *-rings, liberally seasoned with such axioms of a general nature as are needed to make the arguments work. From here on, most arguments entail qualitative distinctions between Baer *-rings; a particular argument will generally apply only to certain kinds of Baer *-rings (with or without extra axioms-usually with). (Some analogous qualitative considerations in group theory: commutativity, finiteness, solvability, decomposability, simplicity, etc.] Such qualitative distinctions are the basis of structure theory. By structure theory we mean the description of general Baer *-rings in terms of simpler ones. (The best-loved model of a successful structure theory describes finitely generated abelian groups in terms of cyclic ones.) When we say that a ring-or a class of rings-is simpler, we mean, vaguely, that less can happen in it. An inventory of the things that can happen in a Baer *-ring will lead off with annihilation, commutativity, projection lattice operations, and cquivalencc of projections; for structure theory, the most important happenings are commutativity and equivalence. {These are, in a sense, opposite sides of the same coin; equivalence is interesting only when there exist partial isometries w for which w* w and w w* are different.) Structure theory comes in two grades, fine and coarse; in both cases the center of the ring, aptly, plays a central role. In fine structure theory, we seek to describe general Baer *-rings in terms of factorial ones (i. e., Baer *-rings in which 0 and 1 are the only central projections), accepting whatever equivalence behavior the factors may exhibit. In the coarse structure theory, we accept general centers (i. e., we do not insist on factors) but seek direct sum decompositions into summands in which equivalence behavior is limited is1 various ways. Followi~lgin the wake of a colllplete structure theory is a classification theory, i. e., a full listing of the various kinds of objects that can occur, with a specification of when two objects are isomorphic. By these standards, the structure theory of von Neumann algebras, despite intensive cultivation for ncarly four decades, remains incomplete even for S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

88

Chapter 3. Structure Theory of Baer *-Rings

separable Hilbert spaces; for Baer *-rings, there is barely a beginning. Chapter 7 is devoted to the fine structure of one special class of Baer *-rings (namely, the Baer *-rings in which u* u = 1 implies u u* = I, and LP(x)-RP(x) for all x); this isn't much, but it's all there is at the present state of the subject. The present chapter is devoted to the coarse structure theory of Baer *-rings. As far as it goes, this theory goes remarkably smoothly; the general Baer *-ring theory is not essentially harder than the special case of von Neumann algebras. (Remauks. However, we should not give the impression that practically all coarse structure theory carries over from von Neumann algebras to Baer *-rings. We cite here three examples to the contrary. (1) The coarse structure theory of von Neumann algebras, when applied simultaneously to an algebra and its commutant, leads to spatial isomorphism invariants for the algebra [cf. 23, Ch. 111, 9 6, No. 4, Prop. 101; the lack of an appropriate notion of 'spatial' for Baer *-rings (or even for A W*algebras) is a natural boundary for the theory. (2) The coarse structure theory of von Neumann algebras of 'Type 1' leads to a complete system of *-isomorphism invariants for such an algebra, the invariants consisting in a set of cardinal numbers together with a corresponding set of commutative von Neumann algebras (cf. [23, Ch. 111, 9 3, Prop. 21, [8l, Part 11, Th. 101); for AW*-algebras of Type 1 there is a partial theory of this sort ([48], [49]), fully satisfactory in the 'finite' case, but for 'infinite' algebras the theory bogs down in unresolved questions of cardinal uniqueness [cf. 48, Th. 41; for Baer *-rings, only a few wisps of such a theory are yet in hand (cf. Section 18). (3) A final example concerns the problem of describing the automorphisms and derivations of a ring; this is a large topic in von Neumann algebras [cf. 23, Ch. 111, $ 91, a small topic in A W*-algebras [cf. 491, and a non-topic in Baer *-rings. For von Neumann algebras, there is a highly developed finc structure theory, called reduction theory [cf. 23, Ch. IT], that is applicable to algebras of arbitrary 'type'; there are limitations to the theory (the reduction is uncanonical and is largely limited to algebras acting on separable Hilbert spaces), but nothing like it exists for general AW*algebras, let alone Baer *-rings.)

5 15.

Decomposition into Types

Throughout this section, A is a Baer *-ring; no additional axioms need be imposed on A . {In fact, there exists an involution-free version: with projections replaced by idempotents, and lattice operations by deft

# 15. Decomposition into Types

89

strategies, the results were originally proved by I. Kaplansky for arbitrary Baer rings [$4, Exer. 41 (see 1541 for an exposition of this theory).) All of the definitions, and many of the propositions (but none of the theorems), can be formulated in an arbitrary *-ring with unity; it is when suprema must be taken that the Baer *-ring condition is required. (In this connection, see the remarks at the end of the section.) The coarse structure theory is cast in terms of the concepts of 'finite projection' and 'abelian projection'.

Definition 1. A is said to be finite if x* x= I implies xx* = I. We agree to regard the ring {O} as finite. If A is not finite, it is called infinite. A projection e c A is said to be ,finite (relative to A) if the *-ring eAe is finite in the foregoing sense. (In particular, 0 is a finite projection.) So to speak, A is finite iff every isometry in A is unitary. Another formulation is that A is finitc iff e --I implies e= I;the key point is that if x * x = I,then ( x x * ) ( x x * ) = x ( x * x ) x *= x x * shows that e = x x * is a projection with e I.

-

Proposition 1. Let e, f be projections with f f'e. Tlzen f is finite relative to A if it is finite relative to eAe. I Proof. ,fAf = f(eAe)f . A projection is finite iff it cannot be 'deformed' into a proper subprojection of itself via a partial isometry in the ring:

Proposition 2. A projection e is ,finite i f e

-f

f'e itnplies f = e.

-

Proof. The projections fie are precisely the projections of eAe. The condition e j'f' e means that there exists a partial isometry w such that w* w= e and w w* = / 5 e [$ I,Prop. 61; such an element w satisfies fw= w= we [$ 1, Prop. 51, therefore w ~ e A e ,thus the equivalence e f is implemented in eAe. The proposition now follows at once from Definition 1. I

-

All projections dominated by a finite projection are finite:

Proposition 3. I f ' e is a fcnite projection in A and is al.vo finite.

-

if'

f

5 e, then

f'

Proof. Say f f ' s e. Since fAf is +-isomorphic to f'Af" [$ 1, Prop. 91, it is no loss of generality to suppose that , f < e. Assuming f g 5 f ; it is to be shown that g = f Let u be a partial isometry such that v* v = J; v v* = g < f: Setting w = v +(e - f ) , we have w*w=e and w w * = g + ( e - f ' ) = e - ( f - g ) < e ; thus e - e - ( f - g ) < e , therefore e - ( f - g) = e (Proposition 2), that is, f - y = 0. I

-

If A is finite, we may put c.= I in Proposition 3:

90

Chapter 3. Structure Theory of Baer *-Rings

Corollary. If' A is finite, then every projection in A is finite. We now parallel the foregoing with 'abelian' in place of 'finite':

Definition 2. A is said to be abelian if every projection in A is central. A projection e e A is said to be abelian (relative to A) if the *-ring eAe is abelian in the foregoing sense. An abelian ring nced not be commutative, but for AW*-algebras thcrc is no ambiguity:

Examples. I . Every divis~on ring with involution is trivially an abelian Bacr *-ring. 2. An AW*-algebra is abelian if and only if it is commutative. (Proof: An AW*-algebra is the closed linear span of its projections [cf. 8 8, Prop. I]. Sce also Exercise 3.)

Tf Baer +-rings are approached through the more general Raer rings (as in [52], [54]), it is appropriate to define a Baer *-ring to be abelian if all of its idempotents are central [54, p. 101 or if all of its idempotents commute with each other [52, p. 5); these definitions are equivalent to Definition 2 (see Exercise 2). Evcry abclian projection is finite; this follows from the fact that in an abclian ring, cquivalcncc collapses to equality:

-

Proposition 4. (i) In an ahelian ring, e j inzplie~ e = f . (ii) Eoery ahelian rirzg L.\ finite. (iii) Eaery ahelian projrct~on is finite.

-

Proof. It is clearly sufficient to prove (i). Suppose e f in an abelian ring, and let w be a partial isometry such that M)* w = c , MJ w* = f . Since f is central, w j ' = ,f'cv = w, therefore e 5 j' [$ 1, Prop. 51. Similarly f j e. I

Paralleling Proposition 1, we have (with identical proof):

Proposition 5. I,et e , f he projections with f ' < r . Then f is uheli~m relutice to A iff it is ahcliun rclrrtive to PAC. Paralleling Proposition 2, abelianness may be characterized as follows:

Proposition 6. The following conditions on a projection e in A are equioulent: (a) P is a h ~ l i u n ; (b) f ' < P itnplies f = r C(,f'). (c) f ' 5 e implies f ' = e h for. sorne central prqjection h in A . Proqf'. Immediate from [$ 6, Prop. 41 and Definition 2. Paralleling Proposition 3:

1

4 15.

Decomposition into Types

91

Proposition 7. If e is an ahelian projection in A and if f 5 e, then f is also ahelian. Proof: We suppose, as in the proof of Proposition 3, that f < e ; then f A f ' c eAe. If q 5 . f then g is central in eAr, therefore also in f A f . I The structure theorems depend on exhaustion arguments whose essence is the following proposition, to the effect that finiteness and abelianness are cumulative, provided that thc projections being combined are very orthogonal:

Proposition 8. If' (e,),,, is a very orthogonal famil~lof finite [(ihelian] projections and if' e = sup e,, then e is also finilc /ahclian]. Proof. Write h, = C(e,), I1 = sup h,. By hypothesis, the h, arc orthogonal, thercforc Iz,e= el [$ 11, Lemma I].Moreover, h = C(e) rtj 6, Prop. 1, (iv)]. Suppose the el are finite. Assuming e - f'5 c., it is to be shown that f = e. For each 1 , h,c h, f < h,e, that is, e, h, f 5 e l ; since r , is finite, h, f = e, = h,e, thus (e- ,f)lz,= 0 for all 1 , therefore (e- f ) h = 0 . But e - f ~ e < C ( e ) = h , thus e - f = ( e - , f ) l 1 = 0 . Now suppose the e, are abelian. Assuming j's e, it will suffice to show that ,f = e C ( f ) (Proposition 6). For each I, 11,f 5 h,e=e,; sincc e, is abelian, h, f = c,C(h,f ) = e , h , C ( j )= h , e C ( f ) ,

-

-

thus (eC(f ) - f)h,=O for all 1 ; therefore (eC(f ) - f ) h = O , since h= C(e), this yields e C (f ) - f =O. I

and,

The direct summands in the coarse structure theory are chosen so as to satisfy one or several of the following conditions:

Definition 3. (I a) Repeating Definition 1, A is said to be finite if x* x= I implies xx* = I. ( I b) A is said to be properly infinite if the only finite central projection is 0 ; in other words, A has no finite direct summand other than { O ) [$ 3, Exer. 41. We agree to regard the ring 10) as properly infinite (thus ( 0 ) is the only ring that is both finite and properly infinite). (2a) Repeating Definition 2, A is said to be abelian if all of its projections are central. (2 b) We say that A is properly nonabeliun if the only abelian central projection is 0 (in other words, thc only abelian direct summand of A is (01.). We agree to regard the ring ( 0 ) as properly nonabelian (thus { O ) is the only ring that is both abelian and properly nonabelian). (3 a) A is said to be semifinite if it has a faithful finite projection, that is, a finite projection c such that C(e)=I.

92

Chapter 3. Structure Theory of Baer *-Rings

(3 b) A is said to be purely infinite if it contains no finite projection other than 0. Abusing the notation slightly, we agree to regard the ring (0) as both semifinite and purely infinite; no other ring can be both. (4a) A is said to be discrete if it has a faithful abelian projection, that is, an abelian projection e such that C(e)= I. (4b) A is said to be continuous if it contains no abelian projection other than 0. We agree to regard the ring jO) as both discrete and continuous; no other ring can be both. A central projection h in A is said to be finite [properly infinite, etc.] if the direct summand h A is finite [properly infinite, etc.].

Remarks. 1. We follow here the terminology of J. Dixmier [19], except for the term 'properly nonabelian', which is ad hoc (but rounds things out nicely). Some authors use the term 'purely infinite' for the condition (Ib), and 'Type 111' for the condition (3b); the 'type' terminology is explaincd latcr in the section. 2. Since every abelian projectioii is finite, the following implications are immediate from the definitions: properly infinite + properly nonabelian; discrete + semifinite; purely infinite 3 continuous. Obviously, abelian + discrete, and finite 3 semifinite. 3. If a Baer *-factor contains a nonzero finite [abclian] projection. then it is semifinite [discrete]. 4. Every finite-dimensional von Neumann algebra is finite [cf. 5 17, Prop. I].If 2 is an infinite-dimensional Hilbert space, then 9 ( X )is properly infinite and discrete (a projection with one-dimensional range is abelian). It is harder to produce examples of (i) rings that are semifinite but not finite or discrete, (ii) purely infinite rings, and (iii) continuous rings that are not purely infinitc (see 1231). Each of the eight classes of rings described in Definition 3 is 'pure' i11 the following sense : Proposition 9. I f A is ,fznite [properly infinite, etc. u.s in Dc:finnition 31 and 11 is u central prqjection in A, then h A is also tinite Iproperljl infinite, etc.1. Proof. For 'finite' and 'abelian' see Propositions 3 and 7. The assertions for 'properly infinite', 'properly nonabelian', 'purely infinite' and 'continuous' follow at once from the fact that the central projections of h A are central in A (cf. [$ 6, Exer. 4 or Prop. 41). Suppose A is semifinite [discrete], and let e be a faithful finite [abelian] projection in A; then h e is finite [abelian] and C(he )= h C(e)= h shows I that h e is faithful in hA, thus h A is semifinite [discrete].

The coarse structure theorems now follow easily from Proposition 8 and the definitions:

# 15. Decomposition into Types

93

Theorem 1. If A is any Baer *-ring, there exist unique central pro,jections h,, h,, h,, h, such that (1) h, A is finite and (1 - h,) A is properly infinite; (2) h, A is abelian and (I - h,) A is properly nonabeliun; (3) h, A is semifinite and (1 - 11,) A is purely infinite; (4) h, A is discrete and (1 - h,) A is continuous. A central projection k is ,finite lf k I h, , and properly infinite iff k < I - h, ; abelian ifJ k 5 h,, and properly nonubelian if k 5 1 - h,; sem$nite if k I h,, and purely infinite iff k < I - h,; discrete iffk I h,, and continuous iff' k I I- h,. Proof: (1) Let (h,) be a maximal orthogonal family of nonzero, finite central projections, and set h = sup h, (if there are no such projections, set h = 0). Then hA is finite by Proposition 8, and (I - h) A is properly infinite by maximality. This proves the existence of a central projection h, satisfying (1). Let k be a central projection. If k I h, then k is finite (Proposition 9). Conversely, if k is finite then k(l -h,) is both finite and properly infinite (Proposition 9), therefore k(1- h,) =0, that is, k < h,. Similarly, - h,. In particular, it is clear that h, is k is properly infinite iff k
Definition 4. A is said to be of Type 1 if it is discrete; Type I1 if it is continuous and semifinite; and Type I I 1 if it is purely infinite. (Thus, A is of Type I if it has a faithful abelian projection; Type I1 if it has a faithful finite projection, but no abelian projections other than 0; and Type TI1 if it has no finite projections other than 0.) We allow the ring (0) to be of all three types; no other ring can be of more than one type. A central projection h in A is said to be of Type v (v=I, 11, 111) if the direct summand hA is of Type v.

94

Chapter 3. Structure Theory of Haer *-Kings

Every Baer *-ring may be decomposed uniquely according to type:

Theorem 2. If A is any Baer *-ring, there exist unique orthogonal central projections h,, h,,, h,,, such that Iz,A is of p p e v ( v = I, 11, 111) and h,+h,,+h,,,=l. A central projection k is of Type v iff k kl 12,. In the notation of Theorem I , h, = h,, lzl,= h,(l - h,), h,,, = 2 - 12,. Proof. Define h, (v = I, 11, 111) by the indicated formulas. Clearly h, is of Type v (for v = I[, cf. Proposition 9). Since h, I h, (cf. Remark 2 following Definition 3), we have h,, = h, - h,, thus the 11, are orthogonal with sum 1. Let k be a central projection. If k is of Type v for v = I or 111, then k 5 h, or k 5 h,,,, respectively (Theorem 1). If k is of Type 11, that is, if k is semifinite and continuous, then k 5 h, and k < I - h, by Theorem 1, therefore k I h,(l - h,) = h,,. Conversely, every direct summand of a ring of Type v is also of Type v (cf. Proposition 9). It remains to prove uniqueness. Suppose k,, k,,, k,,, are cenlral projections such that k, is of Type v and /<, + k,, + k,,,= 1; since k , < h,, the relation (h,- k,) + (h,, - k,,) + (h,,,- k,,,) = 1 1 = 0 -

shows that h, - k, = 0 (v = I, 11, 111).

I

For Type I Baer *-rings with PC, there is a further decomposition into 'homogeneous' rings (see Section 18); this, too, qualifies as coarse structure thcory (but an extra hypothesis is needed, and there are unresolved questions of uniqueness). Meshing Theorem 2 with the decomposition (1) of Theorem 1, we have:

Theorem 3. If' A is a r z l l R L I *-ring, ~ tlzrr~exist unique orthoyonul central projections hl ,L,, hIlnf>I ~ 1 l f , " ~1111,,' hIIl such that: (i) h,,,,, is Type I and fznite, (ii) h,,,, is Type I and properly infinite, (iii) h is Type I1 and ,finite, (iv) hlIInfis Type I1 and properly infinite, (v) h,,, is Type 111, (vi) their sum is 1. Each qf these five projections is maximal in its stated property, that is, it contains every central projections k hulling that property.

,,

9 15.

Decomposition into Types

Proof. In the notations of Theorems 1 and 2, define

The final assertion is clear from Theorems 1 and 2.

1

Remarks. 1. Why the peculiar definition of 'abelian'? One could have proceeded otherwise, as follows. Call a projection e co~nmutative if eAe is commutative. The analogues of Propositions 5 and 7 hold: if f 5 e then f is commutative relative to A iff it is commutative relative to PAP; if e is commutative and j s e , then f is commutative. The analogue of the key Proposition 8 also holds: I f (el) is a very orthogonal family of commutative projections, then e = sup el is also commutative. {Proof: Write h, = C(e,), h = sup h, = C(e). Assuming x, y ~ r A e . it is to be shown that x y = y x . For each 1 , h,x=h,xh,=h,(cxe)h,=e,xe, and h, y = elye,; sincc e,xe, and e,y el commute, we have Iz,(xy y x) = 0 for all L , therefore h(xy -yx)=O, that is, x y - yx=O.} This paves the way for the analogue of Theorem 1: There exists a unique central projection h such that hA is commutative and (1-11) A has no commutative summands; also, there exists a unique central projection k such that kA has a faithful commutative projection and (I-k)A has no commutative projections other than 0. To repeat the question, why the peculiar definition of 'abelian'? A capsule answer is that it keeps division rings together with fields (Example 1 following Definition 2). A fuller answer is to be found in an inventory of examples of Type I rings. Historically, the term 'Type I' was first assigned by Murray and von Neumann [67] to von Neumann algebras isomorphic to 9 ( X ) ,the algebra of all continuous linear mappings on a space with an inner product (complete) taking values in the complex field; algebras of other types appear as subrings of Typc I factors [$4, Def. 51. Subsequently, Kaplansky [49] showed that every Type I AW*-algebra may bc represented as the algebra of all continuous 'operators' on a space with an inner product taking values in a commutative AW*-algebra. (The question of which AW*-algebras may be suitably embedded in a Type I algebra is not yet fully answered; see, e. g. 1261, [38],[79],[90].)A purely algebraic example in this vein: if D is an involutive division ring, n is a positive integer, (4 is the left vector space of n ples x = (A,),v = (p,) over D, and if the inner product n

[x, y]

=

C A,p: 1

is definite (that is, [x, x] = 0 implies x = O), then the

96

Chapter 3. Structure Theory of Baer *-Rings

ring 9 ( b )of all linear mappings on 8 is a (factorial) Baer *-ring [$ 4, Exer. 201 of Type I(viewing 9(Q) as the matrix ring D,, the matrix e with 1 in the northwest corner and zeros elsewhere is abelian-eD,e is *-isomorphic with D-and faithful). In summary, the peculiar definition of 'abelian' (1) keeps together division rings and fields, and therefore (2) includes among the Type I rings the full matrix rings D, just described; a motivation for the dcfinition is the desire, consistent with the classical usage, to ascribe Type 1 behavior to such matrix rings. Another way to put the matter is as follows. The von Neumann factors of Type I are characterized by the property of having a minimal projection. If P is a minimal projection in the von Neumann factor d , then P d P is the field of complex numbers. However, if e is a minimal projection in a Baer a-factor A, then eAe may be a noncommutative division ring-the ring A = D,, described above is an example. (In this connection, see 154, p. 19, Exer. 12; p. 25, Exer. 7; and p. 98, Exer. I].) 2. Let A be any*-ring with unity. Definitions 1 and 2 are meaningful for A. So is Definition 3, provided 'e faithful' is interpreted to mean that the only central projection h such that h e = e is h = 1 [cf. $ 6, Def. I]. Propositions 1-5 and 7 are valid for A . So is Proposition 9, the key point being that if e is faithful in A , then h e is faithful in h A [cf. Jj 6, Exer. 41. Exercises

1A. (i) Every minimal projection in a *-ring is abelian (see [$4, Exer. 31). (ii) In a *-ring with finitely many elements, every nonzero projection is the sum of orthogonal minimal projections. (iii) If A is a Baer *-ring with finitely many elements, then A is finite, of Type I, and is the direct sum of factors. 2A. The following conditions on a Rickart *-ring A are equivalent: (a) A is abelian (in the sense of Definition 2); (b) every idempotent in A is central; (c) all idempotents of A commute with each other. 3A. A Baer *-ring is commut~tiveif and only if it is abelian and is generated by its projections in the sense that (A)" = A. 4A. If (h,) is a family of (not necessarily orthogonal) ccntral projections in a Baer *-ring, such that each h, is finite [properly infinite, etc. as in Definition 31, then sup h, is also finite [properly infinite, etc.]. 5A. In an abelian Kickart *-ring, LP(x)=KP(x) for all x. 6A. If A is a Baer *-ring, e is an abelian projection in A, and jections of e such that f g = 0, then C( f ) C(y) = 0.

/, y are subpro-

7A. If e is any projection in a Baer *-ring, there exists a central projection h such that, on setting e l = h e and e" =(I -h)e, e'Ae' is finite and e"AeUis properly infinite.

97

$16. Matrices

8A. In the notation of Theorem 1, e l h, for all abelian projections e, and J'Lh, for all finite projections f: If, in addition, A satisfies one of the conditions (i), (ii), (iii) of [$6, Exer. 71, then h, = sup {e: e abelian), h, =sup { J': J' finite).

9C. (i) A von Neumann algebra is of Type I if and only if it is *-isomorphic to a von Neumann algebra (on a suitablc Hilbcrt space) whose commutant is abelian (i.e., commutative). (ii) A von Neumann algebra is semifinite if and only if it is *-isomorphic to a von Neumann algebra (on a suitable Hilbert space) whose commutant is finite. 10C. The decomposition into types can be formulated axiomatically in certain complete lattices. 11C. A von Neumann algebra A with center Z is finite if and only if there exists a function tr: A + Z such that (1) tr(a+b)=tr(u)+tr(b) for all a, ~ E A , (ii) tr(z)=z for all ZEZ, and (iii) tr(ab)= tr(ba) for all a, Dt A. Such a function is unique (it is called the center-valued normalized tmcefunction of the finite algebra A) and has, in addition, the following properties: (iv) tr(za)=ztr(a) for all ZEZ, UEA; (v) tr(a*)=(tr(a))* for all a € A; (vi) a 2 0 implies tr(a) 2 0; (vii) if a 2 0 and tr(a)=O, then a=O; (viii) tr is continuous for the ultraweak topology. 12B. Let A be a Baer *-ring, A" the reduced ring of A [$3, Exer. 181. (i) A projection in A is finite relativc to A iff it is finite relative to A". (ii) If A has PC, then a projection in A is abelian relative to A iff it is abclian relative to A". (iii) If A has PC, then the central projections described in Theorems 1-3 arc the same for A" as for A (so to speak, A and A" have the 'same' coarse structure). 13C. (i) If A is an A W*-algebra, then A = A o iff the abelian summand of A is 0. (ii) If A is any A W*-algebra and if e,f are projections in A, then e - J iff e /. (iii) A von Neumann algebra coincides with the algebra generated by its projections iff its abelian summand is finite-dimensional.

"

14A. If A is a Baer *-factor and e is a nonzero abelian projection in A (in particular, A is of Type I), then e is minimal and eAe is a *-ring without divisors of zero.

5 16. Matrices The elementary material on matrices presented here could just as easily have been included in Part 1-it does not depend on type-but the main role of matrices is in structure theory. Matrix rings have already occurred in several of the exercises; for the record, here is the official definition : Definition 1. If B is a ring and n is a positive integer, we write Bn for the set of all n x n matrices x=(x,,), y=(yii) with entries in B, equipped with the usual ring structure:

If, in addition, B has an involution, we regard Bn as a *-ring, with *-transposition as the involution: x* =(zij), where zij=x;.

98

Chaptcr 3. Structurc Theory of Baer *-Rings

It is generally quite hard to show that a matrix ring is a Baer *-ring (see Chapter 9). The question of when a Baer *-ring is a matrix ring is (at least, piecewise) easier; for example, it is shown in Section 17 that if A is a properly infinite Baer *-ring with orthogonal GC, then, for every positive integer n , A is *-isomorphic with A"; in Section 18, every finite Baer *-ring of Type I with PC is 'decomposed' into matrix rings over abelian rings; in Section 19, it is shown that every continuous Baer *-ring A with PC is, for each positive integer n, *-isomorphic to B, for a suitable Baer *-subring B of A . We now discuss the elementary algebra underlying such matrix representations. Suppose B is a *-ring with unity, n is a positive integer, and A = B, as in Definition 1. For i= I, ...,n, let e, be the ith diagonal unit, that is, the matrix with 1 in the ( i , i)position and zeros elsewhere. The original ring B may be recaptured by 'compression': B is *-isomorphic to e,Ae,, via the correspondence

-

under which ~ E isB paired with the matrix having b in the (i, i) position and zeros elsewhere. Observe that e, t; (via the partial isometry w,, having 1 in the (j, i) position and zeros elsewhere). Thus e l , .. . ,e, are orthogonal, equivalent projections in A with sum 1 (the identity matrix). Important for structure theory is the converse: Proposition 1. Let A be a *-ring with unity, let n he a positive integer., and suppose that A contain.^ n orthogonal, equivalent projection.~e,, . . . ,e,, with sum 1. Then A is *-isomorphic with the rinq ( e , A e l ) , of' all n x n matrices over e , Ae, . Explicitly, for i= I , ...,n let w, be a partial isometry such that w: w, = el and w, w: = el. Then the inupping cp : A + ( e l A c , ), defined bv

Proof. If a e A then w:u~v~,,te,Ae, for all i, j, thus cp(a)~(e,Ac~,),. Clearly cp(a+h)=q(u)+cp(h) and cp(rr*)=rp(u)*. cp is injective: ~f w,*oct;=O for all I, j , then r , n r , = w , ( w ~ a n ; )=O w~

C

cp is surjective: if ( a i i ) ~ (Ael),, c, define a = wiaijwT; noting that i,j wT wj = 0 when i f j , the calculation

shows that ~ ( a=)(aii).

5 16. Matrices

Finally, cp(ah)= cp(a)cp(b) results from the calculation

As an application, we cxplain in matricial terms a calculalion made in the proof of [ji 11, Lemma 31:

-

Proposition 2. Let A he u weaklv Rickart *-ring and let e, f be projections in A with e j = 0. Then P J if o n ~ only i if' there exists a projection g such that Proof: If such a projection g exists then, defining w = 2 J'q e, wc havc and similarly w w* = f , thus e -- f (the hypothesis e l ' = 0 is not nccdcd here). Conversely, suppose e J'. Dropping down to ( r + , f j A ( r + ,fj, we can suppose that A is a Rickart +-ring and e + ,f = 1 (see [$ I , Prop. 41, [ji 5, Prop. 61). Let w be a partial isometry such that w* w = e , w w * = j' = I- e. Define w, = e, w, = w and consider the *-isomorphism

-

cp: A

+

(eAe),

given by cp(a)= (w: a u;) as in Proposition I. For a € A write a,, = w: a w, ( i j = l 2 ) . Since w,,=ewe=O, w,,=eww*=O, w,,=~c*~.l.c=e, w,,=w*ww=ew=O, wc have

It follows that (or similar computations yield)

It is more suggestive to write 1 for the unity element of PAL' and identify A with (eAe),; then

Note that if a=(aij)eA then

Chapter 3. Structure Theory of Baer *-K~ngs

100

Since 2 is invertible in A [$ 11, Lemma 21, we may define

Evidently g is a projection, and direct calculation yields the desired relations 2 e g e = e, etc. {Note, incidentally, that 2 g = e + f + w + w* (cf. the proof of [S; 11, Lemma 3]).} 1 Exercises 1A. If A is a Rickart *-ring and if, for some positive integer n, A is *-isomorphic with the *-ring B, of n x n matrices over a *-ring B with unity, then n l is invertible in A. In fact, (n!)l is invertible in A.

2A. Let A be a ring with unity and let n be a positive integer. If I is a subset of A, write I,, for the subset of A, consisting of those matrices whose entries are all in I. Then 1-t 1, maps the set of all (two-sided) ideals of A bijectively onto the set of all ideals of A,, and A,,/I,, (All), for every ideal I. In particular, I is maximal in A iff I, is maximal in A,; A is simple iff A, is simple. If A is a *-ring, then I I, pairs the *-ideals of A with those of A,.

-

3A. Let B be a ring with unity, let n be a positive integer, and let B, bc the ring of all n x n matrices over B. For b t B write diag (b) = diag (b, b, . ..,b) = (a,, h) for the 'scalar' matrix with b on the diagonal and zeros elsewhere. For any subset S of B, write D(S) = {diag (s): s t S} ; also, write S, for the set of all n x n matrices with entries in S. (i) If S is any subset of A containing 0 and 1, then ( S a y = D(S'). (ii) Let A be a subring of B such that A=A". Then A,=(D(A'))'; in particular, A, is a subring of B, such that A,= A:. Moreover, D(A')=(A,)', thus A,, and D(Ar)are each others' commutants in B,,. Similarly (A'),= (D(A))', D(A) = ((A'),)'. (iii) Assume, in addition, that B is a *-ring, and let A be a *-subring of B such that A = A . Then each of A,,, (A'),, D(A), D(A') is a *-subring of B, that coincides with its bicommutant in B,. If B,, is a Rickart *-ring, then A, and (A'), are also Rickart *-rings (with unambiguous RP's and LP's); if B, is a Baer *-ring, then A, and (A'), are Baer *-subrings of B,. (iv) Related remarks: If B is a von Neumann algebra on a Hilbert space A", it is easy to show that B, may be regarded as a von Neumann algebra on the direct sum of n copies of 2. If B is an AW*-algebra, then so is B, [fj 62, Cor. 1 of Th. I]. For B a Baer *-ring, the situation is complicated, and largely unknown (see Section 55).

4A. Let B be a Baer *-ring, A a *-subring of B such that A= A", Z = A n A' the center of A, and n a positive integer. Assume that B, is also a Baer *-ring (cf. Exercise 3, (iv)). If xeA,,, x't(A1),, the following conditions are equivalent: (a) xxl=O; (b) there exists a projection ceZ, such that xc=O and c x f = x ' ; (c) there exists c t Z , such that xc=O and c x f = x ' ; (d) xD(Z')xr=O, where D(Z') is the set of all matrices diag (z') with z' in 2'=(A nA')' =(A u A')" (see Excrciw 3).

5 17.

Finite and Infinite Projections

5A, C. (i) If , Iis a complex number with 1/11 ber a, 5 5 a 51, such that the matrix

101

54,then there exists a real num-

:-3

*

is a projection. (ii) If A= B,,, where n 2 2 and B is a complex *-algebra with unity, then the reduced ring A" [$ 3, Exer. 181 is a *-subalgebra of A . If, in particular, B is the field of complex numbers (or even a C*-algebra), then A = A .

§ 17. Finite and Infinite Projections

Throughout this section, A denotes a *-ring with unity (often, but not always, a Baer *-ring). Recall that A is said to be finite if x * x = I implies xx* = I,and a projection P E A is called finite if the ring e A r is finite [$ 15, Def. I]. Nonfiniteness may be characterized as follows: Proposition 1. (i) I f A is an infinite *-ring with unity, then there exists a sequence f n of orthoyonal, rnutually equivalent, nonzero projections. (ii) Conversely, if A is a Rickart *-ring that contains such a sequence, und if every sequence of orthogonal projrction~ in A has a supremum, then A is infinite.

-

Prooj. (i) By hypothesis, there exists a projection e # 1 such that I e (see the remark following [$ 15, Def. I]). Let w be a partial isometry such that w* w = 1, w w* = e, and let cp: A+eAe be the *-isomorphism defined by y(x)=wxw*. In particular, y is an orderpreserving bijection of the projections of A onto the projections of eAe. Define el = 1 and, inductively, en+,= y(e,) for n = 1,2,3, ... 111 particular, e, = q(1) = e. Since el 2 e, and el # e,, an application of cp to the inequality yields e, y e , , e, # e,. Continuing inductively, we see that the sequence of projections en is strictly decreasing. Defining f n = en- en (n= 1,2,3,...), we have an orthogonal sequence of nonzero projections; moreover,

+,

-

shows that ,fn ,f,+, [$ 1, Prop. 91. (ii) Suppose f , is an orthogonal sequence of nonzero projections such that j; f; 1, . .. . By hypothesis, we may define e = sup { fn : n 2 I j and f =sup{J,:n>2). Then f ; j'=O, and r=f;+,f by the associativity of suprema. We have e f' [$ 11, Prop. I ] , where f 5 and e - j'= f ; # 0, thus e is not a finite projection [$ 15, Prop. 21; it follows that A is not finite [$ 15, Cor. of Prop. 31. 1

- - -

-

102

Chapter 3 Structure Theory of Bacr *-Rings

It is useful to have some terminology to describe orthogonal families such as those occurring in Proposition 1 :

Definition 1. An orthogonal family of nonzero projections (e,) is called a partition, with terms el; if e= sup el exists, it is a partition of e. Two equipotent partitions (e,),,,, (j;),,, are equivalent if e l - , f ; for all L E I . A partition (e,) is homogeneous if its terms are mutually equivalent. (Warning: The word 'homogeneous' has another meaning in the context of Type I rings (see the next section).} A homogeneous partition (el) is called maximal if it cannot be enlarged, that is, there does not exist a projection e such that P el and e c., = O for all 1 .

-

Remarks. 1. We admit partitions with finitely many terms; but the message of Proposition 1 is that infinite homogeneous partitions correlate with nonfiniteness of the ring. 2. If e f' and (el)is a partition of P , then there exists a partition (j',) o f f that is equivalent to (e,)[$I, Prop. 91. If, in addition, (e,)is homogeneous, then so is (f,). 3. Every homogeneous partition can be enlarged to a maximal one (a routine application of Zorn's lemma). Wc remark that if ( r , ) is a homogcneous partition of e, then ( e l ) can be enlarged if and only if r , S l - e ; if (el)is maximal, the leftover scrap I-6. can sometimes be absorbed by dropping down to a direct summand:

-

Proposition 2. I f A is a Baer *-ring with orthogonal GC, and zf (e,) is a maximal homogeneous partition in A w ~ t h~n/initelytnany tc.rtn\, then there exists a nonzero central projection h and a hotnogeneour partztion ( f ,) of h that i~ equivalent to (he,). Proof. Fix an index I , , ; for simplicity, writc 1 , = 1. Sct c= sup el. Since 1 -e and el are orthogonal. by hypothesis thcrc exists a central projection h such that (*)

h(1 - e ) 5 h p l

and (I - h ) e , 5 (1 - h ) ( l -r). Necessarily h e , # 0; for, Itel = O would imply el = (1 - h ) ~5, ( 1 - h ) ( l - e) 5 1 - e, contrary to the maximality of (e,). Since (he,)is a homogeneous partition of he, it will suffice, by Remark 2 above, to show that h e h. In view of the Schroder-Bernstein theorein ([$I, Th. 11 or 1512, Prop. I ] ) , it is enough to show that 11 5 he. Let f =sup { e l :r f I j; thus r = / + e , . Since the family (el) is infinite. we have P f 1 1 , Prop. I ] , therefore

-

-

s; 17.

Finite and Infinite Projcctioils

adding (*) and (**), we have The hypotheses of Proposition 2 can be fulfilled in any infinite Baer *-ring with orthogonal GC (apply Remark 3 above to the sequence J, whose existence is ensured by Proposition I), leading to a direct summand with a certain property. When A is properly infinite, the argument can be pursued to exhaustion: Proposition 3. If A is a proprdy infinite Baer *-ring witit orthogonal GC, then there exists a sequence en that is a honlogeneous partition of 1. Proof. By Proposition 1 there exists a homogeneous partition with infinitely many terms, which we can suppose to be maximal (Remark 3 following Definition 1). Invoking Proposition 2, Lhere exists a non7ero central projection h that possesses a homogeneous partition (f,),,,with infinitely many terms. Since No card 1=card I, the index set 1 can be written as the union of a disjoint sequence of equipote~ltsets I,,

-

Defining f , = sup ( f ,: i E I,) ( n= 1,2,3, ...), we have ,f, ,f, for all H I , n by orthogonal additivity of equivalence [$11, Th. I],and sup.fi= h by the associativity of suprema. Summarizing, there exists (in every infinite Baer *-ring with orthogonal GC) a noilzero ccntral projection 11, and a sequence f , that is a homogeneous partition of h. Let (Iz,),,~, be a maximal family of orthogonal, nonzero central projections such that, for each M E A ,there exists a sequence r,, that is a homogeneous partition of iz,. Defining

-

we have en, en for all rn, n, and sup en= sup h, (for thc reasons cited above). It will suffice to show that sup h,= I . Assume to the contrary that 1 - suph, # 0. Then ( I - suph,) A is infinite (because A is properly infinite); by the first part of the proof, it contains a nonzero central projection k and a sequencef, that is a homogeneous partition of h. This contradicts the maximality of the family (h,),,,. I In the conclusion of the above proposition, the en are mutually equivalent; this can be achieved by making them equivalent to 1 :

Theorem 1. Let A be a properly infinite Burr *-rlng ~villz orthogonal GC. (1) There exists an orthogonal sequence o f projections f n such that sup f,=l and /,-I f o r a l l n .

Chapter 3. Structure Theory of Raer *-Rings

104

-

(2) For each positive integer tn, there exist orthogonal projections +...+g, = 1 and g, 1 for all i.

g,, .. .,g, such that g,

Proof. By Proposition 3, there exists a sequence e, that is a homogeneous partition of I . (1) Write the index set 1 = (1,2,3,.. .) as the union of a disjoint sequence of infinite subsets, and define ,fn = sup ( e i :i~ I , } . The f , are mutually orthogonal, sup f , = 1, and ,f,- 1 [jj 11, Prop. I ] . (2) The proof is similar, based on a partition of 1 into infinite subs t I . .I . I A matricial interpretation of ( 2 ) : Corollary. If A is a properly infinite Baer *-ring with orthogonal GC, then, for each positive integer m, A is *-isomorphic to the ring A, of all m x m matrices over A.

-

Proof. By the theorem, there exists a partition e,,. . . , e m of 1 such that e, 1 for all i. Thus A is *-isomorphic to (e,Ae,), 1916, Prop. 11. The equivalence e l - 1 induces a *-isomorphism of e,Ae, with A [tj 1, Prop. 91, which in turn induces a *-isomorphism of (e,Ae,), with A,. I Another application of Theorem 1 (see Theorem 3 for a more general result) : Theorem 2. Let A he a Baer a-ring with GC, satisfying the parallelogram law (P). If e , j are finite projections iiz A , then e u f if also finite (thus the finite projections in A form a lattice).

Proqf: {We remark that in a Baer *-ring satisfying the parallelogram law, the hypotheses GC, PC, and orthogonal GC are equivalent [# 14, Prop. 71.) The subring ( e uf ) A ( e uf ) is also a Baer *-ring with GC and satisfying (P), in which e , J' are finite projections; dropping down to it, we can suppose e u f = I. Citing (P), we have I - e = e u , f ' - e - f - e n , f ' < J; thus 1 - e d f ; since f is finite, so is 1 -e [jj15, Prop. 31. Thus I =e+(l -e) is the sum of orthogonal finite projections. Changing notation, we can supposc ef=O, e + f = I . Assume to the contrary that A is not finite. Then, by structure theory, there exists a nonzero central projection k such that k A is properly infinite [$ 15, Th. I ] . Clearly k A has GC and satisfies (P), and

6 17.

Finite and Infinite Projections

105

ke, kf are orthogonal finite projections with sum k. Dropping down to kA, we seek a contradiction from the following conditions: A is a properly infinite Baer *-ring with GC and satisfying (P), and e, f are orthogonal finite projections such that e + f = I.By Theorem 1 (with m = 2) there exists a projection g such that g I l-g. Apply [$ 14, Prop. 81 to the pair g, e: there exists a central projection h such that

--

since e and I-e

are finite, it follows from the relations

that hg and (I - h)g are also finite. Since h y and (I -h)g are very orthogonal finite projections, their sum g is also finite [$ 15, Prop. 81. is, A is finite-contrary to Then 1 -g shows that 1 is finite-that supposition. I

Remarks. 1. Theorem 2 applies to any Baer *-ring satisfying the (EP)-axiom and the (UPSR)-axiom [$ 14, Th. I]-in particular, to any AW*-algebra. More generally, it applies to any Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom [$ 14, Exer. 51. 2. The parallelism between finite and abelian projections is evident in Section 15. Theorem 2 marks a parting of the ways: if e and f are finite [abelian] projections, then e u f often [hardly ever] has the same property. We now consider some conditions, on a *-ring A with unity, closely related to (and usually equivalent to) finiteness; these conditions are sometimes more convenient to apply than the definition of finiteness. The following list is not exhaustive (see, e. g., Exercises 4-6 and 10-12): (1) A does not contain an infinite homogeneous partition. (2 a) e f implies e, f are unitarily equivalent, that is, ueu* = f for a suitable unitary u. (2b) e - f implies I - e - I f. (3 a) A is finite (that is, x* x = I implies xx* = 1). (3b) e - I implies e = l . (3c) e - f l e implies f = e .

-

- -

Proposition 4. Let A be a *-ring with unity. (i) The implications (2 a)e (2 b) = (3 a) (3 b) (3c) and (1) 3 (3 a, b, c) hold. (ii) If A has GC then (3c) => (2 b), thus the conditions (2), (3) are equivalent and (1) implies them. (iii) l f ' A is a Rickart *-ring in which every orthogonal sequence of projections has a supremum, then (3a) 5 (1).

Chapter 3. Structure Theory of Biicr *-K~nga

106

(iv) lf A is a Rickart *-ring with GC. in which every orthogonal sequence of projections has a supremum, then the six conditions (1)-(3 c) are equiuulent. PvooJ: (i): (3 a) o (3 b) is remarked following [ji 15, Def. I]. (3c) * (3b) is obvious. (3 a) (3c): Suppose A is finite and e f < e. Since e is finite [ij15, Cor. of Prop. 31, it follows that J'= e [ij 15, Prop. 21. Thus (3 a) o (3 b) o (3c). (2 a) * (2b): Suppose e j'. By hypothesis, there exists a unitary u such that ueu* = J'. Then w = u(l -e) satisfies w* w = I -e, ww*=u(l-e)u*=uu*-ueu*=I-,f', thus 1 - e - 1 - f . (2 b) * (2a): Suppose e J'. By hypothesis, I- e -1 - j'. Let w, v be partial isometries such that w* w =e, w w* = f and v* v = I -e, v v* = 1 - f . Since wcJ'Ae and v ~ ( 1 -f ) A ( l - e), it is routine to verify that u = w + c is unitary and u e u * = , f . Thus (2a) o (2b). (2 a) .=> (3b) is obvious. (1) => (3 a) is covered by Proposition 1, (i). (ii) Assuming A has GC, suppose (3c) holds. Given e -f, we are to show that 1 - e -1 - j'. Applying GC to the pair 1 -e, 1 -f', there exists a central projection h such that

-

-

-

h(1-e) - f " < h(1- f ) ,

(*)

(I-h)(l- f)-e's(1-ll)(l-e). Since he hj; addition with (*) yields

(**)

-

2 I = h e + h ( l -e)

- hf + f ' I h,

therefore hf'+ f"= h by (3c), that is, f ' =h(l - f ) . Thus (*) reads h(l -e)-h(1 - f ) , and ( I - h) (1 - f ) - ( 1 -h) (1 -e) follows similarly from (**); adding these equivalences, we have 1 - P 1 - f'. (iii) Quote Proposition 1, (ii). I (iv) Immediate from (i)-(iii).

-

An application (more generally, see Exercise 8): Proposition 5. Let A be a Burr *-ring with GC, satisfying tlzeparallelothat e r r f , then gram law (P). If e, f are finite projections in A .SUC/Z lGP-l- f .

Proqf. The subring B=(e u f )A(e u f ) is also a Baer a-ring with GC and satisfies (P), and it is finite by Theorem 2. Applying Proposition 4, (ii) in B, there exists a unitary v in B such that vev*= f ; then u = v + ( l - e u f ) is unitary in A and u e u * = J ; therefore I - e - I f' (Proposition 4, (i)). I

# 17. Finitc and Infinite Projections

107

The final result, a generalization of Theorem 2, is for application in Section 58; it can be omitted by the reader who is omitting Chapter 9:

Theorem 3. Let A be a Rickart *-ring with GC, sutisfying theparullelogram law (P), such that every sequence of orthogonal projections in A has a supremum. If e, f are finite projections in A , then e u f is also finite (thus the finite projections in A form a lattice). Proof. As argued in the proof of Theorem 2, we can suppose ej'= 0, e + f =I. Assume to the contrary that A is not finite, and let g, be a sequence of orthogonal, equivalent, nonzero projections (Proposition 1). Define g = sup(g,:n=1,2,3 ,...}, g'

= sup

{g,: n odd),

g" = sup jg,: n even}. Then yrg"=O, g = g l + y", and g'-g -g" [$ 11, Prop. I ] . Applying G C to the pair g'n e, g" n f , there exists a central projection h such that h ( q 1 n e ) 5h ( g U n f ) ,

(1)

(1 - h) (y" n f ) 5 (1 h) (y' n e).

(ii)

-

Citing (P), we have h(gf- g ' n e )

(iii)

-

h(g'u e - r ) .

The left sides of (i) and (iii) are obviously orthogonal; prior to adding them, we check that the right sides are orthogonal too. Indeed, q r ' n , f ' and g ' u e are orthogonal; for, g" n f is orthogonal to g' (because g' g" = 0) and to e (because f e = 0), hence to y' u e. Adding (i) and (iii), we have h g ' = h ( g f n e ) + h ( g ' - q ' n e ) 5 h(g" n f ' ) + h ( g l u e - e ) ; since g " n f < f and g ' u e - e < I - e = hq-hq' we have

J; this yields hg'5.f; and since

h ,9 5 f'.

(*)

Again citing (P), (iv)

(1 -h)(gl'-gUnf')-- ( 1 - h ) ( y U u f - J ' ) ,

and (ii) and (iv) yield, by a similar argument, From (*) and (**) we see that h g and (1 - 1z)y are finite (by [$15, Prop. 31, which holds in any *-ring). But hg is the supremum of the sequence

108

Chapter 3. Structure Theory of Baer *-Rings

hg, of orthogonal, equivalent projections; it follows from Proposition 1, (ii) that hg,=O for all n, thus hg=O. Similarly (I -h)g=O. Then g = hg + ( I - h)g = 0 , a contradiction. (One could also argue that g = h g + (I - h)g is finite [cf. 9 15, Prop. 81, contradicting the relations Y " Y ' I ~ g, ' f 9.: 1 Theorem 3 applies, in particular, to any Rickart C*-algebra with GC ([s 8, Lemma 31, [$ 13, Th. I]). Exercises 1A. If a *-ring with unity has finitely many elements, then it is finite (in the sense of [§ 15, Def. I ] ) . 2A. If h is a finite central projection in a *-ring, and if h 5 e, then h 5 c.. (Cf. [$ 1, Exer. 151.)

3A. Let A be a *-ring with unity and GC. If e, f' are finite projections such that e - f , and if e ' 5 e, f '< j' are subprojections such that e'- f'', then e-e'- f - f ' . 4A. Let A be a symmetric *-ring with unity [$ 1, Exer. 71 satisfying the (WSR)axiom (for example, let A be a C*-algebra with unity). If A is finite, then y.x= I implies x y = l. 5A. Let A be a Rickart *-ring satisfying the (WSR)-axiom. If A is finite, then y x = l implies x y = l . 6A. Let A be a Rickart *-ring such that LP(x)-RP(x) for all x in A. If A is finite, then yx= l implies x y = l . 7 A . If A is a properly infinite Baer *-ring with orthogonal GC, then n l is invertible for every positive integer n, thus A may be regarded as an algebra over the field of rational numbers.

-

-

8A. Let A be a Rickart *-ring satisfying the conditions of Theorem 3. If r , j are finite projections in A such that r f , then 1 - e 1 - f . 9A. Theorem 3 remains true with 'Rickart *-ring' replaced by 'weakly Rickart *-ring'. In particular, the finite projections in a weakly Rickart C*-algebra with GC form a lattice.

10A. (i) If A is a finite Kickart *-ring satisfying then the projection lattice of A is modular, that is, if e l g, then ( e u f ) n g = e u ( f n g ) . (For a result in Exercise 11.) (ii) The finite projections form a modular latticc satisfying the conditions of Theorem 3.

the parallelogram law (P), e, j; g are projections with the converse direction, see in a weakly Rickart *-ring

11'2. If A is an AW*-algebra whose projection lattice is modular (see Exercise l o ) , then A is finite. 12A, C, D. Let A be a Rickart *-ring. Projections e , , j in A are said to be perspective if they admit a common complement, that is, there exists a projection g such that e u g = l , e n g = O and f ' u g = l , f n g = O .

# 17. Finite and Infinite Projections

109

(i) Let h be a projection in A , and let e , f be projections in IzAh. If e , f are perspective in hAh, then they are perspective in A ; the conversc is truc if h is a central projection of A. (ii) Let h be a projection in A, and let e , f be projcctions in hAh. Suppose that the projection lattice of A is modular (cf. Exercises 10, 11). If e , f are perspective in A then they are perspective in hAh. (iii) If e , f are in position p', then they are perspective. (iv) Suppose A satisfies the parallelogram law (P). If e , f are perspective, then they are unitarily equivalent. (v) Suppose A satisfies the parallelogram law (P). If e - j' implies e , 1' are perspective, then A is finite. (vi) If e f' and ef = 0 then e , f are perspcctivc. (In fact, it suffices to assume that the orthogonal projections e , f are algebraically equivalent in the sense of [jj 1, Excr. 61.) (vii) Suppose A satisfies the conditions of Theorcrn 3. If e , f are finite projections in A such that e - f ; then e , f' are perspective. (In view of (iv), this yields another proof of Exercise 8.) (viii) Hcre are somc applications of (vii). Consider the following conditions on a pair of finite projections e , j (a) e, f are perspective; (b) a, f are unitarily equivalent; (c) e f'. Then ( a ) o ( b ) o (c) under any one of the following four hypotheses: (1) A satisfies the conditions of Theorem 3. (2) A is a Baer *-ring with GC, satisfying the parallelogram law (P). (3) A is a Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom. (4) A is an A W*-algebra. (ix) Suppose A is either a Rickart C*-algebra with GC, or a Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom. If e , f are finite projections in A such that e f , then there exists a symmetry exchanging cJ and f: (x) If A is a von Neumann algebra, and e , f are projcctions in A (not neccssarily finite), then e , f' are perspective iff they are unitarily equivalent. (xi) Problem: Docs (x) gcneralizc to AW*-algebras?

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13A. Let A be a finite Rickart *-ring with GC, and let n be a positive integer. Then, under either of the following two hypotheses, the matrix ring A, is also finite: (1) A, is a Rickart *-ring, satisfying the parallelogram law (P), in which every sequence of orthogonal projections has a supremum. (2) A is a Rickart C*-algebra and A, is a Rickart *-ring (hence also a Rickart ('*-algebra).

.-

14A. Let A be a *-ring with unity and GC. Suppose that e , f are projections and ( j ; ) , ,,, are homogeneous partitions in A, n is a positive integer, ( e J l of e and , f , respectively; supposc, in addition, that e f and that e is finite (hcncc J; e,, ,f, are also finite). Then e , - f,. (Cf. Exercise 15.) 15A. Let A be a Baer *-ring with GC, satisfying the parallelogram law (P). Let e be a projection in A, and suppose e = e , + e , = Jl+ f , are orthogonal deThen e , - j ; . (Cf. Exercise 14.) compositions of e with e , - e , and f , - , f 2 . 16A. Let A be a Rickart *-ring satisfying the conditions of Theorem 3, and assume that A is properly infinite [S; 15, Def. 31. If e, f are finite projections in A , then e 5 I- f . In particular, e 5 I- e for every finite projection e. (Cf. [jj 18, Prop. 61.) 17A. Let A be a Baer *-ring with G C and let e , f bc projcctions in A such that e - f .

Chapter 3. Structure Theory of Bacr *-Kings

110

(i) Therc exist orthogonal decompositions e= e, + e,, f = f ; + J, such that e, and ,fi are unitarily equivalent (i=1,2). (ii) When A is a von Neumann algebra, it follows that e,, ji are perspective (i= 1,2). 18A. Let A be a properly infinite Baer *-ring with GC, satisfying the parallelogram law (P), and suppose A contains a nonzero finite projection g (thus A is not purely infinite). Then there exist a nonzero central projection lz and a hornogencous partition (f;),,,of h with h g -1; (I is necessarily infinite). 19A. Let A be a semifinite Baer *-ring with PC, and let e be any nonzero projection in A . Then (i) e is the supremum of an orthogonal family of finite projections; (ii) eAe is semifinite; (iii) if e is faithfirl, then (. contains a [aithful finite projection. (Cf. [$ 18, Exer. 21.) 20C, D. (i) Let A be a Baer *-ring satisfying the (EPj-axiom and the (SR)-axiom, and let A" be the reduced ring of A [$3, Exer. 181. If e, f a r e finite projcctions in A , then the following conditions are equivalent: (a) e -1; (b) e, f can be exchanged by a symmetry 29-1, g a projection; (c) 2 f: In particular, in a linitc Bacr *-ring satisfying the indicated axioms, the relations e -f and c 2 f are equivalent. (ii) If A is an A W*-algebra, and e,f a r e any projcctions in A, thcn r f'iff e 2 J: (iii) Problem: Does thc conclusion of (ii) hold for the rings of (i)'!

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21B. If A is a properly infinite Baer *-ring with PC, thcn (i) partial isometries in A are addable, and (ii) A has GC. 22C, D. (i) In a properly infinite von Neumann algebra, evcry unitary element is the product of four symmetries. (ii) Problem: Does (i) generalize to A W*-algebras? 23A. Let A be a finite Baer *-ring with GC. If e is a projection in A such that, for cvcry positivc intcger n, A contains n orthogonal projections equivalent to 0, thcn e=O. 24D. Problem: If A is a Baer *-ring with GC and if e,J are finite projections in A, is e u j' finite?

5 18.

Rings of Type 1; Homogeneous Rings

Recall that a Baer *-ring is said to bc of Type I (or discretc) if it contains a faithful abelian projection, that is, an abelian pro-jection e such that C(e)= I 15, Def. 31. Consider, for example, the Baer s-ring 1"(~Y') of all bounded linear operators on a Hilbert space .% [54, Prop. 31. Let (t,),,,bc an orthonormal basis of 2 and let (E,),,, be the corresponding family of onedimensional projections, that is, E,i; =(i;l[,)<, for all [€.%'. If I , xt 1 then El and E, are unitarily equivalent via the obvious permutation operator, thus (E,),,, is a homogeneous partition of the identity operator, in the sensc of 1517, Dcf. I]; moreover, E,y(YZ') E, is the set of scalar multiples of El, thus El is an abelian projection in .Y(.YZ'). This situation is a special case of the following:

[a

5 18. Rings

of Typc I ; Homogeneous R ~ n g s

111

Definition 1. A Baer *-ring A is said to be homogeneous if there exists a homogeneous partition of 1 whose terms are abelian projections; that is, there exists an orthogonal family (e,),,, of abelian projections such that sup el = 1 and el e, for all i and x. Though card 1 is in general not known to be unique, we speak of A as being homogeneous of order card I.

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Remarks. 1. A homogeneous Baer *-ring is of Type I ; for, in the notation of Definition 1, we have [ij 6, Prop. I] C(r,)= C(c,) for all 1 and x, therefore 1 = C(1) = C(supe,)= sup C(e,)= C(e,) for any x. 2. In the wake of the operatorial example given above, it should be emphasized that the rings el Ae, in Definition 1 are assumed to be abelian, but need not be commutative (cf. the remarks at the end of Section 15). 3. For a homogeneous von Neumann algebra, the order is unique; more generally, the order is unique for a homogeneous AW*-algebra whose center satisfies a certain decomposability condition (sec Exercise lo), but the question of uniqueness in general is opcn. However, when the order is finite, the tendency to uniqueness is strong (see Proposition 2 below). 4. A Baer *-ring A is homogeneous of order n, n a positive integer, if and only if A is *-isomorphic to the ring B, of n x n matrices over an abelian Baer *-ring B; this is immediate from [$16, Prop. I]and the discussion preceding it. In such a ring, I is the sum of n finite (evcn abelian) orthogonal projections; we do not know if A must bc finite (Exercise 12), but A is finite if it has G C and satisfies the parallelogram law (P) 1917, Th. 21--in particular, when A satisfies the (EP)-axiom and the (SR)-axiom [ij14, Exer. 51. 5. In a Baer *-r~ngwith PC, the conditions el-e, in Dcfinition 1 may be replaced by C(e,) = C(e,) ( = I), in vicw of the following proposition (cf. Exercise 1): Proposition 1. Let e, j be ahrlirln projections in a Baer *-ring A, .such that C(e)= C(J). Then e- j under either of the following hvpolhe,te.t: (i) A ha^ GC, or (ii) e f = 0 and A ha.7 PC. Proof Under either hypothesis, e and j are generalized comparable (see [$14, Prop. 51). Let h be a central projection such that he 5 h f and (1 - h) f 5 (1 - h) e. Say h e f ' 5 hJ. Citing [tj 6, Prop. I], we havc

-

thus C ( f l ) = h C (f ) . But f ' = f C (f ' ) since f ' i s abelian [$15, Prop. 61, thus f'=.f'[hC(f)l = h.f'C(f)= hJ; therefore he-h f . Similarly (I - h)e -(I -17) f .

I

112

Chapter 3. Structure Theory o f Baer *-Rings

Proposition 1 implies a slight generalization of itself: Corollary. Let e, f be projections in a Baer *-ring A, such that C(e)

5 C(f), and suppose e is abelian. Then e 5 f under either of the fi~llowing

hypotheses: (i) A has GC, or (ii) e J = 0 and A has PC.

Proof: As in the proof of Proposition 1, let h be a central projection such that h e 5 h f and (I - h)f 5 (1 - h ) e . It clearly suffices to show that (1 - h)e- (1 - 1 z ) f ; since both projections are abelian [$15, Prop. 71, and their central covers are equal by the calculation

they are equivalent by Proposition 1.

I

If a finite Baer *-ring is homogeneous, then the order is necessarily finite [g 17, Prop. I]; in the presence of generalized comparability, we can show that it is unique: Proposition 2. Let A be a finite Baer *-ring with GC, alzd suppose A is homogeneous of order n ( n a positive integer). Say 1 = e , + ... + en, where the ei are orthogonal, equivulent, abelian projections. Then: ( 1 ) Everyfaithful abelian projection in A is equivulent to the P i . (2) If (f,),,, is any orthogonal ,fumily cffuithful abelian prejec-tions, then card 1 < n . (3) n is unique. (4) The abelian ring e,Ae, is unique up to *-isomorphism. Thus, if A is a finite, homogeneous Baer *-ring with GC, then A is *-isomorphic to an n x n matrix ring B,, where B is an abelian Baer *-ring; these conditions determine n uniquely, and B uniquely up to *-isomorphi.snz. Proof. (1) Quote Proposition 1, (i). (2) Assume to the contrary that there exist n + I orthogonal, faithful abelian projections f , ,...,f,,, f,, . By (I), e, - , f , (i= I, ..., n), thus

,

l = e l + . . . + e , - f 1 +...+,f, # 1 , contrary to the finiteness of A. (3) Immediate from (2). (4) Immediate from (1) [S; 1, Prop. 91. 1 Note that the following definition is limited to n finite and A finite: Definition 2. Let n be a positive integer. A Baer *-ring A is said to bc of Type I, if (i) A is finite, and (ii) A is homogeneous (hence Type 1) of order n. Remarks. 1. A Baer *-ring A is of Type 1, (n a positive integer) iff A is finite and is *-isomorphic to the ring B, of n x n matrices over an

ij 18. Rings of Type 1; Homogeneous Rings

113

abelian Baer *-ring B (Remark 4 following Definition 1). If A has GC, then n is unique and B is unique up to *-isomorphism (Proposition 2). When A has GC and satisfies the parallelogram law (P), the assumption that A is finite is redundant (Remark 4 following Definition 1). 2. For N an infinite cardinal, 'Type I,' is defined in Definition 3; for the present, we proceed without this concept.

Warning. 'Type I,' (n a positive integer) means 'finite and homogeneous of order n'; it is conceivable that a homogeneous ring of order n may fail to be of Type I, (i.e., it might fail to be finite). {Problem: Can this happen?} A homogeneous Baer *-ring is of Type I (Remark 1 following Definition 1); it follows that if a Baer *-ring A possesses an orthogonal family (h,) of central projections such that suph,= I and Iz,A is homogeneous for all a, then A is of Type I [$15, Th. 1,(4)]. In the converse direction, we show in Theorem 1 below that every Type I Baer*-ring with PC possesses such a family (h,); the incisive result in this direction is as follows: Proposition 3. Let A be u Buer *-ring with PC, and suppo~rA c,ontuin.s a nonzero abelian projection e (that is, A is not continuous). Then A has u homogeneous direct summand. More preciselev, there exists a nonzero centrul projection h and a homogeneous partition of 11 having he us one of its terms (thus lz A is homogeneou.~). For clarity, we separate out two lemmas.

Lemma 1. If A is u Baer *-ring of' Type I with PC, then every nonzero projection in A contains a nonzero abelian projection. Proof. Let J' be a nonzero projection, e a faithful abelian projection. Then C(e)C(,f)= C(f )# 0, therefore eAf # 0 [$6, Cor. 1 of Prop. 31; in view of PC, there exist nonzero subprojections e, 5 e, jo i fsuch that e, f, [$14, Def. 31. Since fb 5 e, f, is abelian 1515, Prop. 71. 1

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Lemma 2. IfA is a Baer *-ring qf Type I with PC, and f is any faithful projection in A, then f contains a,faithful abeliun projection. Pro($ Let (f,),,, be a maximal family of nonzero abelian projections such that f ; I f and C(f,) C(,f,) = 0 when L # x (get started by Lemma 1).Set f f = s u p f , ; f ' is abelian [tj 15, Prop. 81, thus it will suffice to show that C(,fl)=I.Set h = 1 -C(fl) and assume to the contrary that h #O. Since f is faithful, h f#O. By Lemma 1, h f contains a nonzero abelian projection f,; since C(f,) 5 C(hJ') 5 lz, C(f,) is orthogonal to every C(f,), contrary to maximality. I

114

Chapter 3. Structurc Theory of Haer *-Rings

Proof of' Proposition 3. The direct summand C(e)A obviously has PC, and e is a faithful abelian projection in it; dropping down to C(e)A, we can suppose A is of Type I and C(e)= 1. Set e , = e and expand {el) to a maximal orthogonal family (e,),,, of faithful abelian projections. Set f'=supe,. By maximality, I- f does not contain a faithful abelian projection; by Lemma 2, I-f is not faithful. Setting h = I - C ( l -f'), we have h # 0 and h(l -,f) = 0. Thus

If r # x then, since C(e,)= C(e,) = 1, we have e,- e, by Proposition 1, (ii), therefore he,- he,; then (*) shows that (he,),,, is a homogeneous partition of h with h el = h e abelian. I In a ring of Type I, Proposition 3 can bc pursued to exhaustion:

Theorem 1. I f A is a Baer *-rinq of Type I with PC, there e x i ~ t san orthogonal family (h,) of nonzero central projection.c such that sup ha = I and every h, A is homogeneous. Proof. Let (ha) be a maximal orthogonal family of nonzero central projections such that every h, A is homogeneous (get started by Proposition 3). Set h = I- suph, and assume to the contrary that h # 0; then hA is also a Baer *-ring of Type I with PC, and an application of PropoI sition 3 to it contradicts maximality. Attached to each homogeneous summand Iz,A there is a cardinal number, namely, the cardinal number of the index set of a homogeneous partition of ha with abelian terms. As remarked in Definition 1, no uniqueness is claimed for this cardinal number; nevertheless, homogeneous summands with same cardinal number can be lumped together:

Lemma. If A is a Baer *-ring and (Ii,),,, is an orthogonalfamily of central projections such that every h, A is homogeneous of order N,then (sup h,)A is also homogeneous of order N. Proof: For each a, let (e,,),,, be a homogeneous partition of h, with the e,, abelian and card 1 = N. For each 1 E 1, define

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The e, are abelian [§ 15, Prop. 81 and mutually orthogonal, r , e, for all r and x [$11, Th. I], and C(e,)= sup, C(e,,) = sup, h,.

I

We spread out the applications so as to avoid the pitfall indicated following Definition 2.

5 18. Rings

of Type 1; Homogeneous Rings

115

Theorem 2. Ij'A is ufinite Baer *-ring qf'Type 1 with PC, tl~ercexists a .scJquence h, of' central prqjections suc.1~that (i) the h, ure ortlzogorzal, (ii) suplz,= 1, and (iii) h,A is either O or o f ' Type I,. I f ; in addition, A hrrs GC, then condition (i) is redundant and the h, are unique. Proof. If hA is a homogcneous dircct summand of A, then the order of IzA must be finite [$17, Prop. I]; since h A is a finite Baer *-ring, the conditions of Definition 2 are met, that is, hA is of Type I, for some positive integer n. An application of Theorem 1, and a lumping togethcr of the various orders via the lemma, produces the desired sequence h,. Suppose, in addition, that A has GC. Let h be any central projection h,,. In such that h A is homogeneous, say of order m. We assert that h I other words, if n # m we assert that hh,=O; if, on the contrary, 1211,, wcrc nonzero, then the summand h h, A would be homogeneous of both orders m and n, contrary to Proposition 2. {Thus, h, may be described invariantly as the largest central projection k such that hA is homogeneous of order m (modulo the possibility that h =O).f The final assertion of the theorem follows at once. I

Remark. Let A be a Baer *-ring of Type I with PC, and write A as the direct sum of a finite ring and a properly infinite ring [$15, Th. I]. Theorem 2 may be applied to the finite summand. The properly infinite summand can be decomposed via Theorem 1; as warned following Definition 2, finite ordcrs could conceivably occur hcrc-but not whcn A has GC and satisfies the parallelogram law (P). Theorem 3. Let A be a Baer *-ring o f Tvpr 1 with PC. Conside/. cardinal numbers N I card A. There exists an orthogonal family (h,),. A of central projections such that (i) sup h , = 1, and (ii) h , A is either 0 or honzogeneous of order N. I f ; in addition, A sati,Sfies the parallelogram law (P), then, for N finite, the h , are uniquely determined. Proof. {The choice of card A is not critical; it is simply a convenient upper bound for the cardinal numbers that can occur.) The existence of a family (12,) satisfying (i) and (ii) is immediate from Theorcm 1 and the lemma to Theorem 2. If, in addition, A satisfies (P), then A has GC [$14, Prop. 71. It follows that if K is finite, then h , A is finite (Remark 4 following Definition 1); thus, setting h=sup {h,: K finite), we conclude that hA is finite [$15, Exer. 41. On the other hand, if K is infinite then h , A is properly infinite; for, if (e,),,, is a homogeneous partition of h,, with the e, abelian and cardl = K , and if k is any nonzero central projection in It, A , then (kc,),,, is a homogeneous partition of k with infinitely many terms,

116

Chapter 3. Structure Theory of Bacr *-Rings

therefore k(h, A) = kA is infinite [rj 17, Prop. 11. Since I- h = sup {h, : K - h) A is properly infinite [rj 15, Exer. 41. Thus infinite), it follows that (I A=hA+(l -h)A is the unique central decomposition of A into finite and properly infinite summands described in [915, Th. I]. The final assertion of the theorem now follows from Theorem 2 applied to hA. I We cautiously augment the notation of Definition 2:

Definition 3. If A is a homogeneous Baer *-ring of order N, N infinite, we say that A is of Type I, (but we do not imply that K is uniquely determined by A). [Such a ring is of Type I (Remark 1 following Definition 1) and is properly infinitc (see the proof of Theorem 3).} Remarks. 1. Suppose A 1s a homogeneous Baer *-ring. If A is finite, If A is infinite, it is then it can only have finite order [$17, Prop. I]. conceivable that A has both finite order and infinite order (see the warning following Definition 2); to put it another way, it is conceivable that A is not of Type 1, for any N, finite or infinite. However, it is a feature of Definition 2 that A can't be both Type I, (n finite) and Type I, (N infinite). 2. If A is a Baer *-ring of Type I, with GC and satisfying the parallelogram law (P), Theorem 3 provides a 'decomposition' of A into rings of Type I, (K I cardA). This result is applicable to a Baer *-ring of Type I satisfying the (EP)-axiom and the (UPSR)-axiom [rj14, Th. I] (see also [rj 14, Exer. 5]), in particular to an AW*-algebra of Type I. Better yet, for AW*-algebras the 'central direct sum' alluded to in Theorem 3 is an honest C*-sum [$lo, Prop. 21, and we have proved the following theorem: Theorem 4. If A is an AW*-algebra of Type I, then A is the C*-sum of a family of homoyeneous A W*-algebras. In detail, there exi,\ts an orthogonal family (h,) ,,,,,, oj central projections such that (i) suph, = I,and (ii) h, A is either 0 or of Type I,; thus A is the C*-sum of homogeneous algebras h, A. For Nfinite, the h, are uniquely determined.

,

We conclude the section with miscellaneous results for later application. The first is a sharpening of Proposition 2, part (2):

Proposition 4. If A is a Baer *-ring of Type I, (n finite), with PC, then A does not contain n + I orthogonal, equivalent, nonzero projections. Proof By hypothesis, A is finite and homogeneous of order n (Definition 2). Let el, ..., en be a homogeneous partition of 1 into abelian projections.

# 18. Rings of Type I; Homogeneous Kings

117

Assuming J;, ...,fi+ are orthogonal, equivalent projections, it is to be shown that j;=O. It will suffice to show that e , A f ; =O. (This will imply that O= C ( e , )C ( j ; ) =C(f ; ) , therefore f ; =O.) Assume to the contrary that e , A f ; f0. By PC, there exist nonzero Since e , is abelian, subprojections e, 2 e,, f;,5 f; such that e, -f,. e,=he, for a suitable central projection h 1815, Prop. 61. Then, for each i = l , ..., n, we have thus he, 5 ,J and so h e, 5 h.f;. Then in view of finiteness, this implies h= h J; + ... + h f,, thcrefore h i n + ,=O. Then h f , -...- h f , h,f, + = 0, thereforc h = 0, a contradiction. I

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,

The next proposition is a powerful tool for reducing infinite considerations to finite ones; there are key applications to additivity of equivalence [$20, Prop. 51 and dimension in finite rings ([$28, Prop. I], [$29, Lemma 4, Prop. I], [$33, Th. 4]), and, in modified form, to matrix rings over finite Baer *-rings [$61, Lemma 31 :

Proposition 5. Let A he a finite Baer r-ring of' Tvpe I , with PC, and let (e,),,, be an orthogonal jumily qf'projections in A. There exists an orthoyonal ,family (h,),,, of' nonzero central projections with sup h, = I , such that, for each ~ E A the , set { L E I :h,e, # 0 ) is,f'inite. Proof: By an obvious exhaustion argument, it suffices to find a nonzero central pro-jection h such that he, = O for all but finitely many I . In view of Theorem 2, we can suppose A to be of Type I,, n a positive integer. Assume to the contrary that no such h exists. Applying [$ 14, Prop. 91, we may construct n + I nonzero, equivalent, orthogonal prqjections (recall that the el are orthogonal); this contradicts Proposition 4. 1 The final result is for application in [$20, Prop. 51 (see also Exercise 5):

Proposition 6. Let A be a Barv *-ring of Type I , with GC and satisfying theparallelogvam law ( P ) , andassume that A has no aheliun direct summand. I f e is any abelian projection in A, then e 5 1 - e. Proof. By Theorem 1, there exists an orthogonal family (h,) of nonzero central projections such that sup ha= 1 and each h a A is homogeneous; moreover, by hypothesis, the order of h,A is 2 2 (possibly infinite). It will suffice to show that h , e s h,(l e ) for all a [$ 11, Prop. 21. Dropping down, we can suppose that A is homogeneous of order 2 2.

118

Chapter 3. Structure Theory of Bacr *-Rings

Let f'be a faithful abelian projection, f 5 1 - f (Definition 1 and Remark 1 following it). Since e is abelian and f' is faithful, we have e s f by ,f; since abelian projections the corollary of Proposition 1. Say e g I are finite [# 15, Prop. 41, it follows that I- e I -g [# 17, Prop. 51. Then

-

es,f
-

-

thus e 5 I -e. {Note that when A is finite, the equivalence 1 - e I- g is a consequence of GC [$17, Prop. 41; in other words, (P) can be omitted I when A is finite.) Exercises

1A. Let A be the Baer *-ring of all 2 x 2 matrices over the field of three elements, and let e,f be the projections described in [$I, Exer. 171. Then e,f arc faithful, abclian (cvcn minimal) projections--in particular, A is a finite Bacr *-factor of Type 1,-but e is not equivalent toj: (Cf. Proposition 1.) 2A. If A is a Baer *-ring of Type I with PC, and e is any nonzero prqjection in A, then (i) e is the supremum of an orthogonal family of abelian projections, therefore (ii) eAe is of Type I. (Cf. [$17, Exer. 191.) 3A. Let A be a Baer *-ring with PC, and suppose e is an abelian projection in A. If f'is any projection such that C(f ) 2 C(e), thcn f'contains an abelian projection fo such that C(,fo)=C(e). (Cf. the sharper conclusion of the corollary of Proposition 1 when A has GC.)

4A. Let A be a Baer *-ring of Type I with PC. If e is any projection in A, there e and C(f )= C(e); if, in addition, A has exists an abelian projection f such that GC, then f'is unique up to equivalence.

j's

5A. Let A be a Baer *-ring satisfying the conditions of Proposition 6. If e, f are abelian projections in A, then e s 1 6A. Let A be a *-ring such that every nonzero right idcal contains a nonzero projection. (For example, the (VWEP)-axiom 157, Def. 31 is sufficient.) A nonzcro projection e in A is minimal if and only if eAe is a division ring. 7C. If A is an A W*-algebra of Type I whosc centcr Z is *-~somorphicto a von Ncumann algebra, then A is also *-isomorphic to a von Neumann algcbra. 8C. If A is an A W*-algebra of Type I, with center Z, and if B is a Type I A W*subalgebra of A such that B 3 Z, thcn B = B". 9C. Every A W*-algebra of Type I may be represented as the algebra of all bounded module operators on a suitable ' A W*-module' (thc latter being a generalized Hilbert space whose Inner product takcs values in a commutative A W*algebra). 10C,D. A *-ring H is said to be orthoseparuble if every orthogonal family of orthosepavahl~if there exists an nonzero projections in B is countable; it is Io~ull~v orthogonal family (e,) of projections with sup e,= 1, such that every e,Be, is orthoseparable. (i) Every von Neumann algebra is locally orthoseparable.

$19. Divisibility of Projections in Continuous Rings

119

(ii) If A is an AW*-algebra whose center Z is locally orthoseparable (equivalently, Z is the C*-sum of a family of orthoseparable commutative AW*-algebras), and if A is homogeneous of order K, then K is unique. (iii) If a von Neumann algebra is homogeneous of order K, then N is unique. Thus the term 'Type I,' is completely unambiguous in the context of von Neumann algebras. (iv) Problem: Can local orthoseparability be omitted in (ii)? 11C. Let A be a finite A W*-algebra of Type I, having no abelian summand. If s is any self-adjoint (or skew-adjoint) element of A, there exist orthogonal projectionse,f;gin{x}'such that e + f + g = I , e - l a n d g s r .

12D. Problem: If '4 is a homogeneous Baer *-ring of order n, n a positive integer, is A finite? 13A. Let A be a finite Baer *-factor of Type 1, with PC. (i) All nonzero abelian projections of A are minimal and are mutually equivalent. (ii) Evcry orthogonal family of nonzero projections in A is finite. (iii) Every nonLcro projection in A is the sum of finitely many orthogonal, minimal projections. (iv) Any two projections e,f in A arc comparable (e 5f' or 1.5e). (v) A is of Type I,, n unique; explicitly, if e is any minimal projection in A, then eAe is a *-ring without divisors of zero, and A is *-isomorphic to (eAe),. 14A. (i) If A is a Baer *-factor with finitely many elcmcnts and with 'orthogonal e), then A = K,, for a suitable positivc f or comparability' (ej'=O implies integer n and a finite involutive field K (i.e., a Galois field with a distinguished

es

n

1

automorphism k - t k* such that kT k , = O implies k , = ... = k, =O). (Included 1 is the ring of Exercise I. j (ii) If A is a Baer *-ring with finitely many elements and with orthogonal GC, then A is the direct sum of finitely many matrix rings of the sort described in (i). In particular, A is semisimple (even *-regular). 15A. If A is a Baer *-ring of Type I, (n finite) with PC, then n is unique. 16A, D. Let A be a Baer *-ring of Type I with PC, and let e, f be faithful abelian projections in A. (i) There exist equivalent partitions of e,,f. More precisely, there exists an orthogonal family (ha) of central projections with sup ha= 1, such that h,e h,f' for all a. (ii) Problem: Is e f'? Does it help to assume that A is of Type I, (n finitc)?

-

-

17A. (i) If B is a *-regular ring [$3, Exer. 61 and e is a minimal projection in B, then eBe is a division ring. (ii) If B is a regular Baer *-factor of Type I,, thcn B is *-isomorphic to D, for a suitable involutive division ring D.

5 19. Divisibility of Projections in Continuous Rings Recall that a Baer *-ring is said to be continuous if it contains no abelian projection other than 0. By 'divisibility' of projections, we mean the following:

120

Chapter 3. Structurc Theory of Baer *-Rings

Theorem 1. Let A he a continuous Baer *-ring with PC. If e is any nonzero projection in A and n is any positiz~einteger, there exists a homogeneouspartition of e of length n; that is, there exist orthogonal projections e ,,..., en such that e=e,+...+e, and e l - e 2 - . . . - e ,. The theorem has key applications to additivity of equivalence [$20, Prop. 41, dimension in finite rings [$26, Prop. 151, reduction of finite rings (cf. [$ 36, Prop. I],[#37], [$41, Th. I]), and matrix rings over finite rings (cf. [# 591, [# 60, Lemma I]). We approach the proof through two lemmas, the first of which reformulates nonabelianness:

Lemma 1. Let A he a Baer *-ring with PC, and let e be a projection in A. The following conditions on e are equivalent: (a) e is not ahelian; (b) there exists a pair e', d' of orthogonul, nonzero subprojections of e SUCII that e' e".

-

Proof: (a) implies (b): Since eAe is not abelian, there exists a projection f < e such that f ' # e C ( f ) [#15, Prop. 61, thus C (f ) ( e - f ) # O ; then C(f ) C ( e - f ) # 0 , therefore f A ( e - f ) # 0 [#6, Cor. 1 of Prop. 31. In view of PC, there exist nonzero subprojections e' 5.f;e" I e-,f such that e' e". (b) implies (a) in any *-ring A. For, suppose e', e" are orthogonal subprojections of e such that e' e" (in A). Then e' e" in eAe [$I, Prop. 41. If eAe is abelian, then e', e" are central projections in eAe, therefore el=e" [$ I,Exer. 151; since e', e" are orthogonal, it follows I that er=e"=O.

-

-

-

When e contains no abelian projection other than 0, we may continue inductively :

Lemma 2. Let A be a Baer *-ring with PC, and supposc e is a nonzero projection whose only abelian subprojection is 0. Then, given any positive integer n, there exists a homogeneous partition of lenglh n whose lerms me I e. Proof. It is sufficient to consider n = 2" ( m= 0, 1, 2, 3, . ..). For m = 0 there is nothing to do. Assume all is well with m- I.By Lemma 1, there exist orthogonal nonzero projections e', e" such that e'+e" ~e and el- e". Since e' contains no abelian projection other than 0, the induction hypothesis applied to e' yields 2"-' orthogonal, equivalent, nonzero subprojections of e'; the equivalence e' e" induces an equivalent partition with 2"- ' terms I e". I

-

Proof of Theorem I . Consider nplea of equipotcnt families

# 19. Divisibility of Projections in Continuous Rings

121

of projections such that (a) the e,, are nonzero subprojections of e, (b) eirej,=O if i # j or I # X , and (c) for each L E I , e,,-e,,-...-e nr' Lemma 2 ensures that such an nple exists with I a singleton. We can suppose, by Zorn's lemma, that the family (*) cannot be enlarged (by increasing the index set I), i.e., that it is maximal in the properties (a)-(c). Define e,=sup{e,,: L E I } ( i = l , ..., n ) ; then e l , .. ., en are mutually orthogonal, nonzero subprojections of e, and e , ... en by orthogonal additivity of equivalence [$11, Th. I ] . Setting f = e l +-..+en, it suffices to show that f = e ; indeed, e - f # 0 would imply that e - f contained some homogeneous partition of length n (Lemma 2), which, adjoined to (*), would contradict maximality. I

- -

Corollary. If A is a continuous Baer *-ring with PC, and n is any positive integer, there exists a projection g in A such that A is *-isomorphic to the ring (gAg), of all n x n matrices over gAg. Proof. Applying the theorem to e= 1, the corollary follows at once {We remark that gAg is also a continuous Baer from [lj 16, Prop. I]. *-ring [cf. $ 15, Prop. 51 and obviously has PC.) I Exercises

1A. If A is a finite Baer *-ring of Type 11, with GC, then the projection g of the Corollary is uniquely determined by n up to equivalence, and the ring gAg is unique up to *-isomorphism. 2.4. Let A be a Baer *-ring with PC, having no summand of Type I,, [cf. 9:15, Th. 31. (i) If n is any positive integer, there exists a projection e in A such that A is *-isomorphic to (eAe),. (ii) If A is a complex *-algebra, then the reduced ring A" [$3, Exer. 181 is a *-subalgebra of A. 3

Chapter 4

Additivity of Equivalence

5 20.

General Additivity of Equivalence

We take up again the theme considered in Section 11. Throughout this section, A is a Baer *-ring (satisfying various axioms as needed). We assume given a pair of equipotent families of projections (er)rtl,( f;)lsl such that (i) the e, are orthogonal, (ii) the f; are orthogonal, and (iii) el j; for all 1 E 1. We write

-

e = sup e,,

f =sup 1;

Thus, (el),,,, (j;),,, are equivalent partitions of e, ,j' [$ 17, Def. 11. For each L € 1 , we denote by w, a fixed partial isometry such that w: w, = e,, wIWT = j ; . Our aim in this section is to prove the following results (Theorem 1): (1) If A has GC, then e f'; (2) If, in addition, A has no abelian direct summand, then the partial isometries w, are addable, in the sense that there exists a partial isometry rv such that w* w=e, w w* =f (hencc p f ) and we, = w,= f;w for all i E I (cf. Exercise 4); (3) If A is any A W*-algebra, then the partial isometries w, are addable. The plan of attack is to use the structure theory developed in Sections 45-19 to reduce the problem to various special cases. The case that A is abelian is pathological: all projections are central. equivalent projections are equal (therefore e = f ) , and something drastic (e.g., spectral theory in a commutative A W*-algebra) is needed to add up the partial isometries. In all other cases, equivalence counts for something, and the core result is the one proved in Section 11, which we repeat here for convenience:

-

-

Proposition 1. If e f

= 0,

then the partial isometries w, are addable.

More generally, it suffices that e be equivalent to a subprojection of 1 -,f:

Proposition 2. If e 5 I -f ; then the partial isometries w, arc. addable.

S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

5 20.

123

General Additivity of Equivalence

Proof. Let u be a partial isometry such that u* u=e, uu* =f' 5 1 -f. Defining f,' = ue,u*, we have a partition (f;'),,, of f ' [$I, Prop. 91, and f i -f;' via the partial isometry v,=u w:. By Proposition 1, there exists a partial isometry v such that v* 7 1 = , f ; vv* = f " and v,f,=v,=,f;' v for all 1. Then w = v* u is a partial isometry with w* w = e, w w* =f , and it is routine to check that we, = w, =J; w. I Proposition 3. If A is a properly infinite Baer *-ri~zgwith orthogonal GC, then the partial isometries tv, are addable.

-

-

-

--

there exists a projection y such that g 1 1 -y. Proof. By [$ 17, Th. I], Since 1 y we have e e' 5 y for suitable e'. Similarly, 1 1 - g yields f f ' 5 1 -g forsuitable f '.Then e e' 2 g 5 I-, showsthat I' e 5 1 -j". The equivalence f'- f ' induces a partition (f;'),,, of f ' with f , -,f,'. Then el f , f,', and the composcd equivalences e, f,' can be added via Proposition 2. Say v*v=e, vv*= f ' , with ue,=,f,'v the partial isometry implementing e, -f;'. Composing u with the partial isometry implementing f ' - , f , we get a partial isometry w that 'adds' the w,; the routine details are similar to those in Proposition 2. 1

-

-

--

-

A lemma preparatory to disposing of the continuous case: Lemma 1. If h , , . .., h, are orthogonal centrul projections with h , + ... + h,= 1, such thatfor euchj the partial isornetries h j w,are addable, then the w, are also addable. Proof: A routine application of finite addability [$I, Prop. 81.

1

Proposition 4. If' A is a continuou,~Baer *-ring with GC, then the partial i.rometries w, are addable. Protf. Since GC implies PC [$ 14, Prop. 21, projections in A are divisible in the sense of [$19, Th. I]. For each 1 E I write

(1)

el = e: + e','

with el

- ey

(via a partial isometry unrelated to the given partial isometries w,). The partial isometry w, induces a decomposition

(2)

with

f;=.f,'+.f;"

where (3)

e:

(4)

ey

-j;' -

f;"

via

- f;",

1;' MI,ei

,

via w,ey ,

124

-

Chapter 4. Additivity of Equivalence

(and f;' J;" is implemented by a partial isometry in whose geneaology we are not interested). Define e' = sup e:,

e" = sup e:

Since c'e" =0, we infer from (1) and Proposition 1 that

- e"

(5) and similarly

e'

(6)

,f'- f " .

,

We assert that the partial isometries w,e: (see (3)) are addable. To this end, let h be a central projection such that (7)

h e ' s hf ' ,

(8) (I - h ) , f r s (I -h)et. From (7) and (6), we have

he'5h.f'-IzfU
Lemma 2. Suppose that,for each 1 E 1, (e,,),,,, is a partition of' e,. Let (f;,),,,, be the equivalentpavtition of',f, induced by MI,, that is, f i x = w,e,, w:; thus, e,,- f,, via the partiul isometry w,, = w,~~,,.If the doub1.y indexed fumily ofpartial isometries w,, is uddable, then the w, are also addable. Proof. Let w be a partial isometry such that w* w = sup,. ,el, = sup,(sup, el,) = sup e, = e ,

5; 20. General Additivity of Equivalence

225

Ax = f, and welx= w,, = f,, w for all i,x. It is routine to w w* = check that we,= w,= f ; w for all 1 ; for example, we, = w, results from the fact that (we,- w,)e,, =O for all % € K t . I Lemma 1 generalizes to arbitrary central partitions

Lemma 3. If (h,),,, is an orthogona1,familj~qfcentrul prqjections with sup h, = I, such that the doubly indexed fumily qf partial isometries w,, = h, w, is addable, then the w, are also addable.

-

Proof. Set e,,= hue,; for each 1 E I, (e,,),,, is a partition of e,. Set f,, via the partial isometry w,e,, = h, w, = w,,. By hypothesis, the w,, arc addablc, therefore the w, arc addablc by Lemma 2. 1

,fm, = w,e,, w: = h, f,; thus e,,

The most complicated case is properly nonabelian Type Ifin: Proposition 5. Let A befinite, cf Type I, with no abelian direct summand, and assume that A has GC. Then the partial isometries w, are udduble.

Proof. Each el is the supremum of an orthogonal family of abelian projections, by an obvious exhaustion argument based on [fj18, Lemma 1 to Prop. 31. In view of Lemma 2, we can suppose, without loss of generality, that the el (hence the f,) are abelian. {We remark that this is a way of losing countability of the index set.) Let (h,),,, be an orthogonal family of nonzero central projections with suph,= 1, such that, for each a, the set is finite [$18, Prop. 51. By Lemma 3, it suffices to show that the doubly indexed family of partial isometries w,,=lz, w, is addable. Only the nonzero w,, need be considered. For a fixed ~ E A consider , I,= {i,, . .., 1,); thus Iz,ell, ..., h,eln are precisely the nonzero hue, (for this a). By [$ 6, Prop. 51, there exists a finite central partition of h,, say k,, .... k,. such that for each pair of indices j, v , either kj(h,eLy)=O or kj(h,eLv)is faithful in kj(h, A), that is (since kj 5 h,), either kjel,,T O or kje,,, is faithful in kjA. We make mental note of the partial somet tries k wL (k= kl, ..., k,: L = il, .. ., L,,). D o this for each ~ E A In . view of Lemma 3 (or 2), it is sufficient to show that the partial isometries k w, so obtained are addable. Revising the notation, we can suppose that the h, already have the property that h,e, is either 0 or faithful in h, A. Summarizing, we are in the following situation: the el (and ,f;) are abelian; (h,),,, is an orthogonal family of nonzero central projections with sup h, = 1; for each a, the set 1, = { I E I : h, el # 0) is finite; for each pair of indices a, 1, either h,e,=O or h,e, is faithful in h, A. It is to be

126

Chapter 4. Additivity of Equivalence

shown that the partial isometries w,, = ha WJ, are addable. Write J={(u,L):aeA, LEI,). We will show that J can be partitioned into three disjoint subsets, J=J'uJ~uJ~,

(*)

such that for each t (t = 1. 2, 3) the family of partial isometries w,, ((a, L)EJf)is addable; in view of finite addability [$I, Prop. 81, this will complete the proof. For each EEA, let n(a) be the number of elements in I,, and write I,= { L , ..., , L,(,,). (Strictly speaking, L, should be indexed to indicate dependence on a, but enough is enough.) We know that hap,,, ..., h , ~ , ~ ~ ~ , arc faithful abclian projections in h,A; citing [$18, Prop. I], we have hap,, -h,eLLN. - .-h " eI (1) Similarly (or therefore) (2)

haL,

- - ... ha J;,

~

('~

)

h,.f;,,(a,.

{Note that the partial isometries implementing (I), (2) have nothing to do with the given partial isometries w,.) Partition I, into three disjoint subsets, I,= 1; u 1: u 12 , as follows: if n(a) =0, that is, if 1, = a , set I,' = 1; = 12 = @; if ~ ( c I )is an even integer, partition I, into two subsets I,', 1: each with n(a)/2 elements, and set 1: = a ; if n(cc) is an odd integcr, sclcct I: to be any singleton in I,, and partition 1, - 12 into two subsets I,', 1: each with (n(a)- 1)/2 elements. In all cases, card 1: =card 1:

and card 12 5 I.

Define eL=sup(h,e,:1~1i)

(t=1,2,3),

a;

with the understanding that e: = O whcn 1; = the suprema are in fact finite sums, and the sum defining e; has at most one term. Similarly, dcfine f:=~~pjh,e,:r~I:} (t=1,2,3). Note that

and similarly (4)

h, f = j: +j;2+,f2.

5 20.

General Additivity of Equivalence

From (I), and the equipotence of I:, I:,

finite additivity yields

similarly,

fb -f,".

(6)

-

(The equivalences (5), (6) have nothing to do with the w,.) Also, since h,e, h, J, (via h, w,), finite addability yields

(7)

e,1-

(8)

e:-f,Z

Define p1

fa1

=supneb,

j l = s u p , f , ,1

( v i a 1(h, w,:1 E I:)), (vla~{Iz,w,:~~~~)). eZ=sup,e,2,

e3 =supme:,

f2=sup,ja2,

f3=supaf:.

From (3) and (4), we see that e=e'+e2+e3,

f=f1+S2-tf3

by the associativity of suprema. By orthogonal additivity of equivalence (Proposition I), (5) yields similarly, (6) yields {The equivalences (9), (10) have nothing to do with the w,.) The desired partition (*) is obtained by defining

Checking the definitions, we see that (11)

el = sup (h, c., : (a, 1) E J' ) ,

(12)

f =sup ( h , f,: (a, L ) EJt],

for t=1,2,3. We assert that the partial isometries w,, ((a, L)EJ')arc addable (hence r' f I ) . To this end, let h be a central projection such that

-

(13)

he'shl",

(I - h ) f l (14) From (13) and (lo), we have

5 (I -h)el

Chapter 4. Additivity of Equivalence

128

thus h el 5 1 - h f" ; noting the formulas (1I), (12) with t = 1, it follows from Proposition 2 that the partial isometries hw,, ((a, L ) EJ') are addable. Similarly, it follows from (14) and (9) that the partial isometries (1 - h) w,, ((a, L)EJ') are addable. In view of Lemma 1, the assertion is proved. Similarly, the partial isometries w,,((a, L)EJ2) are addable. It remains to consider J3. Note that each 6.5 is an abelian projection (recall that card 1; = 0 or I), thus e3 is the supremum of a very orthogonal family of abelian projections; it follows that e3 is abelian [tj 15, Prop. 81, and similarly f 3 is abelian. We assert that (15) e3- f 3 . Since e3,f are abelian, it suffices to show that c ( e 3 ) = c ( , f 3 ) [$ 18, Prop. 11. Indeed, for any (a, i )J ~ we have hue, h,J;, hence C(h,e,) = C(h,f,); in particular, C(e,3)= C(f:) for all a (recall that e: is either 0 or one of the h,e,), therefore

-

C(e3)= C(sup, e:)

= sup,

C(e;) = sup, C(f:)

= C(f 3,

.

Thus (15) is verified (but the equivalence has nothing to do with the w,). On the other hand, since A has no abelian summand, the abelian projection f satisfies f 3 5 1 -f 3 (see the remark at the end of the proof of [§ 18, Prop. 61); combining this with (1 5), we have (16)

e351-f3.

The equivalence in (16) has nothing to do with the w,, but that does not matter; noting the formulas (II), (12) with t=3, it follows from Proposition 2 that the partial isometries w,, ((a, L ) EJ3) are addable. I The abelian case is half trivial, half pathological:

Proposition 6. If A is ahelian then e = f ; hut the partial isornetries w, may fail to be addable. Proof. Since all projections in A are central, el-f; e,=,fi, thus e=supe,=sup f,= f'. An example of nonaddability is given in Exercise 4.

reduces to

1

An abelian AW*-algebra is commutative (Remark 2 following [$15, Def. 2]), and addability is rescued by spectral theory:

Proposition 7. If A is u commutative AW*-algebra, then the partial isometries w, are addable. Proof. We know that w: w, = w, wT = e,=,f,. Adjoining e,=,f, = w, = I - e to the family, we can suppose e = f = l . We seek a unitary

6 20.

General Additivity of Equivalence

129

element w such that we, = w, for all 1. Since 11 w,ll< I for all 1 E I, the existence of w follows from the fact that A is the C*-sum of the e,A [$lo, Prop. 21. 1 Let us gather up all the threads:

Theorem 1. Equivalence is additive in a Baer *-ring A with GC. Explicitly, let each of' (e,),,,, (j;),,, be an orthogonal family ofprojections, such that e, ,l; ,for all 1 E I, let e = sup el, f = sup f ; and, ,fur each L E 1, let w, be a partial isometry such that w: w,= e,, MI,w: =,/;. Then: (i) e f . (ii) If; moreover, A has no abelian direct summand, then the partial isometries w, are addable. (iii) I f A is any AW*-algebra, then the partial isometries w, are addable.

-

-

Proof. The idea is to break up A into a finite number of manageable direct summands, using Lemma 1 to put the summands back together. The needed central projections are provided by successive applications of the coarse structure theory [515, Th. I ] , as follows. Let h , be a central projection such that

(1) h,A is properly infinitc and (1 -h,)A is finite. Let h, I-h, be a central projection such that (2)

h,A is continuous

1 - h , - h, (and finite) and ( I - h , - h,)A is Type I and finite. Let h, I be a central projection such that h, A is Type I, finite, with no abelian summand, (3) and such that, on setting h, = 1- h , - h, - h,, (4)

h,A is abelian

(2), (3) are covered by Propositions 3, 4, 5, The summands (I), respectively; in these cases, the partial isometries w, are addable, which proves (ii). When the summand (4) is present we may apply Proposition 6 to it, concluding only that e - f , and this proves (i); but in the AW* case, Proposition 7 is available, so the partial isometries are addable on the summand (4) also, and this proves (iii). I An application (see also Exercise 6):

Corollary. Parts (i) and (ii) of Theorem 1 hold when A is a Baer *-ring satisfying the (EP)-axiomand the (UPSR)-axiom. Proof. A has GC [$ 14, Th. I ] .

I

Chapter 4. Additivity of Equivalence

130

The following result is a remarkable application of Theorem 1 to the comparative anatomy of the axioms (see also Exercise 12): Theorem 2. The following conditions on a Baer *-ring A are equivalent: (a) A has GC; (b) A has PC and equiualence in A is additive. Proof: (a) implies (b): Theorem 1 and 1914, Prop. 21. (b) implies (a): This is [$ 14, Prop. 41. 1 Another useful consequence of Theorem I( see also Exercise 2):

-

Theorem 3. If' A is a Baer *-ring with GC satisfying the (WEP)axiom, then LP(x) RP(x) for all x in A. Proof. .[We remark that in a von Neumann algebra, the conclusion comes free of charge with polar decomposition.) We can suppose x # 0. Write e = RP(x), f = LP(x). Let (e,),,, be a maximal family of orthogonal, nonzero projections such that, for each 1 E I, there exists y, E {x*x)" with x*x y: y, = el (the routine Zorn's lemma argument is launched by the (WEP)-axiom). Obviously el also belongs to the commutative ring (x*x)"; replacing y, by y,e,, we can suppose that y,e,= y,, that is, RP(y,) 5 e,. Since e, 5 RP(y,) results from x* x y: y, = el, we conclude that RP(y,) = e,. From ex* = x*, we see that e, 5 e for all 1. We assert that supel= e. Write g =sup el; at any rate, g 5 e, and g E jx* x)" 144, Prop. 71. To show that e l g it will suffice to show that x g = x . Assume to the contrary that x(l - g) #O. By the (WEP)-axiom, there exists yE([x(I -g)]*[x(l -g)]}"= {(I-g)x*xffl c (x*x)" such that (I - g)x* x y* y = h, h a nonzero projection. Then g h = 0, thus elh = 0 for all L E I , contradicting maximality. Define w, = xy, (1E I); then w: w, =yr x* x y, = x* xy: y, =el, thus w, is a partial isometry with initial projection e,. Let f i = w, w: =xy,y: x*; thus ,f;= LP(w,) 5 LP(x)= f for all 1. The j; are mutually orthogonal: if L #lc then .Lfx=(xy,yf x*)(xyXy: x*) =xb,y:)(x* x)CyXy:)x* =x e,(y, y:) x* = x e,(exy, y:) x* = 0 .

-

Quoting Theorem 1, we have e = supe, sup J; 5 f . (Thus e 5 ,f, that is, RP(x) 5 LP(x). Applying this result to x* yiclds f 5e, therefore e-,f by the Schroder-Bernstein theorem [§ 1, Th. I]. But the following argument is tidier, and its sharper conclusion is needed in the next section.) Finally, we assert that sup ,f,= f . Write k = I -sup 1;. Since k is orthogonal to every f;, we have O= kAx= k ( x y , y ~ x * ) . x = k x ( y ~ y , ) ( x * x ) = k x e ,

5 20.

General Additivity of Equivalence

131

for all 1 ; since supe,=e=RP(x), it follows that O= k x e = kx, hence kf=O, f 11-k=sup,f,. I An application (see also Exercise 6):

-

Corollary. If A is a Baer *-ring sati.7fying the (EP)-axiom and the (UPSR)-axiom, then LP(x) RP(x) for all x in A. Proof. A has GC [$14, Th. I].

I

This is a convenient place to record the following terminology

-

Definition 1. We say that a weakly Rickart *-ring satisfieer L P if LP(x) RP(x) for every element x.

-

RP

Exercises

1A. If A is a Baer *-ring such that the *-ring A, of all 2 x 2 matrices over A is also a Baer *-ring, then partial isometries in A are addable.

2A. If A is a Baer *-ring satisfying the (WEP)-axiom, then the following conditions on A are equivalent: (a) A has GC; (b) LP(x) -RP(x) for all x in A ; (c) A satisfies the parallelogram law (P). 3A. If A is a properly infinite Baer *-ring with PC, then partial isometries in A are addable. 4A. Let A be the *-ring of bounded complex sequences that are real at infinity [§4, Exer. 141; thus A is a commutative Baer *-ring satisfying the (EP)-axiom and the (UPSR)-axiom. For n = 1,2,3, ... let w, be the element of A whose nth coordinate is i = l / - l , all other coordinates 0; thus w,*w, = w, w,*= enhas 1 in thc nth coordinate, 0's elsewhere. The w, are not addable.

5D. (i) Problem: Is equivalence additive in every Baer *-ring? (Cf. [$ 11, Exer. 31.) {If the answer is yes, then PC and GC are equivalent conditions (see Theorem 2).) (ii) Problem: Is equivalence additive in every Baer *-ring with PC? {Same remark as in (i).} In view of Exercise 3, the question is open only for finite rings. (iii) Problem: Is equivalence additive in every Baer *-ring satisfying the parallelogram law (P)? 6C. The corollaries of Theorems 1 and 3 hold with (UPSK) wcakcncd to (SIC). 7C. If A is a Baer *-ring such that (i) A satisfies the (EP)-axiom, and (ii) partial isornetries in A are addable, then A also satisfies the (SR)-axiom.

8A. Let A be a Baer *-ring with GC, and let e,f be projections in A such that C(e) < C(J), eAe is orthoseparable, and fAf is properly infinite. Thcn e 5 f .

9A. Let A be an orthoseparable Baer *-ring of Type 111, with orthogonal GC. (i) If e is any nonzero projection, there exists a nonzero central projection h such that he-h. (ii) If e is any projection with C(e) = I,then e I. (iii) e C(e) for every projection e. (iv) e - f'iff C(e)=C(j').

-

-

132

Chapter 4. Additivity of Equivalence

(Of course (iv) implies (i)-(iii), but it is convenient to prove them in the indicated order.) The conclusion (namely, (iv)) remains true with orthoseparability replaced by the following condition: there exists an orthogonal family (h,) of central projections with sup he= 1, such that h,A is orthoseparable for each a. 10A. If A is an abelian Rickart *-ring, then LP(x)=RP(x) for all ~ E A . 11D. Let A be a Rickart C*-algebra. Problems: (i) Does LP(x)-RP(x) hold for all X E A? (ii) Are partial isometries NO-addable in A [3 11, Def. I]? (iii) Is equivalence KO-additivein A? (iv) ~ o eitshelp to assume GC?(Recall that (P) holds in every Rickart C*-algebra [$13, Th. I].)

-

12C, D. Equivalence in a Baer *-ring A is said to be centrally additive if, in the notation at the beginning of the section, e f whenever there exists an orthogonal h, for all r € 1 . family of central projections (h,),,, such that e,,j , are I (i) The following conditions on a Baer *-ring A are equivalent: (a) A has GC; (b) A has PC and equivalence is additive; (c) A has PC and equivalence is centrally additive. (ii) Problem: Is equivalence centrally additive in every Baer *-ring? In every Baer *-ring with PC?

5 21.

Polar Decomposition

The concept of 'polar decomposition' (or 'canonical factorization') has several axiomatic mutants. Before pinning down a specific definition, we exhibit three results of this genre.

Proposition 1. Let A be a Baer *-ring satisfying the (WEP)-axiom and the (WSR)-axiom,and assume that partial isometries in A are addable. Let X E A and choose r~ {x*x)" with x*x=r*r. Then: (i) There exists a partial isometry w such that x = w r, w* w = RP(x), w w* = LP(x). (ii) This factorization of' x is uniquely determined by r, in the sense that i f also x = vr with v* v =RP(x), then v = w. (iii) w*x=r. Proof. {Note that A has PC [ji 14, Prop. 31. If, in addition, A satisfies the parallelogram law (P), then it has GC 14, Prop. 71; in this case the addability hypothesis is redundant when A has no abelian summand [ji 20, Th. I].) (i) For brevity, write C = ( x * x ) " . We adopt the notation in the proof of [ji 20, Th. 31. In particular, x*xy:y,=e,, where y,, el are in C and y, el =y,. First, we revise the y, slightly. Since r* r y: y, = e,, with all elements lying in the commutative ring C, clearly re, is invertible in e,C, with invcrse z, = r* y: y, . Then z , e,~C c C and e, = (re,)z, = r z,, therefore

[s

5 21.

Polar Decomposition

133

r* rz:z,= (rz,)*(rz,)= el; thus z, has the properties of y,. Changing notation, we can suppose, in addition, that r y , = el. Write e=RP(x), f = LP(x), and set w,=x y,. As noted in the proof of [$20, Th. 31, e= supe, and the projections f,= w, w: are also orthogonal, with supf;=,f. By the addability hypothesis, there exists a partial isometry w such that w* w=e, ww* =f and we,= w,=,f;w for all I. Let us show that x = wr. For all 1, we have

therefore ( x- w r)e = 0; since e = K P ( x )= RP(r), we conclude that x-wr=o. (ii) If also x = vr with v* v = e , then 0 = w r -vr = (w- v)r, thcrcfore O=(w-v)e=we-ve=w-v. (iii) In the notation of (i), w* x = MI* w r = e r = I . I When unique positive square roots are available, the result is sharper (see also Exercise 11) :

Proposition 2. Let A he a Baer *-ring satisfying the (EP)-axiom and the (UPSR)-axiom, and assume that partial isometries in A are addable. Let X E A and let r be the unique positive square root of x * x . Then: (i) There exists a unique partial isometry w such that x = w r und w* w = R P ( x ) . (ii) In addition, w w* = L P ( x ) and w* x = r. (iii) If also x = v s with s 2 0 and v* v = R P(s), then .s = r and z. = w. Proof. (i),(ii) Let e= RP(x),J'=LP(x). By Proposition 1, we may writc x = wr with w* w=e, w w* =f , and w*x= r. The uniqueness assertion

follows from (ii) of Proposition 1. (iii) Suppose also x = us with the indicated properties. Then x * x = s v* v s = sRP(s)s= s 2 , therefore s = r by thc uniqueness of positive square roots. Then v* v = R P(s)= R P(r)= R P ( x ) , therefore t > = MI by the uniqueness assertion of (i). I

The next result is elementary, but instructive as to the algebra of such factorizations:

Proposition 3. Let A he a weakly Rickart *-ring, let ~ G A and , suppose there exists an element r e A such that x ~ A r , x*x=rYr

and LP(r)=RP(r)

Then one can write x = wr with w* w = R P ( x ) and w w* = L P ( x ) ; the partial isometry w is uniquely determined h,v r, and one has w* x = r .

134

Chapter 4. Additivity of Equivalence

Proof. Let e=RP(x), f = LP(x). Thus e =RP(x)= RP(x*x)=RP(rYr) =RP(r)= LP(r). Say x = a r . Then x=a(er)=(ue)r; setting w=ue, we have x = w r with we=w. We assert that w* x = r. Indeed,

therefore O=e(w*x-r)=(we)*x-er=w*x-r.. Next, w*w=e. For, 0 = w * x - r = w * ( w r ) - e r = ( w * w - e ) r , therefore O=(w*w-e)e=w*w-e. Since the involution of A is proper [ $ 5 , Prop. 31, it follows that w is a partial isometry [$2, Prop. 21, thus g = w w* is a projection; we now show that g =f. Since j'=LP(x) = LP(w r) 5 LP(w)=g, g -f is a projection ;moreover, (g -f ) w r = g x -f'x = x,- x = 0, therefore 0 = (g -f )we =(g- f)w; thus 0=(g- f)g=g-,f. If also x = vr with v* v=e, then v = w as in the proof of Proposition 1. 1 Motivated by Propositions 1-3, and the fact that LP(r)=RP(r) for every normal element r in a weakly Rickart *-ring, we define 'polar decomposition' as follows:

Definition 1. Let A be a *-ring. We say that A has polar decomposition (PD) if, for each ~ E A there , exists an element r such that ~E(x*x}", r*=r,

x*x=r2

and x ~ A r

(in particular, A satisfies the (SR)-axiom). We say that A has weak polar decomposition (WPD) if, for each ~ E A there , exists an element r such that r ~ { x * x ~ " ,x * x = r * r and x ~ A r (in particular, A satisfies the (WSR)-axiom). We may also speak of individual elements x having PD or WPD, even when not all elements of A do. Even weak polar dccomposition is a strong hypothesis:

Proposition 4. If A is a weakly Rickart *-ring with WPD, then (i) A satisfies LP -RP, and therefore (ii) A satisfies the parallelogram luw (P). (iii) If, in addition, A is a Buer *-ring, then A has GC. Proof. (i) The r of Definition 1 is normal (r* r = r r*), therefore LP(r) =RP(r); quote Proposition 3. (ii) See [$ 13, Prop. 21. (iii) See [$ 14, Cor. 2 of Prop. 71. 1

6 21.

Polar Uecornpos~t~on

135

Exercises

1C. If A is a Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom, and if A has no abelian summand, then A has PD. 2A. Every commutative *-ring with unity trivially has WPD 3A. The *-ring of bounded, complex sequences that are real at infinity [cf. $20, Exer. 41 has WPD but not PD. 4A. Let A be the C*-algebra of all compact operators on a separable, infinitedimensional Hilbert space, and let A , = { a + L l : a e A , L complex) as in [$3, Example I ] . Then A , satisfies the (UPSR)-axiom, but there exist elements ~ E A such that a $ ~ , ( a * a ) In f . fact, A , does not have WPD. 5A. In a *-ring with PD, every ideal is self-adjoint (i.e., is a *-ideal).

6A. If A is a *-ring with unity, satisfying the (WSR)-axiom, then every invertible element of A has a WPD. 7A. In a finite Rickart *-ring with WPD, y x = I implies x y = I.

8A. Suppose A is a Baer *-ring with W PD. Let X E A and choose rE jx*x)" with x ~ A rand x*x=r*r. One can write x = u r with u an extremal partial isometry [cf. fj14, Exer. 191. 9A. Let A be an A W*-algebra, X E A, Ilxll< I . There exist extrcmal partial isometries u', u" such that x=$u'+$u8'. 10D. Problem: Does every Rickart C*-algebra have PD? (Cf. 1520, Exer. 111.) 11C. If A is a Baer *-ring such that (i) A satisfies the (EP)-axiom, and (ii) partial isometries in A are addable, then A has PD.

Chapter 5

Ideals and Projections

3 22.

Ideals and p-Ideals

The dominant theme of this book is the interplay, in a *-ring A, between the ring structure and the set 2 of projections of A. For the ring structure, the ideals of A are subsets of central importance; how may the corresponding subsets of A be characterized? This is the question treated in the present section. Throughout this section, A denotes a weakly Rickart *-ring, with additional hypothescs as necdcd; thus 2 is a latticc 155, Prop. 71. A frequent hypothesis is the parallelogram law [$I31 :

e -e n f

(p)

-

euf

-

j for all projections c., j';

this holds, for example, in every weakly Rickart C*-algebra [ij13, Th. I]. Usually, the following stronger condition is needed: (LP

- RP)

LP(x)

- RP(x) for all x in A

This condition implies (P) 1413, Prop. 21; it holds if A has WPD [$21, Prop. 41, or if A is a Baer *-ring satisfying the (EP)-axiom and the (SR)axiom [520, Exer. 61. {Whether the condition holds in a Rickart C*algebra is not known; its validity would simplify some of the results in this section.) A Baer *-ring satisfying L P - R P has GC [ij 14, Cor. 2 of Prop. 71. We remark that if A has PD, then every ideal of A is a *-ideal; this is the case, for example, whcn A is an AW*-algcbra ([ij 20, Th. I], [ij 21, Prop. 2]), or when A is a Bacr +-ring satisfying the (EP)-axiom and the (SR)-axiom and having no abelian summand [$21, Exer. 11.

Proposition 1. Assume A satisfies the parallrloyram law (P). If 1 is any ideal in A, tlze set I" of projections in I has the followiny properties: (i) if ~ E I "and y < e , then g~ I"; (ii) if ~ E I and " g - e , then y ~ I " ; (iii) if e, f G ? then e u f €I". Proqf'. (i) g=ge. S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

# 22. Ideals and p-Ideals

-

137

(ii) If w is a partial isometry implementing g e, then w= e W E 1A c I, therefore g = w * w ~ A c I I. (iii) Suppose e, f ~ f By . (i), e - e n f ~ l ; then e u f - , f - e - e n f yields e u f - ~ E I by (ii), thus e u f = ( e u f - f ) + f ~ l . I

Definition 1. A nonempty subset p of A is called a p-ideal of A if it satisfies conditions (i)-(iii) of Proposition 1. A p-ideal is proper if it is a proper subset of A. {Note that conditions (i) and (ii) can be combined . fact, all three conditions into one: if eEp and g 5 e, then g ~ p In can be combined into one: if e, f ~p and g 5 e u f; then g ~ p . } Proposition 1 asserts that the projections in an ideal form ap-ideal. Proposition 2 is a converse. The following elementary lemma, already used implicitly on several occasions, is worth setting down explicitly in the present context:

Lemma. For all x, y in A, RP(x+y) 5 R P ( x ) u R P b ) and RP(xy) 5 RPCV).

Proposition 2. Let p be a p-ideal of A. Write A p for the set of all fillite sums ale, with a , € A und e , ~ p ,that is, A p is the Iejt ideal generated by p. Then A p = { ~ E AR:P ( x ) € p ) , therefore (A p)" = p. If A satisfies L P and is a *-ideal.

-RP, then A p

=p A

is the ideal generated by p,

Proof: If RP(x)cp, then x = x R P ( x ) A ~ p. Convcrscly, if x ~ A p ,

x n

say x =

aiei, then, citing the lemma, we have

1

therefore RP(x)€p. This establishes the formula for Ap, from which it is immediate that A p and p contain the same projections. Dually, {xEA: LP(x)€p) = p A, the right ideal generated by p. When A satisfies L P -RP, it follows that RP(x)ep iff LP(x)€p, thus A p = p A and is obviously a *-ideal. I Suppose p is a nonempty subset of A, and J is the ideal generated uixib, with x, E p by p. Then J = A p A, the set of all finite sums 1 and a,, b, E A (the essential point is that A p A 2 p because x = LP(x)xRP(x)). If p is a *-subset, then J is a *-ideal. If p is a set of projections, it is obvious that J coincides with the ideal generated by its projections; such ideals have a name:

Chapter 5. Ideals and Projections

138

Definition 2. An ideal J of A is said to be restricted if the ideal generated by J" coincides with J, that is, J=AJ"A ( = the set of all finite sums airib, with ei€J" and a,, b ,A). ~ {As noted above, J is necessarily a *-ideal.)

1

When A satisfies L P as follows:

- RP, its restricted ideals may be characterized

Proposition 3. Assume A satisfis LP - R P . t i o n ~on an ideal J of A are equivalent: (a) J is restricted; (b) X E Jimplies RP(X)EJ; (b') X E Jimplies LP(x)€J; (c) J is the left ideal generated by j; (c') J is the right ideal generated by J".

The following condi-

P~.oof. Write p =J". By Proposition 1, p is a p-ideal; moreover,

by Proposition 2, thus (a), (c) and (c') are obviously equivalent. (a) implies (b): Assuming (a), we have J = A p by the foregoing, therefore J = {XEA:RP(X)EJ} by Proposition 2. (b) implies (a): If X E J then, by hypothesis, RP(x)€p, therefore x = x RP(x) is in the ideal generated by p. Thus (a) and (b) are equivalent; dually, (a) and (b') are equivalent. I Combining the foregoing results: Theorem 1. Let A be a weakly Rickart *-ring sati.~fying LP-RP. For each p-ideal p, write J(p) = {XGA: RP(x) E p]. Then the correspondences P J(P), J - J

-

define mutually inverse bijections between the set qf all p-ideals p and the set of all restricted ideals J (necessarily *-ideals). The correspondences preserve inclusion and, in particular, maximality. Proof: If p is a p-ideal, then, by Proposition 2, J(p) is a *-ideal with (J(p))- = p, and is generated by its projections; thus J(p) is a restricted ideal with (J(p))" = p. Conversely, if J is a restricted ideal, write p = J". By Proposition 1, p is a p-ideal. By Proposition 2, the ideal generated by p is J(p); since J is restricted, this means J = J(p). I

5 22.

Ideals and p-Ideals

139

-

Let us reconsider Proposition 2 in the context of weakly Rickart C*-algebras; whether L P R P in such an algebra is not known, but the closure operation provides a remedy if one is willing to limit attention to closed ideals:

Proposition 4. Let A be a weakly Rickart C*-algebra, let p be a p-ideal in A, and let 1be the closed ideal generated by p. Then

-

and I = p . Proqf'. By definition, I=(ApA)- (the bar denotes closure); to verify (*), it will suffice to prove the second equality, i.e., that the closed left ideal generated by p coincides with the closed right ideal generated , is, by p. Given x ~ A p ,it is sufficient to show that x ~ ( p A ) ~that x* €(Ap)- (recall that the involution is continuous). Given any E > 0, we seek z ~ A psuch that Ilx*-zJl<~. Write x * x = r 2 with r 2 0 , rE (x*x)". Choose y ~ { x * x ) "y, 2 0, such that (i) (x*x)y2=e, e a projection, and (ii) I(x*x-(x*x)el) < &' [99, Cor. of Prop. 41. From (i) we have r2y2 = e, therefore r y = e (by commutativity and the uniqueness of positive square roots). Since e = y2x*x and x ~ A p ,we have e ~ A p ,thus eEp by Proposition 2. Set w=xql. Then w* w=yx*xy = x* x y2 = e; writing f = w w*, we have f' e E p, therefore f E p. Then w* = w* f E Ap. Since, by a straightforward calculation,

-

we have Ilx- wr1I2 = Ilx*x -(x* x)ell< E', therefore writing z = r w*, we have zcr(Ap) c A p, thus the proof of (*) is complete. (If A is an AW*-algebra, the result may be sharpened: A satisfies L P - R P [jj 20, Cor. of Th. 31, therefore A p = p A by Proposition 2.) It remains to show that I contains no new projections; that is, assuming g E l", it is to be shown that g~ p. By (*), there exists a sequence x , A~p such that x,+g. Then x,*X,E A p and g x: x, g+g. Let n be an index such that Ilgx,*x,g -gll< 1; setting y =x,* x,, we have

It follows that gyg is an invertible positive element of gAg; since its inverse is also positive, there exists s ~ g A gs, 2 0, such that (gyg)s2= g. Calculating commutants in gAg, we have s2G {gy g)" [jj 3, Prop. 9, ( 6 ) ] and S E (s2)", thus {s2)"C {gy g)" ;

"

140

Chapter 5. Ideals and Projections

in particular, s commutes with gyg, thus Setting w=y+s, we have w* w= g. Writing f = w w*, we have

therefore f ~ p ;then g- f e p yields g ~ p . I In a sense, closed ideals are the topological analogue of restricted ideals:

Proposition 5. If I is a closed ideal in a weakly Rickart C*-algebra A, then I is the closed ideal generated by its projections (in particular, I is a *-ideal). Proof. Since A satisfies the parallelogram law (P) [$13, Th. I], it follows from Proposition 1 that I" is a p-ideal. Let x ~ l Given . any E > 0, it will suffice to produce an element x' in the ideal AI"A generated by I", such that Ilx -xlII < E. Choose y e (x*x)" with y 2 0, x* xy2 = e a projection, Ilx* x - (x* x)ell< s2 [$9, Cor. of Prop. 41. Write x* x = r2 with r e {x*x)", r 2 0; setting w= xy, we have w*w=e. Let j =ww*=LP(w). Since e = y 2 x * x e ~ l c l ,we have f - e I", ~ therefore f ~ f then ; fw = w shows that w r = f wv is in the ideal generated by I", and, setting x' = wr, we have Ilx -x'll < E as in the proof of Proposition 4. (It is true, more generally, that every closed ideal in a C*-algebra is a *-ideal [cf. 24,§ 1, Prop. 1.8.21. In an A W*-algebra, it follows immediately from polar decomposition that every ideal is a a-ideal: in the notation of [§ 21, Prop. 21, x* = w*x w*. The above argument uses a sort of 'approximate polar decomposition'. We remark that I is in fact the closed linear span of its projections (Exercise 7).) 1 Combining Propositions 4 and 5, we get a topological analogue of Theorem 1:

Theorem 2. If A is a weakly Rickart C*-algebra, then the covvespond- ences p++Ap=*A, define mutually inverse bijections between the set of all p-ideals p and the set of' all clos~dideals 1 (in particular, the latter are neces.suri~*-idecrls). The correspondences preserve inclusion and, in particular, maximality.

I"=

Proof. If p is a p-ideal, then (A p)p (Proposition 4).

= (p A)-

is a closed *-ideal I with

9 22. Ideals and p-Ideals

141

Conversely, suppose I is a closed ideal. Set p=I"; as noted above, p is a p-ideal. By Proposition 4, ( A p ) - = ( P A ) - is the closed ideal generated by p, thus it coincides with I by Proposition 5. 1 Exercises 1A. The condition LP-RP in Theorem 1 can be dispensed with by limiting the classes of ideals and p-ideals that are paired, as follows. Let A be any weakly Rickart *-ring. I RP(x)eI. If I is _a (a) A strict ideal of A is an idgal I such that ~ E implie? strict ideal of A, then (i) g < e e l implies g e l , (ii) RP(x)EI implies LP(x)eI, (iii) e,fel" implies e u j ' ~ f ,and (iv) I is a *-ideal. One has I = { x t A : R P ( x ) E ~ ) . A strict ideal is a restricted ideal. (b) Let p be a nonempty set of projections in A satisfying the conditions (i) g < e e p implies yep, (ii) RP(x)€p implies LP(x)ep, and (iii) e,f ' ~ pimplies e u f ep. Since LP(w) RP(w) for any partial isometry w, clearly p is a p-ideal. Call such a set a strict p-ideal. If p is a strict p-ideal, then

-

is a strict ideal I such that l"= p. (c) The correspondences 1-I" and p-1 described in (a), (b) are mutually inverse bijections between the set of all strict ideals and the set of all strict p-ideals. (d) If A satisfies LP-RP, then all p-ideals and restricted ideals are strict, and (c) concides with Theorem 1. 2A. If A is a weakly Rickart *-ring and I is a strict ideal of A (Exercise I), then I is also a weakly Rickart *-ring (with unambiguous RP's and LP's). 3A. Let A be a weakly Rickart C*-algebra and let p be a p-idcal in A that is closed under countable suprema (that is, if e,, is a sequence in p, then sup e , ~ p ) . Define I = { X E A : R P ( X ) E ~ )Then . I is a closed, restricted ideal with I = p ; I is itself a weakly Rickart C*-algebra (with unambiguous RP's and LP's). (Cf. [b3, Example 21.)

4A, C. (i) Let A be a Baer *-ring in which every nonzero left ideal contains a nonzero projection (a condition weaker than the (VWEP)-axiom). If L is a left ideal that contains the supremum of every orthogonal family of projections in it, then L = Ae for a suitable projection e. (ii) If A is a von Neumann algebra and L is a left ideal that is closed in the ultrastrong (or ultraweak, strong, weak) topology, then L = Ae for a suitable projection e. 5A. Let A be a Rickart *-ring, A" its reduced ring 153, Exer. 181. If I is an ideal in A, write I" = I nA". (i) Although A" is generated by its projections (as a ring), it does not follow that every ideal of A" is restricted. (ii) If I is an ideal of A, then I" is an ideal of A". (iii) If A satisfies e f iff e z,f'[cf. 4 17, Exer. 201, then A and A- have the same p-ideals; if, in addition, A satisfies L P RP, then the correspondence I rt I pairs bijectively the restricted ideals of A and A". (Application: A any A W*-algebra.)

-

-

6A. If A is a Banach algebra and I is an ideal in A, thenihas no new idempotents (that is, every idempotent in i is already in I).

Chapter 5. Ideals and Projections

142

7A. Let A be a weakly Rickart C*-algebra. (i) If p is any p-ideal in A, then ( A p ) - = ( P A ) = (C p ) (the closed linear span of p). In particular, (ii) every closed ideal I is the-closed linear span of its projections, that is, I = ( e l ) - . (iii) If J is any ideal, then J = (CJ ) - (and (J)" = J).

8A. Let A be a Rickart *-ring in which any two projections e , j'are comparable ( e 5 f or f 5 e). In particular, A is factorial. (i) The p-ideals of A form a chain under inclusion, that is, if p , , p, is any pair of p-ideals in A , then either p, c p, or p, c p , . (ii) If A satisfies L P R P then the restricted ideals of A form a chain under inclusion. (iii) If A is a C*-algebra then the closed ideals of A form a chain under inclusion, and the center of A is one-dimensional.

-

9A. Let A be a finite Baer *-ring with GC, and let p,, p, be p-ideals in A . (i) The set { e ~ A : e < e , u e , for some e , E p 1 , e 2 ~ p 2 )

is the smallest p-ideal containing both p, and p,. (ii) If p, is maximal and p, $ p,, then I= e , u e , for suitable e i € pi. 10D. Problem: Does every Rickart C*-algebra satisfy LP

5 23.

- RP?

The Quotient Ring Modulo a Restricted Ideal

Throughout this section, A denotes a Rickart *-ring satisfying LP-RP, and I denotes a proper, restricted ideal of A [cf. 922, Prop. 31. We study AII, equipped with the natural quotient +-ring structure; the canonical mapping A-tAII is denoted x-X.

Lemma. If x , y ~ Aand e=RP(x), then RP(xy)=RP(ey). Proof. It suffices to observe that xy and ey have the same rightannihilators. I Proposition 1. (i) AII is a Rickart *-ring. (ii)RP(1) = (RP(x))" and LP(1) = (LP (x))" for all x E A; in particukar, every projection in A I I has the Jorm 8, with e a projection in A. (iii) ( e v f ) " = P v f and ( e n f ) " = Z n f for allprojections e,,f' in A . (iv) e f implies 8 f, and e 5 f implies P 5 f . (v) A11 sat is fie.^ L P -RP. (vi) If e,f are prqjections in A such that- 8-f,- then there exist suhprojections e, Ie, f, I f such that 8, = d, f', =f and e, f,.

-

-

-

Proof. Note that, since I is restricted, X = O iff RP(x)€ I [522, Prop. 31. (i), (ii) If XEA and e=RP(x), then, citing the lemma, Xy=0 iff x y ~ iff I RP(xy)cI iff RP(ey)€I iff e y ~ iff I EJ=O, thus the right-8) AII. This shows that A I I is a Rickart annihilator of 1in AII is (I *-ring and that RP(1) = d = (RP(x))". If, in particular, 1is a projection, then 1=RP(.F)=P.

# 23. The Quotient Ring Modulo a Restricted Ideal

143

(iii) This is immediate from (ii) and [§ 3, Prop. 71. (iv) Obvious. (v) Immediate from (ii), (iv) and the fact that A satisfies LP-RP. (vi) By assumption, there exists x E A such that f* 2 = P, X X* =f . Then 1= fXI=( fxe)- ; replacing x by f x e , we can suppose f x = x = x e . Let e, =RP(x), fb = LP(x). Then e, I e, ,f, I f ; e, J, and, citing (ii), we have

-

-

and similarly f,=,fl. {Warning: If w is a partial isometry implementing e o - f o , it does not follow that ii, implements the original equivalence P-f (that is, I%need not equal 1)) I

Proposition 2. (i) If u, v are projections in A/I such that u I v, and if v = f with f a projection in A, then there exists a projection e in A such that u=P and e l f . (ii) If u, is an orthogonal sequence of projections in A l l , their there exists an orthogonal sequence of projections en in A such that u,=P, for all n. Proof. (i) Write u = 8,g a projection in A. Then u = u v yields u = (gf )" ; setting e=RP(g j'), we have e l f and u=E by Proposition 1, (ii). (ii) Let el be any projection in A with u1=PI. Since u, 5 1 -u, = ( I - e,)-, by (i) there exists a projection e, 5 1 -el such that u, = d,. Since u, 5 1 - (u, + u,) = (1 -el - e2)-, there exists a projection e, 5 1 -el - e, such that u, = e", , etc. I

Proposition 3. If A has GC (e. g., i f A is a Baer *-ring), then so does All. Proof. If u,v are projections in A / [ , lift them to projections e, f in A, apply GC to e,f and pass to quotients (note that if h is a central projection in A, then /? is central in AII). For example, if A is a Baer *-ring, then it follows from LP-RP that A has GC [$14, Cor. 2 of Prop. 71. 1

Proposition 4. If A isfinite and has GC, then A/I isfinite. Proof. If u,v are projections in A/I such that u- v, it will suffice to show that 1 - u - 1 -v [$17, Prop. 4, (i)]. Write u=P, v = f with e-- f (Proposition 1). Since A has GC and is finite, it follows that 1 -e 1 - f ; passing to quotients, 1 - u 1 - v. I

-

-

Proposition 5. If A has GC (e.g., ij A is a Baer *-ring),then every central with h a centrtrl projection in A. projection in A/I has the form

144

Chapter 5. Ideals and Projections

Proof: Let u be a central projection in All. Write u = 0, e a projection in A, and let h be a central projection in A such that

Passing to quotients in (*), we have h"(1 - u ) S h u ; since h"u is central, it follows - that h"(1- u) 5- h"u, therefore h(1- u) = 0. Similarly, (**) yields (I-h)u=O, thus u = h . I

Definition 1. We call I (or the p-ideal f)factorial if All is a factor, that is, if the only central projections in A/I are 0 and 1. Corollary. I f A has GC, then the following conditions on 1 are equivalent: (a) I is jactorial; (b) if h is any central projection in A, then either h € I or I - h € I . Proof: (b) implies (a): If u is a central projection in A/I then, by Proposition 5, there exists a central projection h in A such that u =I$; by hypothesis, h ~ orl 1 - h ~ l , thus u = O or 1. It is obvious that (a) implies (b). I Exercises 1A. Let A be any weakly Rickart *-ring and let I be a strict ideal of A 1 5 22. Exer. I]. Equip A/I with the natural *-ring structure, and write x-% for the canonical mapping A + A / I . (i) A/I is a weakly Rickart *-ring. (ii) RP(j?)=(RP(x))" and LP(j?)= (LP(x))- for all x t A ; in particular, every projection in A/I has _the form t with e a projection in A . (iii) (e u f)" = t u f and ( e n f )- = i? nf for all p~ojectionse, f' in A. (iv) e f implies F f , and e .
-

-

2B. Let A be a Rickart C*-algebra, I a closed ideal of A , and write x - X for the canonical mapping A + A / [ . (i) I is a *-ideal of A, and A/I is a C*-algebra. (ii) If u is a projection in A / [ , then there exists a projection e in A such that u=e. (iii) If u, v are projections in A/I such that u I v, and if v = ,T with f ' a projection in A , then there exists a projection e in A such that u=F and p < f . (iv) If u, is an orthogonal sequence of projections in A/I, then there exists an orthogonal sequence of projections en in A such that u,=< for all n. 3A. With notation as in Exercise 2, suppose, in addition, that A is an AW*algebra.

6 23.

-

The Quotient Ring Modulo a Restricted Tdcal

145

,z

(i) Let u, v be projections in A//, say u=F, v = with e, f' projections in A . If u o, then there exist su~projections e, 2 e, ,f, < f and a partial isometry WEA, such that u = G , v = f,,, w* w=e,, ww*=fb and E is the given partial isometry implementing u v. (ii) If A is finite, then equivalent projections in A// are unitarily equivalent, thereforc A// is finite. 4A. If A is a fzctorial Baer *-ring satisfying L P RP, then every restricted ideal in A is factorial. 5A. Consider the category whose objects are the weakly Rickart *-rings, and whose morphisms are the *-homomorphisms cp satisfying RP(q(x))=q(RP(x)). (i) The subobjects are the *-subrings containing RP's. {Examples: [S; 5, Props. 5, 61.) (ii) The kernels of morphisms are the strict ideals (Exercise 1). 6A. Consider the category whose objects are the Baer *-rings, and whose morphisms are the *-homomorphisms cp satisfying IiP(cp(x))=cp(RIJ(.u)) and cp(supe,)= sup cp(e,) for families (e,) of projections. (i) The subobjects are the Baer *-subrings [$ 4, Def. 31. (Examples: [$ 4, Props. 6, 71.) (ii) The kernels of morphisms are the direct summands. In a sense, the reduction theory [g 44, Th. I]mixes certain objects of this exercise with certain morphisms of the preceding one.

-

-

7A. Consider the category whose objects are the Baer *-rings, and whose morphisms are the *-homomorphisms q satisfying RP(cp(x))=q(RP(x)) and cp(supe,)= supcp(e,) for orthogonal families ( e , )of projections. The subobjects, and kernels of morphisms, are the same as for Exercise 6 [cf. 3 4, Exer. 271. 8A. Consider the category whose objects are the Baer *-rings satisfying the (WEP)-axiom, and whose morphisms are the *-homomorphisms cp such that cp(supe,) = sup cp(e,) for orthogonal families (e,) of projections. (i) If A is an object, then the subobjects of A are the objects B such that B is a *-subring of A containing orthogonal sups (as calculated in A ) ; it follows that B is a Baer *-subring of A [S; 7, Exer. 81. (ii) The kernels of morphisms are the direct summands. (iii) Every morphism cp satisfies RP(cp(x))=q(RP(x)). A subcategory: the AW*-algebras, with same morphisms [cf. S; 7, Exer. 91.

9C. Consider the category whose objects are the von Neumann algebras (on all possible Hilbert spaces), and whose morphisms are the *-homomorphisms cp such that q ( l ) = I (I stands for the identity operator on the respective spaces) and cp(sup E,)=sup q(E,) for orthogonal families (E,) of projections. (i) If d is an object-say .d is a von Neumann algebra on the Hilbert space 2-then the subobjects of .d are the von Neumann algebras % on .Y such that $8 c d . (ii) The kernels of morphisms are the direct summands. (iii) Every morphism cp satisfies RP(y(T)) = cp(RP(T)). (iv) The morphisms are the *-homomorphisms cp, with cp(l)=I, that are normal (ultrastrongly continuous, ultraweakly continuous). (v) Every morphism cp satisfies cp(sup E,)= sup q(E,) for arbitrary families (E,) of projections. With 'von Neumann algebra on .X" replaced by 'weakly closed *-subalgebra of 2 ( X ) ' ,the condition cp(l)=I can be dispensed with.

Chapter 5. Ideals and Projections

146

3 24.

Maximal-Restricted Ideals, Weak Centrality

-

In this section A is a Baer +-ring satisfying L P RP, and Z is the center of A . We pass freely between restricted ideals of A and p-ideals [i322, Th. I ] . A maximal-restricted ideal of A is a proper, restricted ideal that is maximal among such ideals. Since A has GC [$ 14, Cor. 2 of Prop. 71, the results of Section 23 (except Proposition 4) are applicable to A.

Proposition 1. I f 1 is a maximal-restricted ideal of'A, then I is factorial. Proof. Write Y = f ; thus Y is a maximal p-ideal. Assuming h is a central projection of A such that h $ Y , it suffices to show that 1 - h ~ , a [lj 23, Cor. of Prop. 51. Let . a 1 = { e € A : he^^); obviously Y c Y' and I -h E.%', so it will suffice to show that .f = 9'. In turn, it is enough (by the maximality of 9)to show that 9 ' is a proper p-ideal. If f S e E Y ' , then h f 5 h e ~ 9 therefore , Iz f ~ . f ,thus f E Y ' . If e, f ~ 4 ' , then h ( e u f ) = h e u h f ~ Y ,thus e u f ~ . f ' . Finally l $ Y 1 (because h $ 9 ) , thus Y ' is proper. I The next lemmas lead up to the proposition that . f bijectively the maximal p-ideals of A with those of Z.

H

,f n Z pairs

Lemma 1. Let Y l , .f2 be proper p-ideals such that .al nZ c ,a2 nZ , and suppose that Yl is factorial. Then Yl 9; or Y2c Y l , and in either case Y l n Z = c f 2 n Z . Proof. If Y2c
+

Since e € Y 1 , it results from (**) that (1 - h)f ' ~ . f; ~since f$,Y,, it follows from f = h f + ( I - h)f that h f $ Y l . All the more, h $ , f l ; since Yl is factorial, I-he,P1 [lj 23, Cor. of Prop. 51, thus ; citing (*), we have h e 5 h f I~ ' E Ythere~ , Then also (1 - h ) e ~ , A also, fore - h)e, therefore their thus Y2 contains both h e and (I sum e. Thus .a, c .q2. It remains to show that .a, nZ c ,PI nZ . Suppose

S; 24. Maximal-Restricted Ideals, Weak Centrality

147

h ~ 9n , Z. If, on the contrary, h&.F, nZ, then I- h€.F1by factoriality, , Z c 9, nZ; thus Y2 contains both h and 1- h, therefore I-h ~ 4n I therefore I €Y2,contrary to the hypothesis that .a2is proper.

Lemma 2. Let XI, 4, be proper p-ideals with .a, n Z c .& nZ, and suppose that Yl is factorial and Y2 is maximal. Then 4, c and Y1nZ =Y2nZ. Proof. By Lemma 1, 9, nZ = 9, nZ and either .a, c .a2 or Y2c 9, ; in the latter case, 9, = Y, by maximality, thus c Y2 in both cases. I The following result is known as weak centrality (see also Exercises 2 and 4):

Proposition 2. If 9,, 9, are maximal p-ideals with 4, n Z c Y2nZ, then Y1=Y2. Proof. Since is factorial (Proposition 1) and Y2 is maximal, Lemma 2 yields 4, c .F2, therefore 9, = 4, by maximality of 4, . I For ideals, Proposition 2 means the following:

Corollary 1. I f I,, I, are maximal-restricted ideals o f A with I,n2 c 12n2,then [,=I2. Proof. Writing 4i = fv(v = 1,2), the hypothesis reads 4, nZ c Y2nZ, therefore .a,= .F2 by Proposition 2; since a restricted ideal is generated I by its projections, I , = I,.

Corollary 2. If 4, is a factorial p-ideal, then there exists a unique maximal p-ideal ,a2such that Yl c 9,. Moreover, Yl n Z = .a2n Z. Proof. By a straightforward application of Zorn's lemma, there exists a maximal p-ideal .a2 such that .al c 9,. Necessarily % . , nZ =9, nZ by Lemma 2. If also Y1c ,a;';, 9 ; a maximal p-ideal, then 9; nZ = Yl nZ= 4 , nZ, therefore 9 ; = 92by Proposition 2. 1 We now relate the p-ideals of A to those of Z. In a commutative ring (or even in an abelian ring), equivalent projections are equal; in such a ring, a nonempty set of projections is a p-ideal iff (i) along with e, it contains every g _< e, and (ii) along with e, f , it contains e u f = e +f'- ef. If 4 is a p-ideal of A, it is obvious that 9 n Z is a p-ideal of Z and that .% is proper iff 4 n Z is proper. In the next proposition, we show that the correspondence 9 H 9 nZ pairs bijectively the maximal p-ideals of A with those of 2.

Lemma. If j is a proper p-ideal of Z and if ,f = { ~ E AC " :( e ) ~ j ) , then 4 is a proper p-ideal of A such that Y n Z = 2.

Chapter 5. Ideals and Projections

148

Proof Since C(e u f') = C(e)u C(f ) and since g 5 e implies C(g) 9is a p-ideal. Obviously 9n Z = 2 ;in particular, I$9. I

I C(e),

Proposition 3. The correspondence .a H ,a n Z maps the set of maximal p-ideals of A bijectively onto the set of all maximal p-ideals of Z. Proof. (i) Suppose .f is a maximal p-ideal of A. Then .fn Z is a proper p-ideal of Z ; Ict us show that it is maximal. Assuming 2 is a proper p-ideal of Z with 9 n Z c 2 , it is to be shown that ,9 n Z = f . By the lemma, there exists a proper p-ideal.9' of A such that 9' n Z = 9. Let 9 " be a maximal p-ideal of A with 9' c .Y". Then .8" n Z 3 .Y' n Z = 2 3 .an 2,therefore 9"=.a by Proposition 2, thus f = .F n Z . (ii) The mapping .$ H . f n Z is injective by Proposition 2. (iii) To show that it is surjective, suppose # is any maximal p-ideal of Z. As argued in (i),there exists a maximalp-ideal 9 of A with .9 n Z 3 cy; since 9 n Z is a proper p-ideal of Z, .9 n Z = f by the maximality ofy. I Proposition 3 has topological overtoncs; the details are as follows. The projections of Z form a complete Boolean algebra 2 ; let 9"be the Stone representation space of 2. Thus 3 is a Stonian space (see Section 7) whose Boolean algebra of clopen sets is isomorphic to 2. Let us identify a projection ~ E with Z the characteristic function of the clopcn set in .%"to which it corresponds. If G E T , set { ~ E . Zh(0)=0) : ; then O H 2, maps 3 bijectively onto the set of all maximal p-ideals of Z. It follows from Proposition 3 that to each a€.OR;' thcrc corrcsponds a unique maximal p-ideal .auof A such that ,Yen Z = y u , and o t t 9 , pairs X bijectivcly with thc sct of all maximal p-ideals of A. This is a good place to preview Chapter 7. Suppose A is a finitc Baer *-ring satisfying L P -RP. It is shown in Chapter 6 that there exists a 'dimension function' e H D(e) on A", with values in C(.%"),having various pleasant properties (see Section 25). The technical core of Chapter 7 is the proof that .au= {PEA:D(e)(o)=O) fa=

for each OE% (Section 39); writing (this is the typical maximal-restricted ideal of A), it results that All, is a Baer *-ring (finite and factorial). This, together with the fact that

5 24. Maximal-Restricted Ideals, Weak Centrality

149

(proved in Section 36), constitutes the 'reduction theory' of such a ring A [$44, Th. I]. Exercises

1A. A factorial p-ideal need not be maximal.

2.4. Let A be an AW*-algebra with center Z . If M,, M, are maximal ideals of A such that M, n Z c M, n Z, then M, = M,. (See also Exercise 4.) 3C. Let A be a Banach algebra with unity. (i) If M is a maximal ideal of A, then M is closed and AIM is a simple Banach algebra. (ii) If P is a primitive ideal of A, then P is closed and AIP is a (primitive) Banach algebra; the center of AIP is one-dimensional. (iii) If I is an ideal of A such that A11 is a semisimple ring, then I is thc intersection of a family of primitive ideals, thus I is closed and AII is a semisimplc Banach algebra. (iv) If A is a C*-algebra, then every primitive ideal of A is a closed *-ideal, and every closed ideal is the intersection of a family of primitive ideals. 4C. Let A be a Rickart C*-algebra with GC (for example, let A be any AW*algebra), and let Z be the center of A. (i) If I,, I, are closed, proper ideals of A such that I, n Z c I, n Z , and if 0, 1 are the only central projections of All, (as is the case when I, is maximal--or, more generally, when I, is a pr~mitiveideal), then either I, c I, or I , c I, ; in either case, I, n Z = I, n Z . (ii) If P is a primitive ideal and M is a maximal ideal such that P n Z c M n Z , then P c M and P n Z = M n Z . (iii) If MI, M, are maximal idcals such that M, n Z c M, n Z, then M, = M,. (iv) If P is a primitive ideal, there exists a unique maximal ideal M such that P c M (in other words, AIP has a unique maximal ideal). Necessarily PnZ=MnZ. (v) If N is a closed, proper ideal of Z, then there exists a closed, proper ideal 1 of A such that I n Z=N (in particular, I 2 N ) . (vi) The correspondence M --t M n Z maps the set of all maximal idcals M of A bijectively onto the set of all maximal ideals of %. 5A. Let A be a Rickart *-ring, with center Z , and let 3 be the Stone representation space of the Boolean algebra Z. (Thus !f is a totally discon~ected,compact space whose Boolean algebra of clopen sets is isomorphic to Z.) Identify each h c Z with the characteristic function of the clopen set in .?Z to which it corresponds. Fix OEX. We say that x c A vanishes in a neighborhood of' CT if there exists h ~ with 2 h(a)=l and xh=O. Let J be the set of all such x. (i) J is a_ strict ideal of A (cf. [§ 22, Exer. I], [$ 23, Exer. I]). (ii) J n Z = { h f Z : h(a)=O}. (iii) If A has GC, then J is factoriaj. (iv) If A is a Baer *-ring, then e~ J iff C ( e ) c j .

6A. Let A be a Rickart *-ring, .9 a proper, strict p-ideal of A [$ 22, Exer. I]. (i) If h is a central projection such that hg.9, and if Y'= { c ~ e iIzetY), : then .a' is a proper, strict p-ideal such that .a c .F and 1 -he./'. (ii) If 9 is maximal-strict (that is, maximal among proper, strict p-ideals), and if A has GC, then 4 is factorial. This generalizes Proposition 1.

Part 3: Finite Rings

Chapter 6

Dimension in Finite Baer *-Rings Throughout this chapter (except in some of the exercises), A denotes a finite Baer a-ring with GC. For the most part, the Type I and Type I1 summands of A require different techniques and are treated separately. A salient feature of the exposition is that virtually all results are obtained without assuming the parallelogram law (P); it is only in the final section on modularity (Section 34) that (P) is invoked. We write Z for the center of A. The projection lattice 2 is a complete Boolean algebra; we write .!?" for its Stone representation space, and C(X) for the algebra of continuous complex-valued functions on X (but only real-valued functions are needed in this chapter); thus 9"is a Stonian space, 2 may be identified with the complete Boolean algebra of clopen sets in .% (by identifying a central projection h of A with the characteristic function of the corresponding clopcn subset of .%), and C(X) is a commutative A W*-algebra [§ 7, Prop. I]. In general, the rings Z and C ( X )have in common only their projection lattices (also, certain rational-linear combinations of projections may be regarded as common to the rings, as described in Section 26 below). {If Z is an A W*-algebra-. as it is when A is an AW*-algebra-then it is the closed linear span of its projections [tj 8, Prop. 31 and we may identify Z with C(Y). Nevertheless, the considerations of the present chapter would not be materially simplified by assuming that A is a finite AW*-algebra.)

5 25.

Statement of the Results

The central preoccupation of the chapter is with C(.%)-valued functions of the following kind:

Definition 1. A (jinite) dimension function for A is a function e H D(e) defined on the set 2 of projections of A, with values in C(X), such that (Dl) e f implies D(e)= D( f ) , (D2) D(e) 2 0 , (D3) D(h) = h when h is central, (D4) ef = O implies D(e+ j ) = D(e)+ D ( j ) .

-

S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

154

Chapter 6. Dimension in Finite Baer *-Rings

(In connection with (D3), recall that we are identifying h with the characteristic function in C(%) corresponding to it.) Our principal aim is to show that such a function exists, and is uniquely determined by the properties (Dl)-(D4). Along the way, we develop a number of properties that are implicit in these four, of which the most striking is complete additivity (a generalization of (D4) to orthogonal families of arbitrary cardinality). Assuming, in addition, that A satisfies the parallelogram law (P), it is shown in Section 34 that the projection lattice of A is modular, and it follows from complete additivity of dimension that the lattice operations are 'infinitely distributive' in a certain sense (the official term is that they are 'continuous'); thus the projection lattice of a finite Baer *-ring, satisfying GC and (P), is a 'continuous geometry' in the sense of von Neumann. The actual order of events is as follows. $ 26. Rudimentary dimension theory for the 'simple' projections, i.e., the projections that neatly divide their central covers; proof that there exist sufficiently many simple projections to serve as building blocks for the general dimension theory. 5 27. First consequences of (Dl)-(D4); proof that the general case may be reduced to the Type I and Type I1 cases. $ 28. Type I case: complete additivity and uniqueness of dimension. $ 29. Type I case: existence of dimension. $ 30. Type I1 case: dimension theory for the 'fundamental' projections. 5 31. Type I1 case: existence of a completely additive dimension function. $32. Type I1 case: uniqueness of dimension. $33. Dimension in an arbitrary finite Baer *-ring with GC. 5 34. Modularity, continuity of the lattice operations (assuming the parallelogram law).

5 26.

Simple Projections

For the first part of the section, no restrictions are needed on the type of A (a finite Baer *-ring with GC); the deeper results at the end of the section require separate discussions for Types I and TI.

Proposition 1. If e is a nonzero projection and if e=el

+...+em=fl +...+f n

are orthogonal decompositions with ei

- fj for all i and j, then m

= n.

Proof: Say m 2 n. By finite additivity of equivalence [sf, Prop. 81, e=e,+..-+em-- f l + . - . + f m i e ,

5 26.

Simple Projections

155

therefore f , + ... + f , = e by finiteness [$17, Prop. 41; it follows that m = n (m < n would imply ejn=O). I

Definition 1. Let e and f' be nonzero projections. We say that f' divides e (n times) if there exists an orthogonal decomposition e= f ;

+.-.+ji

with f'- ji for i= I, ..., n. The integer n is unique by Proposition 1 ; it is denoted (e: f ). To indicate briefly that f divides e, we say that '(e:f ' ) exists'. Whenever we write ( e :f ) , it is understood that the condition 'f divides e' is either being assumed or is on the verge of being verified.

In Definition 1 it is not required (nor in general possible) that f =ji for some i, or even that f I e; divisibility is really a property of equivalence classes of projections, as the next propositions show. Proposition 2. l f (e :f ) exists and f = (e : g).

-

Proof. Immediate from Definition 1. Proposition 3. l f (e :g) = (f : g) then e

g, then (e: g ) exists and (e :f )

-

I

f

Proof: Immediate from finite additivity of equivalence. Proposition 4. l f (e : g ) exists and e

= ( f:g).

-

I

f ; then ( f : g ) exists and (e : g)

-

Proof. Let n = ( e :g) and suppose e= el + ... +en with g ei for all i. The equivalence e - f induces an orthogonal decomposition ,f =fl + ..- +ji with e, g [$ 1, Prop. 91. 1

- -

Generalized comparability makes its entrance here:

Proposition 5. If (e :f ) = (e: g) then f

- g.

Proof. Say e=f,+...+,fn=gl+...+gn with f - j i and g - g i all i. In view of GC, there exists a central projection h such that

for

h.f15hg1, ( l - h ) ~ 1 5 ( 1 - h ) f ' 1 . For i = l , ..., n we have h.h-hflShg, - h a , say hji

-

gi I hg,; adding the latter equivalences, we have

n

therefore x g j = 1

n

C hg, by 1

finiteness. It follows that g;=hgi for all i,

156

-

Chapter 6. Dimension in Finite Baer *-Rings

-

thus hf; h g,. Similarly (I -h ) f , ( 1 - h)g,, therefore J;

- gi.

I

Proposition 6. Suppose ( e :g ) and ( f : g ) exist. (i) Zf' ef = 0, then (e+ j' : g) exists and (ii) If e 5 f ; e # f , then ( f - e : g) exists and Proof. (i) Obvious. (ii) Write m = (e :g), n = ( f : g). Say e=el +...+em,

f = f ' ,+...+,fn

- -g. Necessarily m < n ; for, n

with e, f j

5 m would imply

f =f,+...+ fn-e,+...+en<e<

f',

-

therefore e = f by finiteness, a contradiction. Set e'= f,+...+,f,; e e', therefore +...+f n f -e-f-el=J,+,

then

[ij 17, Prop. 4, (ii)]. Citing Proposition 4, we have ( j - e :g)=( f-e':q) =n-m. I

Propositions 3 and 4 show that i f (e:g ) and ( f : g ) exist, then (e:g) = ( f ' : g ) i f f e- f. More generally, Proposition 7. If (e:g ) and ( f :g ) exist, then (e:g) 5 ( f :8 )

iff e 5 .f.

g ) 5 ( f : g ) by PropoProof. I f e 5 , f , say e-e' 5 j; then (e:g)=(ef: sitions 4 and 6. The converse is obvious from finite additivity o f equivaI lence.

Proposition 8. If (e:f ' ) and ( f : g ) exist, then (e:g ) exists und (e:g) =(e:f)(f:g). Proof. Let m=(e:f ), n = ( f : g ) . Say e= f',+...+ f,,

f = g , +...+g,

with f - f , and g-gj for all i and.1. For each i, the equivalence f - f i induces an orthogonal decomposition o f fi into n projections equivalent t o g, thus e is the sum o f mn projections equivalent to g. I Conversely: Proposition 9. If (e:f ) and (e:g ) exist, and if (e:,f) divides (e:g), then ( f :g) exists and (e:g) = (e:,f) ( f :g).

$26. Simple Projections

Proof. Say (e:f ) = m, (e:g ) = mn. Write

.fi =

C gij

. . . , m) . (i= I,

j= 1

-

Since (,fi:g)=n for all i, we have f , -...- fm by Proposition 3 ; then e = f , +...+fm shows that (e:f ,)=m=(e:f), therefore f l j' by Proposition 5. Since ( j ' ,:g) =n and f , f; we infer from Proposition 4 that ( f :g) exists and is equal to n. I

-

The most important case of Definition 1 is when the 'numerator' is central:

Proposition 10. I f lz is a central projection and (h:e) exists, then h = C(e) and there exists an orthogonal decomposition h = el +...+en, n=(h:e), with e - e , for all i and e=e,. Proof. Say h= f , +...+fn with e-.fi for all i. Then C(e)=C ( f i ) , thus h=C(h)=C(flu...ufn)=C(f,)u..~uC(f,)=C(e)

-

[§ 6, Prop. 11. In particular, e < h, therefore e f , implies h - e

-

h -f , equivalence induces an orthogonal decomposition of h -e into n - I projections equivalent to e. I =f ,

+...+A;the latter

Definition 2. A nonzero projection e is called simple if it divides its central cover, that is, (C(e):e) exists. The integer (C(e):e) is called the order of e. Suppose e is a simple projection, h = C(e), n = (h:e). Write h = el +...+en with e= el and e-e, for all i (Proposition 10). Since h A is *-isomorphic to the n x n matrix ring (eAe), [§ 16, Prop. 11, one can show that nh is invertible in h A 1516, Exer. 11, thus ( l / n ) h may be defined (as the inverse of nh in h A). Actually, it is more convenient to identify h with the characteristic function in C(%) corresponding to it in the Stone representation; then ( l / n ) h is just a scalar multiple of the function h.

Definition 3. If e is a simple projection, say h = C(e),n = (h:e),we define

(the notation T alludes to the 'normalized trace' of a matrix); by the preceding remarks, T(e) may be regarded as an element of Z or of

Chapter 6. Dimension in Finite Haer *-Rings

158

C(%). We also define T(O)=O. Thus T(h)= I 2 for every central projection h.

Proposition 11. If e is simple and e -,f, then f is simple and T(e)= T( f ) . Proof: Let h = C(e), n=(h:e). Then C(f ) = C(e)=h and ( h : f ) =(h:e)=n by Proposition 2, thus T ( f )=(l/n)h= T(e). I Conversely:

Proposition 12. I f e, f are simple and T(e)= T ( f ' ) , then e- f Proof: Writing h = C(e),m = (h:e) and k = C(f'), n = ( k f: ), the hypothesis is that (l/m)h=(l/n)k. It follows easily (for example, from the functional representation) that h = k and m = n; thus (11:e)= rn = n = (h:f'), therefore e f by Proposition 5. 1

-

Proposition 13. I f e is simple, then T(he)= h T(e) for every central projection h. Proof. If he = 0, then h C(e)= O and the desired relation reduces to T(O)=O. Assuming he#O, let k=C(e),n=(k:e) andwrite k=el +...+en with e-e, for all i ; then 11 k= h e , +...+hen with he-he, for all i, thus he is simple and 7(he)= ( l l n )h k = h T(e). I We conclude the section with the key results on the existence of simple projections. The Type I and Type I1 cases are treated separately.

Proposition 14. If'A is a jinite Baer *-ring of Type I , with GC, and if e is any nonzero projection in A, then there exists an orthogonal ,furnily (e,),,, of' simple abelian projections such that e = sup el. Proof: By an obvious exhaustion argument, it suffices to show that e has a simple abelian subprojection. We can suppose A is homogeneous [§ 18, Th. 21, say of order n. Write 1 =el +..,+en with el --... --en and the e, abelian. From e#O and C(e,)=l it follows that eAe,#O [$6, Cor. 1 of Prop. 31; in view of GC, there exist nonzero subprojections f 5 e, f l 5 el such that f - f l [§ 14, Prop. 21. Let h= C(f ,)=C ( f ) . Since el is abelian,f , = h el [tj 15, Prop. 61. Define f = h ei for i= 2, .. ., n. Then

,

h=h(e,+...+en)=.fl

+...+f , ,

and f - f 1 = h e , -he, =,fi for i= 2, . .. , n, thus f is simple. Moreover, el shows that f is abelian 15, Prop. 71. I

[a

Definition 4. A simple projection is called fundanzentul if its order is a power of 2. (In particular, a nonzero central projection is fundamental of order 2 O . ) The concept is germane only to the continuous case:

159

$26. Simple Projections

Proposition 15. Let A be a finite Baer *-ring of Type TI, with GC. If e is any nonzero projection and r is any nonnegative integer, there exists a subprojection j of e such that ( e :j ) = T . In particular, if e is central then f is fundamental and T ( j )= 2-'e. Proof. Immediate from [$19, Th. 11.

I

Lemma. Let A be as in Proposition 15. Suppose

where g,-...-

gn and q 5 q 1 . Then ql has a ,fundamental subprojection.

Proof. Let r be a positive integer such that 2' > n + 1, and write m=2'. By Proposition 15, there exists an orthogonal decomposition

with dl -...-dm.

Let h be a central projection such that

(1) hdl 5 h u 1 , ( 1 - h ) g 1 5 ( 1 -h)d1 . Say h d , -go Ih g l ; it will suffice to show that h d , (and therefore go) is m

fundamental. Since h = E h dj with h dl -... that h #O. therefore

1

-

h dm, it is enough to show

Assume to the contrary that h = 0. Then gl 5 d l by (I),

(2)

g i 5 d i for i = l , ..., n .

Also, Adding (2)and (3), we have

contrary to finiteness.

I

Proposition 16. Let A be ajinite Baer *-ring of Type IT, with GC. If e is any nonzero projection, there exists an orthogonal family (el),,, of jundumental projections such that e = sup e,. Proof: It will suffice to show that e has a fundamental subprojection. with Expand { e ) to a maximal homogeneous partition-necessarily finitely many terms [$17, Prop. 1, (ii)]-say e l , ..., en, where e= e l . Set

Chapter 6. Dimension in Finitc Baer *-Rings

160 n

,f = 1 - x e i and let h be a central projection such that 1

hfshe,

(1-h)ed(l-h).f.

Necessarily h # 0 (h = 0 would imply e 5 f ; contradicting maximality). n Then h=hf+h(l - f ) = h f + E h e , , 1

where h e , -...- h en and h f 5 h e = h e l . Applying the lemma in h A (which satisfies the same hypotheses as A), we see that he contains a projection g that is fundamental relative to h A-hence relative to A. I Exercises

1A. Let A be a finite Baer *-ring with GC, of Type I,. If j' is any nonzero abelian projection in A, then f' is simple of order n. 2A. Proposition 5 fails in the ring of 2 x 2 matrices over the field of three elements [cf. jj 1, Exer. IS].

3A, D. Proposition 14 holds with GC weakened to PC. Problem: Does Proposition 26?

5 27.

First Properties of a Dimension Function

In this section we develop some of the direct consequences of the defining properties (Dl)-(D4) of a dimension function [lj25, Def. I ] (see also Exercise 1 ) ; the deeper consequences (complete additivity and uniqueness) come later.

Proposition 1. Let A be a finite Baer *-ring with GC, and suppose D is a dimension function for A ; thus ( D1 ) e f implies D(e)= D(,f ), ( D 2 ) D(e)>O, ( D 3 ) D(h) when h is central, ( D 4 ) ef=O implies D ( e + f ) = D ( e ) + D ( f ) . Then D also has the following properties: (D5) O ~ D ( e ) l l , (D6) D(he)=hD(e) when h iscentral, ( D 7 ) D( f )= T ( f ) when f is simple, (D8) D(e)=O iff e = 0 , ( D 9 ) e - f !IT D(e)= D ( f 1, e 5 f iff D(e) l D(.f1. Proof. (D5) 0 I D(e)< D(e)+ D(1- e)= D(e + ( I - e))= 1. More generally, e I f implies D(e) s D( f ); indeed, f = e + ( f - e) yields D( f ) =D(e)+D(f-e), thus D ( f ) - D ( e ) = D ( f - e ) > 0 .

-

tj 27.

First Properties of a Dimension Function

3 61

(D6) Since h commutes with e, we have h u e-e= h -he Prop. 31; by the preceding remark,

[$I,

D(h u e)- D(e)= D(h)- D(h e)= h - D(he), and multiplication by h yields

(*I

hD(hue)hD(e)=h-hD(he).

Since h e 5 h, we have D(h e) 5 D(h)= h, therefore h D(h e)= D(h e) by the functional representation. Also, h 5 h u e implies h = D(h)ID(h u e) 5 1, and multiplication by h yields h = h D(h u e). Thus (*) simplifies to h-hD(e)=h-D(he), which proves (D6). (D7)Let h= C ( f ) ,n=(h:,f) and write h = f l all i. Then n

+...+f,,

with f - f i for

h=~(h)=xD(f~)=nD(f), 1

thus D ( f ) = ( l / n ) h = T ( f ) . (D8) If e#O there exists a simple projection f such that f < e [§ 26, Props. 14 and 161, therefore D(e)2 D( f ) = T(f ) # 0. On the other hand, D(0)= 0 by either (D3)or (D4). (D9) Suppose D(e)= D ( f ) . Let h be a central projection such that heshf,

(I-h)j's(l-h)e.

Say h e - f ' < hf: Then

) D(hf ) , D(hf -f ' ) = 0, hence h f -f ' = 0 by (D8). Thus thus D ( , f l = h e - f ' = h f . Similarly (I-h)e-(I-h).f, therefore e - f . (D10)If e s , f , say e-el s f , then D ( e ) = D ( e , ) lD ( f ) . Conversely, suppose D(e)5 D( f ) . Let h be a central projection such that

Then D ( h f )I D(he)=h D(e)I h D( f ) = D(hf ) , thus D(he)= D(h,f), therefore he-h f by (D9); adding this to the first relation in (**), we haveesf. I The deeper properties of dimension depend on the fact that the set of all real-valued functions in C(X)-that is, the real algebra CR(*Y)is a boundedly complete lattice with respect to the usual pointwise ordering. To put it another way, the positive unit ball of C(X)-that is,

162

Chapter 6. Dimension in Finite Baer *-Kings

the set of all continuous functions c such that 0 I c I I-is a complete lattice. These completeness assertions concerning the real function lattice are equivalent to the extremal disconnectedness of ,% by Stone's theory [SS]. (Caution: The lattice supremum of an infinite set of functions is 2, but in general #, the pointwise supremum.) If cj is an increasingly directed family in C, (X), bounded above by some element of C,(%) (equivalently, by some real constant), and if c = supcj (in the lattice sense just described), we write cj f c. The following is a sample of the kind of elementary facts about such suprema that we shall need: Lemma 1. For positive functions in C(X): (i) I f c i r c then a c , f a c . (ii) I f cjf c and d, f d, then cj + dj f c + d . (iii) If c j l c and dk f d, then c j + dk 7 c + d .

Proof. (i) For all j, a c j I ac, therefore s= supacj exists and s s a c . The assertion is that s = a c. Assume to the contrary; then there exists an e > 0 and a nonzero projection h in C(%) such that a c -s 2 eh, therefore h a is invertible in hC(3); let h be the element of hC(X) such that b(ha)= h, that is, h a = h. For each indexj,

multiplying by b, we have (ba)c 2 (ba)c,+ehh, thus

on the other hand, adding (*) and (**), we have e 2 c j + E h. Thus cj i c - s b for all ,j, therefore c 2 c - e b; this implies b = 0, a contradiction. (ii) Let a = sup(cj+ dj); obviously a 5 c + d and c j + d j l a. The assertion is that a= c+d; it is enough to show that d 5 a - c. Fix an index J ; it will suffice to show that d j < a - c. For all k 2j we have ck
,

,

Let (a,),,, be a family in C,(X) such that a, 2 0 for all each finite subset J c I , write a ~ 1= % ; raJ

L I Z1.

For

5 27.

First Properties o f a Dimension Function

163

with the indices J ordered by inclusion, the family (a,) is increasingly directed. If the family (a,) is bounded above by some element of C,(.%"), we write a, for the supremum of the a,; thus

1

~€1

~ a , = s u p { ~ a l : ~Jfinite cl, LEI

Such sums enter dimension theory via the following lemma:

Lemma 2. If D is a dimension ,function for A, (el),,, is an orthogonal family of projections in A, and e = sup el, then C D(e,) exists and is < D(e). re1 Proof: If J is any finite subset of I, write eJ= C el; then e, is a projection < e, thus rtJ

Definition 2. A dimension function D for A is said to be completely additive if, in the notation of Lemma 2, one always has D(e,)=D(e).

2

LEI

{It is shown later in the chapter that every dimension function is completely additive [§ 33, Th. I].) Our study of general finite Baer *-rings with GC will be reduced to the Type I and Type I1 cases by writing A= hA +(I - h)A, where h is the unique central projection such that hA is Type I and (I - h ) A is continuous [§ 15, Th. 1, (4)]. The pieces are put back tegether by means of the following proposition; although the result is needed only for the case of two summands, that is, for the case A = {I, 2), it is more instructive-and not essentially harder-to consider arbitrarily many summands :

Proposition 2. Let (ha),,, be an orthogonal family of nonzero central projections in A such that sup ha= 1. (i) If; for each ~ E A D, , is a dimension junction for h,A, then the D, may be assembled in a natural way to form a dimension ,function D for A. If every D, is completely additive, then so is D. (ii) If one knows that each h,A has a unique dimension function, it follows that the dimension function of A is also unique. Proof. {The main task of the chapter is to remove the "iP"s; thus, eventually, the proposition has content for every finite Baer *-ring with GC.) For each a, let P, be the clopen subset of 3 corresponding to ha in the Stone representation of the complete Boolean algebra 2 (thus h, is the characteristic function of 4). The central projections of h,A are the projections in h,Z, that is, the projections h in Z such that h < ha;

164

Chapter 6. Dimension in Finite Baer *-Rings

these correspond to the clopen sets in X that are contained in P,, in other words, to the clopen subsets of the Stonian space P,. Thus we may identify P, with the Stone representation space of the complete Boolean algebra of central projections of h,A. At the same time, C(4) may be identified with the ideal h,C(X) of C(.%): a function in C(P,) is regarded as a function in h,C(X) by defining it to be zero on 9"-P,; and a function in h,C(X) is regarded as a function in C(P,) by restricting it to P,. Since suph,= I (in other words, the union of the P, is dense in .OR^), the commutative AW*-algebra C(X) is the C*-sum of the h,C(%) [$10, Prop. 21, which in turn may be identified with the C*-sum of the C(P,); this is the technical core of the proposition. (i) Suppose that for each a, D, is a dimension function for Iz,A. If e is any projection in A, then h,e is a projection in h,A for all a € A , and we may define D(e)= (Da(hae)),,, 3

an element in the C*-sum of the C(P,). In view of the identifications made above, D(e) may be regarded as the unique element of C(X) such . follows routinely that D is a that h,D(e)= D,(h,e) for all ~ E A It dimension function for A. Suppose, moreover, that every D, is completely additive. Let (e,),,, be an orthogonal family of projections in A, e = sup e,. Write

(these sums exist by Lemma 2). By hypothesis, c,= D,(h,c) for each a € A ; it is to be shown that c=D(e). Citing Lemma 1, (i), we have

thus h,(D(e) - c) = 0 for all a, thereforc D(e)- c = 0. (ii) Suppose Dl and D, are both dimension functions for A. For each a, the restrictions of Dl, D, to the set of projections of h,A evidently define dimension functions for h,A; by the assumed uniqueness for h,A, we conclude that D1(12,e)= D2(h,e) for all projections e in A. Thus h,D,(e)=h,D,(e) therefore Dl (e)= D2(e) for all e. I

for all a and all c,

In principle, one could use Proposition 2 to reduce the Type I case to the Type I, case (n=1,2,3, ...) [cf. 9 18, Th. I]; it turns out to be

# 28. Type I,,,,: Complctc Additivity and Uiliqueiless of Dimension

165

simpler to give a unified treatment of the Type I case, thereby avoiding the necessity of putting the homogeneous pieces back together. Exercises 1A. Let A be a Baer *-ring with GC, and suppose there exists a function e- D(e) defined on the set of projections of A , with values in a group (notated additively), satisfying conditions (Dl), (D3) and (D4) of [$25, Def. I].Then A is finite. 2A. Let A be a purely infinite (i. e., Type 111) Baer *-ring with orthogonal GC. Assume, moreover, that A is orthoseparable [$ 18, Exer. 101. Then e - f if and only if C(e)=C(f). 3A. Notation as in Proposition 1. (i) For every projection e, C(e) is the right projection of D(e). (ii) D(e) is a projection iff e is a central projection. 4A. The ring A of all 2 x 2 matrices over the field of three elements is a finite Baer *-factor of Type I, that possesses a dimension function D, although A does not have GC. If e , f are the projections described in [Ej I, Exer. 171, then ~ ( e ) =D(f ) = f but e and j' are not equivalent.

5D. Problem: Does every finite Baer *-ring have a dimension function?

8 28.

ripe I,.,,: Complete Additivity and Uniqueness of Dimension

Proposition 1. Let A he a ,fiMite Baer *-ring of Type I, with GC. Suppose D is a dimension ,function ,for A [§ 25, Def: I ] . Then D is completely additive; that is, jf (e,),,, is any orthogonal jumily ofprojections and e= supe,, then Die)= Die,).

1

LEI

Proof. For each finite subset J c 1, write e,= c=

1 D(e,). Thus

1 el,

and let

LEJ

D(e,) c and c 5 D(e) [$27, Lemma 21 ; the problem

,€I

is to show that c= Die). Let (ha)be an orthogonal family of nonzero central projections with sup h, = 1, such that for each a, the set of indices is finite [# 18, Prop. 51; it will suffice to show that h,c Fix an index a. We have

= h,

Die) for all a.

thus h, D(e)= h, DieIa) 1927, Prop. 1, ( D 6 ) ] ;the proof will be concluded by showing that

166

Chapter 6. 1)imension in Finite Raer *-Rings

At any rate,

[$27, Lemma 1, (i)]. If J is any finite subset of I such that J 3 I,, then h,e,=O for LEJ-I,, thus h,eJ=h,eI, and so h,D(e,)= h,D(e,J ; passing to the limit along J (the right side remains constant), we infer from (**) that h, c = h, D(e,J, which is the desired relation (*). I Uniqueness in the Type I case follows at once:

Proposition 2. Let A be a,finite Baer *-ring of Type I, with GC. If Dl and D, are dimension functions for A, then Dl = D,. Proof. Let e be any nonzero projection, and let (e,),,, be an orthogonal family of simple projections such that e=supe, 1526, Prop. 141. Since D,(e,)= T(e,)= D,(e,) for all 1 [$27, Prop. I],it follows from Proposition 1 that

5 29. Q p e I,,,:

Existence of a Dimension Function

Throughout this section, A is a finite Baer *-ring of Type 1, with GC. A feature of the exposition is that we do not explicitly resort to a decomposition of A into homogeneous summands (implicitly, this was done in the proof of [$26, Prop. 141). Lemma 1. If' el, ..., en are orthogonal, simple abelian projections,

x n

then

T(e,) 5 I .

1

h,

Proof. There exist orthogonal central projections h,, . . ., h, with 1, such that, on setting

+ ... +h,=

eori=haei (u=I, ..., Z; i=1, ..., n), either e,,=O or C(e,,)=h,

[tj 6, Prop. 51. It will suffice to show that

n

for each a, h,

n

T(h,ei) 5 h, [$ 26, Prop. 131.

T(ei) 5 h,, that is, 1

1

Dropping down to h,A and changing notation, we can suppose that the e, are faithful. We are reduced to showing that if el, . . . , en are orthogonal simple

x n

abelian projections such that C(ei)= I for all i, then

1

T(ei)5 1 . We

jj 29. Typc I,,,,: Existence of a Dimension Function

167

have el -...- en [$18, Prop. I ] , therefore T ( e l ) = ... = T(en) [$26, Prop. 111. Let r= ( I : e,); then T ( e i )= ( l / r )I and

: e l ) = r we may write The problem is to show that n < r. Since (I l = f l + . - . + , j ; with e , - j j ( j = l ,..., r); ifonehad r < n , then would contradict finiteness.

I

Lemma 2. If (e,),,, is an orthogonal jamily cf simple ahelian proT(e,) exists and is I I. jections, then

C

1st

Proof: This is immediate from Lemma 1 and the definition of such sums (cf. the discussion preceding [$27, Lemma 21). 1 Lemma 3. I f ' el +..-+em= J; + ... +h, where the ei and fi are simple abelian projections, the ei are orthogonal and theji are orthogonal, then

Proof. We can suppose, without loss of generality, that the ei and

fj are faithful (cf. the proof of Lemma 1). Then ei -,f;: for all i and , j [$ 18, Prop. I], therefore m = n and T(ei)=T ( f i ) [$26, Props. 1 and 111.

I

Lemma 4. I f e = supel= sup f,, where each of' (el),,, and (f;),,, is an orthogona1,family of simple ahelian projections, then

Proof. Write c=

C T(e,), d= C l t l

T ( f , ) (the sums exist by Lemma

xtK

2). Let (h,) be an orthogonal family of central projections with sup ha= I , such that for each a, the sets of indices I , = { L E I :hael# O } ,

K a = ( x € K : h,.f,#O)

are both finite [ $ I S , Prop. 51. It will suffice to show that Iz,c=h,d all a. Fix an index a. Clearly

for

Chapter 6. Dimension in Finite Baer *-Rings

168

applying Lemma 3 in h,A, it follows that

If

i

4 I, then h,e,=O, therefore h, T(e,)= T(hae,)= 0 ; it follows that

and similarly

h,d=

thus hac=had by (*).

x

ntK,

T(h,.fn),

I

Lemma 5. I f { c i j :i ~ l , , j ~ is J )a doubly indexed family in C,(%) such that 0 I cij5 1 ,for all i and,],then sup (cij:i~ I, j~ J ) = sup (sup c i j ) it1

jtJ

Proof. A routine argument, valid in any complete lattice-in thc present instance, the lattice of continuous functions c, on the Stonian space X, such that 0 I c I 1. I Proposition 1. If' A is a ,/kite Baer *-ring of Type I, with GC, then there exists a unique dimension function D for A. Moreover, D is completely additive.

ProoJ: Uniqueness and complete additivity are covered by Section 28; we are concerned here with existence. Define D(O)=O. If e is any nonzero projection, write e= supe, with (e,),,, an orthogonal family of simple abelian projections [$26, Prop. 141 and define C T(e,); re1

the sum exists and is I 1 (Lemma 2), and it is independent of the particular such partition of e (Lemma 4). It remains to verify properties (Dl)-(D4) of [§ 25, Def. I]. (D2) By construction, D ( ~ )C(%), E 0I D(e) 5 1 . ( D 1 ) If e is partitioned as above and if e f, then the equivalence induces a partition (f,),,, off such that el f, for all L ; since T(e,)= T ( f , ) [$26, Prop. 111, it follows that

- -

(D3) If h is a nonzero central pro-jection, it is to be shown that D(h)=h. Since the definition of D for h A is consistent with that for A,

# 29. Type I,,,,: Existence of a Oimensio~lFuilction

169

we can suppose without loss of generality that h = l . Thus, assuming (e,),,, is an orthogonal family of simple abelian projections such that sup el = I,and setting c = T(e,) (that is, c = D(l)), it is to be shown LEI that c = I. Let (h,) be an orthogonal family of nonzero central projections with sup h, = I,such that for each a, the set of indices

1

l , = { i ~ l :h,e,+O)

is finite [$18, Prop. 51; it will suffice to show that h,c=h, For any fixed index a,

for all a.

and

(sec the proof of Lemma 4);thus it suffices to consider the case that the family (el)is finite. Changing notation, we are in the following situation: we are given orthogonal, simple abelian projections e,, .. ., en such that n

and we are to show that

1T ( e i ) = l . 1

Dropping down to each of a

-

finite number of direct summands (as in the proof of Lemma I ) , we can suppose that the ei are faithful. Then el ... -- e [ $ I S , Prop. I ] and it follows from (*) that (1: ei)=n ; thus T(ei)= ( l / n )I,

n

T(ei)=I. 1

(D4) Assuming e, f are nonzero projections such that ef=O, it is to be shown that D(e+ f')= D(e)+ D ( f ) . Write where (ei)i,I and ( j J j , , are orthogonal families of simple abelian projections. Let K be the disjoint union of 1 and J ; for k g K , define g, = ek or ,h according as k ~ orl k~ J. By the associativity of suprema, (gk)ksKis a partition of e + f into simple abelian projections. Write E, F, G for generic finite subsets of 1, J, K , and write

170

Chapter 6 . Dimension in Finite Baer *-Rings

by the definition of D, we have Die)= sup a,,

D(e +J') = sup cG .

D(f )= sup b,,

E

F

G

Each G can be written uniquely in the form G = E u F, and one has cG=aE+b,; conversely, given any E and F we may define G = E u F. Thus, the cG's may be regarded as being doubly indexed by E's and F's:

Citing Lemma 5 at the appropriate step, we have D(e+ f ) = sup c,= G

= sup E

(a,

+ sup b,) F

Exercises 1A. If A is a finite Baer *-ring of Type I,, with GC, and if e is any projection in A , then D(e) is a simple function on X whose values are contained in the set {O, l / n , 2/n,. ..,(TI- l ) / n , I}. 2A. Proposition 1 yields an alternative proof of the additivity of equivalence for the case of a Baer *-ring of Type I,,, with GC [cf. 4 20, Th. I].

5 30. mpe IT,,,:

Dimension Theory of Fundamental Projections

Throughout this section, A is a finite Baer *-ring of Type I1 with GC. The techniques developed in this section are the continuous substitutes for the special properties of abelian projections (notably [# 18, Props. 1, 4 and 51). The exposition is patterned after that of J. Dixmier [18] (see also [2, Appendix 1111). Lemma 1. Suppose that T(f) +

P

n

21 T(&)5 2 Tie,) 1

3

# 30. Type TI,,,,: Dimension Theory of Fundamental Projections

171

where each of the projections f ; f,, . .., ,&, e l , . . ., en is either 0 or ,fundamental; assume that the ei are orthogonal, the f j are orthogonal, and that

Then f

5 xei1

Cf,. 1

Proof. For any central projection h,

[$26, Prop. 131; therefore, dropping down to each of a finite number of direct summands, we can suppose that each of the given projections is either 0 or faithful [56, Prop. 51. If f =O there is nothing to prove; assume f # 0. Those e,,f) that are 0 contribute nothing to the hypothesis or the conclusion; discarding them, we can suppose that the e,,f j are nonzero (but we accept the possibility that no f,'s remain-in

which case, both

1T ( , f j ) and 1f j P

P

1

1

are interpreted as 0). (We write out the proof assuming that the 1)are present; if they are not, the same proof works provided all reference to the f j is suppressed.) Say (1:e,)= 2rz,( 1 :fj) = 2"1, (I :f ) = 2". The hypothesis reads

thus (i) Let r be a positive integer greater than all of ri,s j , s, and let g be a projection such that (1:g)=2' [§26, Prop. 151. Since (1 : g ) is divisible by each of (1:e,), (I :f j ) and (I :f ), it follows from [$26, Prop. 91 that y divides each of e,, fj, f and that (f.: 1

)=T-"j,

(f:y)=2'-".

Multiplying through (i) by T , we have P

(ii)

n

) C(ei:~). (f :g)+C ( , f j : g 1 1

1

172

Chapter 6. Dimension in Finite Baer *-Rings

x P

2

Setting f = f, and Z= ei, the inequality (ii) may be written ( f g) 1

+(Fg) 1@: g)

1

[$26, Prop. 61, thus

(iii)

(f:g)s@:g)-(T:g);

moreover, since , T s F it follows that y divides F - x written (f:g)<@-.Fg), therefore f

SF-71926, Prop. 71.

and (iii) may be

1

Lemma 1 is the first step of an induction:

Proposition 1. Suppose that

where each of g,, j,,ei is either 0 or fundumental; assume that the ei are orthogonal, the j,are orthogonal, and that

-

Then there exist projections dl, .. ., d,, orthogonal to each other and to the f,, such that d, g, (k = 1, . ..,q) and

Proof: For q = l this is the lemma. Assume inductively that all is well with q - 1, and consider q. The given inequality implies that

-

By the induction hypothesis, there exist orthogonal projections d l , .. . , d,-, such that d, g, (k = 1, . .. , q - 1) and (ii)

dtc+Cfi~ei

Since T(d,) = T(g,) (k = 1, ..., q - I)the , given inequality may be written (iii)

5 30.

Type II,,,: Dimension Theory of Fundamental Projections

173

in view of the orthogonality of the projections on the left side of (ii), the lemma applied to (iii) yields

thus g, is equivalent to a projection d, meeting the requirements.

I

Lemma 2. If c€C(%), c 2 0, c#O, then there exists a fundamental projection f such that T(f ) 5 c. Proof: Since 3 is extremally disconnected, there exists an E > 0 and a nonzero projection h in C ( 3 ) such that c 2 e h. Let r be a positive integer such that 2-' < E and let f be a projection such that (h:f ) = 2' [§26,Prop.I5].Then T ( f ) = 2 - ' h ( c h < c . I Proposition 2. Suppose

where the ei are orthogonal, the f, are orthogonal, and each of'the ei,ji is either 0 or fundamental. Then

Proof. It suffices to show, e.g., that

Assume to the contrary; then there exists an c > 0 and a nonzero central projection h such that

thus

Dropping down to h A and changing notation, we can suppose that P

(*)

n

el+CT(f,)
1

Let f be a fundamental projection such that T (f ) < e I (Lemma 2); from (*), P n ~ (+ f1 ) ~ ( j ;I ) C ~ ( e ,. ) 1

1

Chapter 6. Dimension in Finite Baer *-Rings

174

Citing Lemma 1, we have

thus f =0, a contradiction. n

I

Lemma 3. Zf' e l , . . .,en are orthogonal fundumentul projections, then

zT(e,)lI. 1

Proof. We can suppose, without loss of generality, that the e, are faithful (cf. the proof of [$29, Lemma I ] ) . Say (1 : e , ) = 2 ' ~thus ; T(e,) n

=2-*. 1. Assume to the contrary that C2-" > 1. Let r be an integer such 1

that r > ri for all i and let y be a projection such that (1: y) = 2'. Then

and, arguing as in the proof of Lemma 1, this may be written n

C ( e , : g )> (1 : y ) ; 1

n

setting e= C e , , we have 1

(e:.y)> ( 1 z.4)

(*)

[$26, Prop. 61. From (*) it follows that 1 5 e [$26, Prop. 71, therefore e= 1 by finiteness; but this contradicts (*). I Proposition 3. I f ' (e,),,, is any ortlzogonal ,family of ,fundumental projections, then T(e,) exists and is 5 I.

1

LEI

T(c1,)5 1 by Lem-

Proof: For every finite subset J of I, we have IF

J

ma 3. (Cf. the discussion preceding [$ 27, Lemma 21.)

1

Lemma 4. l f ' (e,),,, is a,fumily of'projections such that h = inf C(e,)# O and if e is any projection such that h e f 0 , then there exists a nonzero central projection k such that k e and the ke, ure ,fuithful in kA. Proof. Let k= h C(e)=C(he). Since k 5 h 5 C(e,) for all C(ke,)=kC(e,)=k;also C(ke)=kC(e)=k. I

1,

we have

The following proposition is the Type I1 substitute for [$IS, Prop. 51 :

C; 30. Type Ilfi,: Dimension Theory of Fundamental Projections

175

Proposition 4. Let (r,),,, be an orthogonal family ofjundamental projections such that the orders (C(e,):e,)are bounded. There exists an orthogonal ,family (ha) o f nonzero central projections with sup h, = I, such that,for each a, the set ( 1 E 1 : h,e, # 0 ) is finite.

Proof. If h is a central projection, then either he,=O or he, is e,) (see the proof of [$26, Prop. fundamental with (C(he,):h el)= (C(el): 131). By an obvious exhaustion argument, it will suffice to find a nonzero central projection h such that { L E Ih: el # 0 ) is finite. Assume to the contrary that no such h exists. Let r be an integer such that 2" > (C(e,):el) for all L G 1, and write m = 2'. We construct m indices I , , ..., 1, as follows. Choose any 1 , € 1 and set h , = C(ell).By supposition, there are infinitely many indices L with h, el # 0 ; let be such an index, 1 , # l 1 ; by Lemma 4 there exists a nonzero central projection h, such that C(h, e l l )= C(h, e,J = h, . Continuing inductively, we arrive at indices L , , ..., 1 , and a nonzero central projection h, such that

L,

L,,

Dropping down to h m A and changing notation, we have the following situation: e l , .. ., em are orthogonal, faithful, fundamental projections, and m=2'>(1:ei) for i = l , ..., nz. Say ( 1 : e i ) = T z ;then T(ei)=2-':I, and Lemma 3 yields the absurdity

The most unpleasant (and the last) computation in the chapter is as follows:

Lemma 5. Suppose thut n

C1 7'( f,) 2 1T(e,)

7

ltl

where (e,),,, is an orthogonal jamily of' ,fundamental projections, and f , , .. . ,f i are orthogonal~fundun~ental projections such thut f l

n

Then

C T(,fj)= C T(e,). 1

+ ... +h5 supe, .

it1

Proof. Assuming to the contrary, Lemma 2 yields a fundamental projection g such that

Chapter 6. Dimension in Finite Baer *-Rings

176

n

In particular, T(g)2

1T(,fj),thus

g 5.f

1

,+...+.f,

by Lemma 1 ; re-

placing g by an equivalent projection, we can suppose that The plan of the proof is to construct an orthogonal family (g,),,,, g,-el, such that gg,=O and 9 , s f l + . . . +f ,

with

for all L E I ;this will imply y supg,=O, and it will then follow from additivity of equivalence [$20,Prop. 41 that whence sup g , = g + supg, by finiteness, g = 0, a contradiction. The construction of the g, is by induction; at the mth stage (m= 0,1,2, ...) one constructs the g, corresponding to those el whose order is 2". For m =O,1,2, ... write I , = ( L E I : (C(e,):e,)=2"}; thus I is the disjoint union of I,, I,, I,, . . . . Note that central. Suppose X G I,. Citing (i) and Lemma 3, we have

L CI ,

iff el is

thus T(g)+e, I 1; it is then clear from the functional representation in C ( X ) that e, T(g)= 0, thus exC(g)= 0, g e, = 0. Moreover, it follows n from (i) that T(e,) C T ( f;) 3

1

n

therefore ex< 1 f j by Lemma 1 ; since e, is central, it results from 1

n

finiteness that e, I

jj [$ 17, Exer. 21. Defining g, 1

an orthogonal family of subprojections of

n

= e,

(xcl,), we have

1f i that are orthogonal to g ; 1

this meets the requirements for the indices in I,. Assume inductively that suitable g , have been constructed for all 1 in I*=l,u... u 1,; thus, for 1 E I*, the g, arc orthogonal subprojcctions n

of

f j , g,g = 0 and g, 1

may be written (ii)

n

-

el.

Since T(g,)= T(e,)( 1 GI*),the inequality (i)

5 30.

Type II,,,,. Dimension Theory of Fundamental Projections

177

(the juggling with infinite sums is justified as in the proof of (D4) in [$29, Prop. I]). If I,,, is empty, there is nothing to be done, and the induction is complete. Otherwise, since I,+, c 1-I*, it follows from (ii) that (iii) 1

The projections (e,),EI,uI,+, have bounded orders (bounded by 2""). By Proposition 4, there exists an orthogonal family (h,),,, of nonzero central projections with suph,= 1, such that for each a, the set

-

is finite; since el y, for I GI*, this means that for each a, the sets are finite. Fix an index a c A . Multiplying through (iii) by h,, we have

and all but finitely many terms in (iv) are 0. Applying Proposition 1 in h,A to (iv), there exist orthogonal projections y: (LEI,, ,)-all but finitely many of them 0-such that, for each LEI,,^, g: h , ~ , ,

x

-

n

y: 2

h,&, and g: is orthogonal to hay and to the h,g, ( x ~ l * ) .

1

D o this for each ~ E A Then, . for each

L

E I,

+

,,define

by additivity of equivalence,

.

Since, for each ~ E A the , y: ( ~ c l , , ,) are orthogonal, it follows that the g, (LEI,,,+ ,) are also orthogonal. n Fix an index 1 E 1, + . Since g: I h, ,fi I ,fi for all a, we have

,

C

s, r Moreover, y, y = 0, because

n

1

C I

E .f; . 1

for all a c A . Similarly, glg,=O for x ~ l * . Thus the g, (1E 1, + ,) have the required properties. This completes the induction, thereby achieving the desired contradiction. I

Chapter 6. Dimension in Finite Baer *-Rings

178

Proposition 5. Let e he u nonzero projection, and suppose where (e,),,, and (fX),,, tions. Then

are orthogonal ,families c?f'fundamentalprojec-

C T(e1)= xCt K

LEI

T ( f x ).

Proof. The sums exist by Proposition 3. By symmetry, it is enough to show that thus, if J is any finite subset of K , it will suffice to show that

Assume to the contrary. Then (as in the proof of Proposition 2) there exists a nonzero central projection h such that

without having equality. Thus xtJ

ltl

equality does not hold in (*), and

this contradicts Lemma 5 (applied in hA).

5 31.

I

5 p e ITfi,: Existence of a Completely Additive Dimension Function

As in the preceding section, A is a finite Baer *-ring of Type I1 with GC.

Definition 1. If e is a nonzero projection in A, let (e,),,, be an orthogonal family of fundamental projections with sup e,= e [$26, Prop. 161 and define D(e)= C T ( e , ); LEI

the sum exists [ji 30, Prop. 31 and is independent of the particular decomposition [$30, Prop. 51, thus D(e) is well-defined. Define D(O)=0.

Proposition 1. If A is a finite Baer *-ring of Type 11, with GC, then the function D defined above is a dimension function ,for A. Moreover, D is coinpletely additive.

6 31.

Type IT,,,,: Existence of a Completely Additive Dimeilsion Function

179

Proof. We verify the conditions (Dl)-(D4) of [$25, Def. I]. (D2) Obvious from Definition 1. (D3) If e is fundamental, it is clear from Definition 1 that D(e)= T(e). In particular, if h is a central projection then D(h)= T(h)=h [$26, Def. 31. (Dl), (D4) The proofs follow the same format as in the Type I case 29, Prop. I],with 'simple abelian projection' replaced by 'fundamental projection'. Finally, suppose e = supe,, where (e,),,, is an orthogonal family of nonzero projections (not necessarily fundamental). We know that C D(e,) exists and that

[a

LEI

[$27, Lemma 21; it is to be shown that equality holds in (*). For each ~ € write 1

e,=sup (e,,:

X E K,)

,

where (e,,),,,, is an orthogonal family of fundamental projections. Then the el, are a partition of e into fundamental pro-jections, therefore D(e) is the supremum of all finite sums of the form where it is understood that x , K," ~ and the ordered pairs distinct. Given such a sum, let

( L , , , x,)

are

thus J is a finite subset of I with

for v = I, ... , n, therefore

(note that the terms on the left are orthogonal); then

Thus

1 D(e,) is

LEI

2 each expression of the form (**), therefore it is 2

their supremum D(e).

I

180

Chapter 6. Dimension in Finite Baer *-Rings

$32. Type lIfi,: Uniqueness of Dimension Proposition 1. Let A be a ,finite Baer *-ring of' Type 11, with GC. If' Dl and D, are dimension functions jbr A, then D l = D, and is completely additive.

Proof. Let D be the completely additive dimension function constructed in the preceding section; it suffices to show that Dl = D. Since every nonzero projection is the supremum of an orthogonal family of fundamental projections [Ej 26, Prop. 161, and since Dl and D agree on fundamental projections [tj 27, Prop. 1 , (D7)], it will suffice to show that Dl is completely additive. Suppose e=supe,, where (e,),,, is an orthogonal family. By [tj 27, Lemma 21, C Dl(el) 5 D,(e). LEI

We assert that equality holds. Set

and assume to the contrary that c#O. Let f be a fundamental projection such that T (f ) 5 c [tj 30, Lemma 21. Since D, (f )= T (f ) , we have thus All the more, if J is any finite subset of I, then

thus

it follows that

[§ 27, Prop. 1, (DIO)],therefore

thus

$ 3 3 . Dimens~onin an Arbitrary F ~ n i t cBaer *-King w ~ t hGC

181

Since J is an arbitrary finite subset of I,

C D(e1) 5 D(e)-D(fi,

LEI

and since D is completely additive this may be written D(ei 5 D(ei - D(.f') ; then D(f)=O, f

5 33.

= 0,

a contradiction.

I

Dimension in an Arbitrary Finite Baer *-Ring with GC

Theorem 1. If A is any finite Buer *-ring with GC, tlzel-e e.xists a unique dimension junction D for A. Moreover, D is completely additive. Proof. Since A is the direct sum of a Type I ring and a Type I1 ring [$ 15, Th. 21, it is enough to consider these cases separately [$27, Prop. 21. For the Type I case, see [$29, Prop. I]. For the Type I1 case, existence is proved in Section 31, uniqueness and complete additivity in Section 32. I An important application of complete additivity (used in thc proofs of [§ 34, Prop. 21 and [9: 47, Lemma I]): Theorem 2. Let A and D he as in Theorem I. If'e, f e then D(e,) D(e) (the notation is explained in the proof). Dually, e, e implies D(e,) J D(e). Proof. We assume that (e,) is a family of projections indexed by the ordinals p < 2, A a limit ordinal; the notation e,, 7 e means that o < p implies e , e,~ and that supe,=e. The notation e,Le is defined dually. In either case, we say that (e,) is a well-directed family. To exploit complete additivity, we replace (e,) by an orthogonal family (,f,), also with supremum e, defined inductively as follows: fl =el, and, for p > I, j;=e,-sup{e,:o
Chapter 6. Dimension in Finite Baer *-Kings

182

any finite subsum. Consider any finite set of indices p , < ... < [ I , . Then fi, <e,,<e,,, for i = l , ..., n , thus

f,, + ... +j;,?,I

x

ePn

n

and therefore

D(j;,,)I D(ePn)I c.

I

i= 1

In the Type I1 case, the following consequence of complete additivity is very useful (see the proof of [§ 41, Th. I ] ) : Theorem 3. Let A and D he 0,s in Tlzeorenl I , und ussurnc thut A is of' Type TI. Given any c~ C,(.??), 0 I c I I, thew exists a projection e in A such that D(e)= c. Proof: We can suppose c#O. Let (e,),,, be a maximal orthogonal family of nonzero projections such that

(*I

1D(e,)I c

for cvery finite subset J of I .

IEJ

(Such a family exists by an evident maximality argument, using 1930, Lemma 21 to get started.) Let e=supe,. By the complete additivity of D, D(e) is the supremum of the finite sums D(e,); in view of (*), we EJ have D(e) 5 c. We assert that D(e)= c. Assuming to the contrary, let f be a fundamental projection such that T ( f )< c - D(e) [$30, Lemma 21. Since D( f ) = T (f ), we have D(f')
+D(e)I c, that is, D(f,)

+

x

rtl

-

j, I I -e. Then f, is ortho-

D(e1)I c

3

contradicting maximality. {So to speak, D assumes all values between 0 and 1. For this reason, I the term p p e IT, is also used to indicate finite Type TI.) The next result could have been proved irnmediatcly following 1530, Prop. 41, but to do so would have interrupted the development of the dimension theory; this is a convenient place to record it: Theorem 4. Let A be a finite Buer *-ring with GC, and let Z he the center of A. Then A is orthoseparable ij and only i j Z is ovtho.sepuruble.

# 33. Dinlension in an Arbitrary Finitc Baer *-Ring with GC'

183

Proof. A *-ring is said to be orthoseparable if every orthogonal family of nonzero projections is countable. Thus the 'only if part is trivial. Conversely, suppose Z is orthoseparable; assuming (e,),,, is an orthogonal family of nonzero projections in A, it is to be shown that 1 is countable. It suffices to consider the cases that A is Type I or Typc I1 [g 15, Th. 21. Suppose first that A is Type 1. Let (h,),,, be an orthogonal family of nonzero central projections with supha= I, such that for each a, the set I , = { l ~ l h,el#O) : is finite 1518, Prop. 51. Since I =

U 1,

(because supha= I) and since,

,€A

by hypothesis, A must be countable, I is the union of countably many finite sets. Suppose finally that A is Type IT. Replacing each e, by a fundamental subprojection of it 1626, Prop. 161, we can suppose the e, to be fundamental. Partitioning I according to the countably many possible orders, we can assume that the el all have the same order. Let (h,),,, be an orthogonal family of nonzero central projections with sup lz, = I,such that for each a, the set I , = ( L E I : h,e,#O) is finite 1930, Prop. 41; the proof is concluded as in the Type I case.

I

As remarked at the beginning of the chapter, a salient fcature of the exposition is an avoidance of the use of the parallelogram law (P). Instead, we make do with GC; perhaps the key technical point is the fact, used in the proof of [$30, Lemma 51, that additivity of equivalence is available in any continuous Baer *-ring with GC (indeed, in any Baer *-ring with GC 1520, Th. I]). {To be sure, the examples of rings with GC offered in Section 14 are obtaincd via (P) [1$14, Prop. 7 and Th. I].) At any rate, without (P) this seems to bc about the end of the line. Exercises 1A. Let A be a finite Baer *-ring with GC. If e, t e (with notation as in Theorem 2) and i f f ' is a projection such that c, 5 f for all p, then e s f .

2A. If A is a finite Baer *-ring with GC, and if A is a factor (hence, for every pair of projections e, f,either e 5 f or f 5 e), then the dimension function D is real-valued and A is orthoseparable. 3A. Let A be a finite Baer *-ring with GC, let Z bc the center of A, let N be an infinite cardinal number, and suppose that every orthogonal family of nonzero projections in Z has cardinality 5 N. Then every orthogonal family of nonzero projections in A has cardinality 5 X.

Chapter 6. Dimension in Finite Baer *-Rings

184

4C. Theorem 2 holds more generally with directed families in place of welldirected families. That is, e, T e implies D(e,) T D(e) for increasingly directed families (e,); dually, e , l e implies D(e,)l D(r).

5 34.

Modularity, Continuous Geometry

We now adjoin the parallelogram law to the foregoing hypotheses: in this section, A is a jinite Baer *-ring with GC, satisfying the parallelogram law (P). (In connection with Baer *-rings satisfying GC and (P), see [$14, Prop. 7 and Cor. 1, 2 ; Th. 1 ; Exer. 5, 9, 21, 221 and [$20, Exer. 21.)

Proposition 1. The projection lattice qf'A is modular; that is, if e, f, g are projections in A and if e I g, then (e u f ) n g = e u ( f n g). Proof: Let h = ( euf ) n g, k = e u ( f n g). Obviously h 2 k, and it is straightforward to check that Invoking (P),it follows that thus h - f n g - k - f n g ; adding f n g , we have h - k . Thus h - k ~ h , therefore k = h by finiteness. I

Proposition 2. Let (e,) be a well-directed family of projections in A. If e, T e then e, n f T e nj for all projections f . Dually, e,l e implies e,uf.leufjorall f . Proof. Suppose e, f e (for the notation, see the proof of [$ 33, Th. 2]), and let j' be any projection. Set y = sup(e, n f ). Obviously e, n f T g 5 e n f ; it is to be shown that g = e n f : Citing (P),we have

therefore where D is the dimension function for A ; thus D(e,) 5 D(e)+D(e,n f ) - D ( e nf ) I D(e)+ D ( g ) - L ) ( e n f ) for all p. Since sup D(e,)= D(e) [§33, Th. 21, it follows that

that is, D ( e n f - g ) 1 0 ,

therefore D ( e n f-g)=O,

enf-g=0.

5 34.

Modularity, Continuous Geometry

185

{A direct proof can be given, avoiding the use of the dimension function [47, Th. 6.51, [54, Th. 691.) 1

Definition 1. A continuous geometry is a complete, complemented, modular lattice, such that if (e,) is a well-directed family then e,Ie implies e, nft e nf for all f , and e, J e implies e, u f Le u,f for all f: Since the projection lattice of a Baer *-ring is complemented (even orthocomplemented, with canonical complementation e-e' = I- e), Propositions 1 and 2 may be summarized as follows:

Theorem 1. If A is afinite Baer *-ring with GC, satisfying the parallelogram law (P), then the projection lattice of A is a continuous geometry. Continuous geometries were invented by von Neumann ([69], [71]), to whom Theorem 1 is due for the case of a factorial finite von Neumann algebra on a separable Hilbert space. For finite AW*-algebras, the theorem is due to Kaplansky [47, Th. 6.51. In a subsequent paper, Kaplansky also showed that if a Baer *-ring A is regular (for each U E A there exists x e A with a x a = a ) then the projection lattice of A is a continuous geometry 151, Th. 31; along the way, he proved that A is finite (in the strong sense that y x = 1 implies x y = I [cf. tj 17, Exer. 4, 5, 61) and that A satisfies the variants of GC and the parallelogram law (P) with respect to 'algebraic equivalence' [§ 1, Exer. 61. Incidentally, one can arrive at Theorem 1 via the following 'shortcut' through the literature: prove modularity as in Proposition 1, and quote Kaplansky's theorem that any orthocomplemented complete modular lattice is a continuous geometry [51]. A more general version of Theorem 1 is given in [54, Th. 691. Exercises 1A. If 2 is an infinite-dimensional Hilbert space, then the conclusion of Proposition 2 does not hold in the projection lattice of 9(A?).

2C. The projection lattice of an AW*-algebra A is modular iff A is finite. 3C. Let L be a complete lattice in which increasing well-directed families have the property indicated in Proposition 2. If e,Te, where (e,) is any increasingly directed family in L, then e , n f t e n f for all f ; that is, one can drop the wellordering of the indices. It follows that if e, ? e and f, f f then e, nf, ? e nf . 4A. The projection lattice is modular for any finite Rickart *-ring satisfying the parallelogram law (P).

Chapter 7

Reduction of Finite Baer *-Rings Throughout the chapter, including the exercises, A denotes a fznite Baer *-ring such that LP(x)-RP(x) for all x in A ; in the final section of the chapter, A is a finite A W*-algebra. (For generalities on the condition LP-RP, see [5 14, Cor. 2 of Prop. 71 and 1520, Exer. 2, 61.)

5 35.

Introduction

The principal result of the chapter is easy to state: If I is a maximalrestricted ideal of A [cf. $241, then the quotient ring A/I is a Baer *-ring (hence is a finite Baer *-factor). In addition, the intersection of all such ideals I is (0) [$36], thus A may be 'embedded' in the complete direct product of the A/I. These two facts constitute the 'reduction theory' alluded to in the chapter heading. The hypothesis LP R P implies that A has GC and satisfies the parallelogram law (P) [$14, Cor. 2 of Prop. 71. As at the beginning of Chapter 6, we write Z for the center of A and Y' for the Stone representation space of the complete Boolean algebra 2, and we identify a central projection h with the characteristic function of the clopen subset of % to which it corresponds. Since A is a finite Baer *-ring with GC, the results of Chapter 6 are applicable to A ; in particular, A has a dimension function D [533, Th. I]. (In view of the parallelogram law, we know in addition that the projection lattice of A is modular-indeed, it is a continuous geometry [534, Th. I]-but the discussion in the present chapter avoids such matters.)

-

5 36.

Strong Semisimplicity

A ring with unity is said to be strongly senzisimple if the intersection of all the maximal ideals (two-sided) is (0). A related concept, relevant to the reduction theory, is as follows: S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

5 36.

Strong Semisimplicity

187

Proposition 1. The intersection of' all the maximal-restricted ideals of A is {O).

Proof. Since, for a restricted ideal 1, x ~ iffl RP(x)EI, it is the same to show that the intersection of all maximal p-ideals is (0) [$22, Th. I]. Suppose to the contrary that some nonzero projection e belongs to every maximal p-ideal. Since e contains a simple projection [Cj 26, Props. 14 and 161, we can suppose without loss of generality that e is simple; then h=C(e) also belongs to every maximal p-ideal. Since h # 0, the p-ideal is proper; let 3 be a maximal p-ideal such that (I -h)A" c~9(Zorn's lemma). In particular, I - h ~ 9 ; but h ~ . 9by the supposition; thus 1e.9, contradicting properness. I The following corollary is not needed for the main line of development, but it is important enough to be noted; it applies, in particular, to finite A W*-algebras : Corollary. Suppose, in addition, that A satisfies the following condition: for each nonzero X E A, the ideal AxA generated by x contains a nonzero projection. Then A is strongly semisimple.

Proof. {We remark that the condition is implied by any axiom of 'EP' type [cf. $7, Def. 31.) Assume to the contrary that A is not strongly semisimple, and let x be a nonzero element that belongs to every maximal ideal M. Choose a nonzero projection E E A X A ( = thc set of all finite sums xa,xb,); then eEAxA c M for every M. A contradiction to Proposition 1 will be obtained by showing that if 1 is any maximalrestricted ideal of A, then e e I. Let M be a maximal ideal such that 1c M (Zorn's lemma). Then ?cM; since M is a p-ideal [$22, Prop. I]and I" is a maximal p-ideal [Cj22, Th. I], it follows that I"= M . Then e~ M = f, achieving the contradiction. I Exercises 1A. A finite AW*-algebra is strongly semisimple.

2C. Every strongly semisimple ring with unity is semisimple. 3A. If A is factorial then it is restricted-simple, in the sense that (0) is the only restricted proper ideal of A . 4A. In any finite Baer *-ring with GC, the intersection of all the maximalstrict ideals is { O ) . (A maximal-strict ideal is one that is maximal among proper, strict ideals [cf. 9: 24, Exer. 61.)

188

Chapter 7. Reduction of Finite Baer *-Rings

5 37.

Description of the Maximal p-Ideals of A : The Problem

Definition 1. If OF.%, we define and

Proposition 1. For each a€%', 9" is a proper p-ideal of A, and .Yg nZ

=A.

Proof. Since A satisfies the parallelogram law (P),we havc for every pair of projections e, f'; in particular, it is clear that if e, , f e 9 , then e uf i y a b . If j'5 e ~ 9 then , D(j")I D(e) shows that f e y g . Thus .Yc is a p-ideal. Since D(h)=h for central projections h, obviously .$nZ =$,; in particular, 1 $9,, thus 9, is proper. I

Definition 2. If o~ $ we write I, for the restricted ideal of A generated by .Po, that is, I,= { X E A: R P ( x ) E.Yo)[§ 22, Th. I]. The problem, alluded to in the section heading, is to show that every 9, is a maximalp-ideal, and that every maximal p-ideal has this form. It turns out that the crux of the matter is to show the following: if 9 is any proper p-ideal, then 9 c Yo for some a e X . Suppose, to the contrary, that 9 is a proper p-ideal for which no such o exists; let us pursue the matter as far as possible by naive techniques. By supposition, for each a€.% there exists an e ~ such 9 that D(e)(o)> 0, hence D(e)> 0 on a neighborhood of a ; the obvious compactness argument produces a finite set e l , ..., e, in 9 such that D(el)+...+ D(e,)> 0 on X . Setting e=e u... u e,, we have ~ € and 9 D(e) > 0 on X . Since X is compact, there exists E > 0 with E I 5 D(e). Suppose first that A is of Type 11; choose a positive integer r with 2-' < E, and a fundamental projection f such that ( 1 : f ) = 2 ' [$26, Prop.151; then D ( f ) = 2 - r 1 1 ~ 1 1 D ( e )thus , f 5 e ~ and 9 therefore , f ~ $ ; since f is simple, 1 = C(f ) ~ 9 a, contradiction. Next, suppose that A is of Type I,, and let f be a simple abelian projection with (1 :f ) = n ; since C(e)= I [$27, Exer. 31, we have f 5 e [Ej18, Cor. of Prop. 11; one argues as above that 1 €9, a contradiction. The discussion extends easily to cover the case that A is the direct sum of finitely many homogeneous rings. There remains the general Type I case, in which A may have homogeneous summands of arbitrarily large order [cf. Ej 18, Th. 21; it is to the solution of this stubborn case that the strategem of the next section is directed.

5 38.

Multiplicity Analysis of a Projection

189

Exercise 1A. Let ./ be a proper p-ideal of A, and let e ~ . / . (i) If A is homogeneous, then C(e) # 1 (in particular, D(e) is singular). (ii) If A is the direct sum of finitely many homogeneous rings, then C(e) # 1. (iii) If A does not have homogeneous summands of arbitrarily large order, then D(e) is singular; it follows that 9 c Yc for some OE.%. (iv) If e is abelian then D(e) is singular. It is shown in Section 39 that D(e) is singular regardless of type [§ 39, Exer. I].

5 38.

Multiplicity Analysis of a Projection

For motivation, see the preceding section.

Definition 1. Iff is a projection in A, h is a central projection, and n is a positive integer, we say that h contains n copies o f f in case there exist orthogonal projections f l , . ..,f, such that f' fl ... ,f, and h 2 ,fl + ... +j". (Since h is central, the latter condition is equivalent to h 2 f [§ 1, Exer. 151.)

- - -

Remarks. Suppose h contains n copies off. 1. If k is any central projection, then k h contains n copies of kd: 2. By the properties of the dimension function, D(f ) ( l / n ) h [$ 27, Prop. I ] . Proposition 1. I f e is any projection in A and n is a positive integer, then exists a (unique) largest central projection h such that h contains n copies of h e (that is, for a central projection k, k < h if' and only i f k contains n copies of k e). Denoting it by h,, we have hl = 1 and h,L 1 - C(e). Proof. If k is any central projection then k contains one copy of ke, thus h , = 1 has the required properties. Assume n 2 2. If no nonzero central projection h contains n copies of he, set h,=O. Otherwise, let (h,),,, be a maximal orthogonal family of nonzero central projections such that h, contains n copies of h,e. Say h,> e,, +... +e,, , where h,e-e,,-...-e,, Then

. Define h=suph, and e,=supe,,(v=I ,..., n).

h e - e l -...-en

LEJ

by additivity of equivalence [§ 20, Th. 11, thus h contains n copies of he. We show that h has the required properties. If k is a central projection with k I h, then k contains n copies of k e (Remark 1 above). Conversely, suppose k is a central projection such that k contains n copies of k e ; it is to be shown that k 2 h. Indeed, since ( I -h)k contains n copies of ( I -h)ke, and since ( I - h)k is orthogonal to every h,, it results from maximality that ( I - h)k = 0, thus k < h. We define h, = h.

190

Chapter 7. Reduction of Finite Raer *-Kings

For all n, 1 - C(e) trivially contains n copies of (1 - C(e))e=O, thus 1 - C ( e )5 h,; writing h' = infh,, we thus have 1 - C(e)I h'. On the other hand, h' I h, implies that h' contains n copies of h'e, thus for all n ; it follows that hlD(e)=O, D(hle)=O, hle=O, hlC(e)=O, h' 5 1 - C(e). Thus h' = 1- C(e). Finally, since h,, contains n + I and therefore n-copies of lz,, e, we have h,+ 5 h,, thus h,i h'. I

,

,

,

-

,

Proposition 2. With notation as in Proposition I , dejine k,= h, - h, , (n = I , 2, 3,. . .). (1) k, is an orthogonalsequence of central projections with sup k, = C(e). (2) For each n, there exists an orthogonal decomposition such that k,e-el-...-em

and g , s k , e

Proof. (1) This is immediate from h,J I - C ( e ) and h , = I [912, Lemma]. (2) Since k, I /I,, we know that k, contains n copies of k,e, say

- - -

with k,e el ... en. Define g, = k, -(el + +en); it will suffice to show that g, 5 k, e. The proof is based on the fact that, since k, 5 1 - h,, no nonzero central projection k 5 k, can contain n + I copics of k e. Apply GC to the pair k,e and k,g,=g,: let lz be a central projection such that (*I hk,e
,,

(1-h)k,g,5(1 -h)k,e.

Setting k = h k,, (*) reads thus k contains n + I copics of k e ; since k 5 k,, it follows, as noted above, that k=O. Thus hk,=O and (**) yields k,g,sk,e, that is, ~nskne. The following term is convenient:

Definition 2. With notation as in Proposition 2, we call the sequence k, of central projections the multiplicity sequence of e. We remark that, in the proof of Proposition 1, the use of the dimension function is easily avoided [cf. 5 17, Exer. 231. The parallelogram law figures in the preceding and the following sections, but the results of the present section are valid for any finite Baer *-ring with GC.

$ 3 9 . Description of the Maximal p-ldcals of A : 'The Solution

5 39.

191

Description of the Maximal p-Ideals of A : The Solution

For motivation, see Section 37. Lemma 1. Let e he a projection in A and let k, he the multiplicity sequence of e [$38, Def. 21. If .a is a p-ideul such that k , e ~ , f for all n, then k , ~ . ffor all n. Proof: This is clear from [$ 38, Prop. 2, (2)].

1

Lemma 2. Let e he a projection in A and let k, he the multiplicity sequence of e. If X and (f k, e ~ & ,for all n, then ~E.Y,. Puoqf. Recall that Yu= { f EA" : D(f)(o)= 0 ) is a proper p-ideal [$37, Prop. I]. By Lemma 1 we have ~,E,Y, for all n, that is, k,(a) = 0; it is to be shown that D(e)(o)=O. By definition, k,=lz,-/I,,+, [# 38, Prop. 21, thus k,+...+k,,=h,-h,=1-/I,;

,

since k,, .. ., k , ~ . f , , it results that I- h,rY,, (*)

h,(a)=l

thus

for n=1,2,3, ... .

By the definition of h, [$38, Prop. I], h, contains n copies of h,e, thcrcfore h, D(e)= D(h,e) 5 (l/n)h,; evaluating this at o, it results from (*) that D(e)(o)
.a, is a

maximal p-ideal.

Proof. At any rate, 9, is a proper p-ideal of A and .F, n Z = f, ~ E Zh(o)=O) : is a maximal p-ideal of Z 1937, Prop. I]. Lct .a be a maximalp-ideal of A with .a,c .f (Zorn's lemma). Then .f, n Z c Y nZ ; since Y n Z is a proper p-ideal of Z, it results from the maximalily of ,A that . Y u n Z = . f n Z . =(

It will suffice to show that .Yu=.f. Suppose c.c.9 and k, is the multiplicity sequence of e. Since k,e~.Y, we have k , ~ 9by Lemma 1. Then k, E .an Z = .a, n Z c 9, for all n, therefore e 6Yu by Lemma 2. 1 Conversely : Lemma 4. 1f.Y is a maximal p-ideal cf A, then there exists u unique G E Tsuch that 9 =Yo. Pvoof: Since 9 n Z is a maximal p-ideal of Z [$24, Prop. 31, there exists o ~ % 'such that Y n Z = f u = f h ~ z : h ( o ) = O ) . But .Yo is also a maximalp-ideal of A (Lemma 3) and YUn Z = &, = ,8nZ 1537, Prop. I], therefore .Y,=,a by weak centrality [$24, Prop. 21. If also Y =.a,, then Y r = . ~ , n Z = . f u n Z = & , , therefore z = o . I

Chapter 7. Reduction of Finite Baer *-Rings

192

Combining Lemmas 3 and 4 with [# 24, Prop. 31, we have the promised description :

Theorem 1. The mapping o HYo is a bijection of 3 onto the set of' all maximal p-ideals of A. Alternatively, writing I,

[g 22, Th. I]:

= ( X E A : RP(x)E&)

[jj 37, Def. 21, we have

Corollary. The mapping o- I, is u bijection o f 3 onto the set qf' all maximal-restricted ideals of A. This is a convenient place to record the following proposition, which is a device for dropping down to direct summands: Proposition 1. Let I be a maximal-restricted ideal of A and let h be a central projection such that h$ I . Then h 1 is a maximal-restricted ideal of hA, and A/I is *-isomorphic to hA/h I . Proof. Write n: A -+ A/I for the canonical mapping. Since h $ I we have 1- ~ E [§I 24, Prop. 11, thus x - h x ~ for l all X E A . It follows that if no is the restriction of to hA, then no: hA + A/I is surjective. The kernel of no is (hA)n I = h I , thus A/1 is *-isomorphic with hA/h I . It is elementary that h l is a restricted ideal of h A [cf. 5 22, Def. 21; it remains to show that there are no restricted ideals properly between h l and hA. Recall that RP(n(x))= n(RP(x)) for all X E A [523, Prop. 11. Some general remarks: Suppose B and C are Rickart *-rings and cp: B + C is a *-homomorphism of B onto C such that RP(cp(x)) =cp(RP(x)) for all ~ E B It. is routine to check that if J is a restricted ideal of B then q ( J ) is a restricted ideal of C ; and if K is a restricted ideal of C, then cpp'(K) is a restricted ideal of B containing the kernel of q (the latter is even a strict ideal of B [cf. # 22, Exer. I]). It follows that the kernel of cp is a maximal-restricted ideal of B if and only if C is 'restricted-simple', i. e., C has no restricted ideals other that {Of and C. Apply the preceding paragraph to the canonical mapping A + A l l : since I is maximal-restricted in A, we infer that A/I is restricted-simple. Then the *-isomorphic ring hA/hl is also restricted-simple; applying the preceding paragraph to the canonical mapping h A + hA/h I, we conclude that h l is maximal-restricted in hA. I Exercises

1A. If 9 is a proper p-ideal of A, then there exists at least one point o€X such that 9 ~ 3 in~ particular, ; for all e ~ 3 D(e) , is a singular element of the Banach algebra C(X). 2A. Let N be a proper ideal of C(X) and define .y,={e~A: D(e)€N}. (i) YN is a proper p-ideal of A such that .aNn Z = B . (ii) If N is closed then YN is the intersection of the maximal p-ideals that contain it.

5 40.

Dimension in A/I

(iii) Let IN be the restricted ideal generated by such that e - f GI,, then D(e)-D( EN.

193

.aN.If e , f are projections in A

3A. Suppose A is of Type 11, and let 9 be a p-ideal of A . Define P= {ctD(e): = P- P+ i P - i P is an ideal of C(X) whose positive part is P, and N = S n Z . a>O, e ~ 9 ) - .Then N

5 40.

Dimension in A11

We fix a maximal-restricted ideal I of A, and write I = I, for a suitable a€%' 1939, Cor. of Th. I]. Reviewing Section 23, we know that A/I is a finite Rickart *-ring with GC, satisfying L P RP, and the canonical mapping x R of A onto A/I enjoys the properties listed in [$ 23, Prop. I]. Moreover, A/I is a factor [$24, Prop. I]. {Alternatively, it is obvious from 1 =I, that, for a central projection h, either ~ E orI I - h ~ l , thus A/I is a factor by [$23, Cor. of Prop. 51.) Our ultimate objective is to prove that A/I is a Baer *-ring. Since A/I is a Rickart *-ring, it will suffice to show that every orthogonal family of projections in A/I has a supremum [$4, Prop. I]. In this section we show, by passing to quotients with the dimension function, that A/I is orthoseparable (that is, only countable orthogonal families occur).

-

-

Lemma. I f e and f are prqjections in A such that e - f ~ l , then D(e)(a)=D ( f )(a). Proqf. Since A satisfies the parallelogram law (P) [$13, Prop. 21, there exist orthogonal decompositions

-

such that e' f ' and e f " =e" f = 0 [§ 13, Prop. 51. Since I contains e- f , it also contains ( e - f)e"=eer'-O=e" and ( f - e ) . f U = f " , thus D(e")( a )= D( f ") ( a )= 0. Since D(el)= D( f ') and D is additive, we have D(e)- D( f ) = D(e")- D( f "), and evaluation at a yields D(e)( a )- D ( , f )(a) =o. I

Definition 1. We define a real-valued function D, on the projection lattice of A/I as follows. If u is a projection in AII, write u = l with e a projection in A ; if also u = ,f, f a projection in A, then e - f 6I , therefore D(e)( a )= D( f ) ( a ) by the lemma. We define (unambiguously) D,(u)=D(e) (a). Thus DAz) = for all projections e in A .

(a)

Chapter 7. Reduction of Finite Baer *-Rings

194

Proposition 1. T h e real-valued function D, on (All)" has the fbllowing properties: (I) 05DI(u)51, ( 2 ) D1(1)=1, ( 3 ) DI(u)=O iff u=O. (4) u v = 0 implies Dl ( u + v) = D, ( u )+ Dl (v), ( 5 ) u v if D,(u) = D,(v), ( 6 ) u 5 z1 $f Dl (u)5 Dl (v).

-

Proof. ( 1 ) and (2) are obvious. ( 3 ) If u=p, e e A , then u=O iff e6?=yn iff D(e) (o)=O, that is, D,(u) = 0 . (4) Suppose uv=O. Write u=Z, v= f with e f =O [#23, Prop. 21. Then e + f is a projection, D ( e + f ) = D ( e ) + D ( f ) and u+v=.?+ f = ( e + f ) " , therefore

-

-

(5), (6) Suppose u v. Write u= d , v = f with e f [$ 23, Prop. I ] . Then D ( e ) = D ( f ) , therefore D,(u)= D(e) (o)=D( f ) (o)= D,(v). Since A / I has GC and is a factor, any two projections u , v in A/I are comparable: u 5 v or v 5 u. Moreover, A11 is finite, thus the proofs of (9, (6) may be completed by the arguments in [# 27, Prop. I ] . I When the proof that A/I is a Baer *-ring is completed, D, will be its unique dimension function [Cj 33, Th. I ] , and in particular, DI will be completely additive. For the present, we are content to exploit finite additivity to prove the following:

Proposition 2. A/I is orthoseparable. Pro$ Lct (u,),,, bc any orthogonal family of nonzcro projections in A / I . For n=1,2,3 ,... write

By ( 3 ) of Proposition 1 , we have

it will suffice to show that each K, is finite. Indeed, if x ~ ,. ..,X, E K , arc distinct, then

thus r < n .

I

$41. A l l Theorem: Type 11 Case

195

Thus, to complete the proof that AjI is a Baer *-ring, it remains to show that every sequence of orthogonal projections in AjI has a supremum. For A of Type 11, this is quite easy (Section 41); for A of Type In, it is nearly trivial (Section 42); the most complicated case, again [cf. S; 371, is that where A is of Type I with homogeneous summands of arbitrarily large order (Section 43).

5 41.

A11 Theorem : p p e TI Case

We assume in this section that A is of Type 11. Fix a maximal-restricted ideal I in A, and write I = I, for a suitable o e Y [S;39, Cor. of Th. 1 1 (for a shortcut, see the discussion in Section 37). Let D, be the dimension function for A/I introduced in the preceding section.

Theorem 1. Suppose A is a Jinite Baer *-ring of Type 11, satisj-ying LP- RP, and let I he a maximal-restricted ideal of A. Then A/I its a finite Baer *+ctor OJ Type 11. Moreover, A/I satisfies LP -RP, and any two projections in A/I are comparable. Proof. From the discussion in Section 40, two things remain to be shown: (1) every orthogonal sequence of projections in A/I has a supremum; (2) A/I is of Type 11. Granted ( I ) , the proof of (2) is easy: if u is any nonzero projection in A l l , say u=& with e e 2 , one can write e= f'+ y with j g [S;19, Th. I]; then u = f +g with g, therefore u is not abelian [S;19,Lemma I ] , thus A/I is continuous [S;15,Def. 3, (4b)l. Suppose u,, u,, u,, .. . is an orthogonal sequence of projections in A / I . The plan is to construct a projection u such that u, c u for all n and

-

r-

C L

1D,(un)=DI(u), and to infer from these properties that 1

u=sup u,.

Let an= DI(un).For all n,

m

defining a = x u i , we have 0 < a < 1. We seek a projection u such that 1

un1u for all n and D,(u)=a. Write u,=t?,, with r, an orthogonal sequence of projections in A [$23, Prop. 21; in particular. a,=D,(u,) = D(en)(a)[$40, Def. I]. Since 0 1 a,, 1 1 and A is of Type 11, there exists, for each n, a projection ~ , E A such that D(J,) * = x,l [S;33,Th. 31. In particular, D(J,)(o)= cc, = D(en)(o), thus D,(J,) = Dl(?,); it follows that P, - , f n [S;40, Prop. 11, hence there exist subprojections g, < en, hn cfnwith - - %-A,, g,=e,=u,, h,=f,,

Chapter 7. Reduction of Finite Baer *-Rings

196

[§23, Prop. 11. Then D(gn)=D(hn)5 D(f , ) = d ; it follows that for every finite set J of positive integers,

Define g=sup y,. Since the en are orthogonal, so are the g,; in view of (*), the complete additivity of D yields

Define u=g. Since g 2 g,, we have u 2 u, for all n. n

We assert that D,(u)= a. For all n, we have

1ui 5 u; then 1

for all n, thus a I D,(u). On the other hand, it follows from (**) that D,(u)=D(g)(o)< a. {By the use of constant functions, we have circumnavigated the fact that the 'infinite sums' in C ( 3 )described in Section 27 cannot in general be evaluated pointwise.) Finally, assuming v is a projection in A/I such that u, 5 v for all n, it is to be shown that u 5 v, that is, u(1- v) = 0. Say v =f ,f a projection in A, set x = g ( l - f ) , and assume to the contrary that Z# 0, that is, ~ $ 1 .Then LP(x)#l (because I is an ideal); writing yo=LP(x), we have go I g, go# I. Thus, setting w =go, we have w 5 u, w f 0, and w = (LP(x))" = LP(X). Note that w is orthogonal to every un; indeed, therefore unLP(2) = 0, that is, unw = 0. Since w, u,, u,, ..., u, are orthogonal subprojections of u, we have

thus D,(w) +

n

1a, I a ;

since n is arbitrary, it results that Dr(w)+a I a,

1

thus D,(w)=O, w= 0, a contradiction.

5 42.

I

A/Z Theorem : n p e I, Case

We assume in this section that A is of Type I, [$IS, Def. 21 (n=1 is admitted, that is, A can be abelian). Fix a maximal-restricted ideal I of A, and write I =I, for a suitable [$39, Cor. of Th. I] (for a

# 42. A11 Theorem: Type 1, Case

197

shortcut, see the discussion in Section 37). Let D, be the dimension function for A/I introduced in Section 40. The main result of this section :

-

Theorem 1. Suppose A is a finite Baer *-ring of' Type I,, satisjying L P RP, and let I be a maximal-restricted ideal qf A. Then A / I is a finite Baer *-factor of Type I,. Moreover, A/I satisfies LP RP, and any two prqjections in A/I are comparable.

-

We approach the proof through two lemmas. Lemma 1. If J' is any abelian projection in A, then J' is simple and D ( f )= ( l l n ) C ( f).

Proof: By hypothesis, there exists a faithful abelian projection e such that (1 :e)= n, that is, there exists an orthogonal decomposition with e

- el --..-en.

1 =el +...+ en Then

-

C(f ), and it will clearly suffice to show with e C ( f ' ) - e , C ( f ) - . . . - e that f eC(f ). Indeed, e C(f') and f' are abelian projections such that C(eC(f ) ) = C(e)C(f ) = C ( f ) [96, Prop. 1, (iii)], therefore eC(f') - , f [§18, Prop. I]. I Lemma 2. I f ' e is any projection in A, tlzen the values of' D(e) are contained in the set {vln:v = 0,1, . .. , n}.

Proof: Write e=sup e,, with (e,),,, an orthogonal family of abelian projections (see [$26, Prop. 141 or [§18, Exer. 21). From Lemma 1 we know that D(e,)(.%)c (0,Iln) for every 1 E J. Let (h,) be an orthogonal family of nonzero central projections with suph,=l, such that for each a, the set J , = { L E J : ~ , ~ , # Ois) finite [jj 18, Prop. 51. Let Pa be the clopen set in .% whose characteristic function is (identified with) h,, thus P, = (z E .%: lz,(z) = I). Since sup h, = I, q!I =.UP, is a dense open set in E. Write F = {vln:v = 0,1, . ..,n). Since GY is dense and D(e)is continuous, it will suffice to show that D ( e ) ( Y )c F; fixing an index a, it is enough to show that D(e)(P,) c F. We have

evaluating at any z E P,,

198

Chapter 7. Reduction of Finite Baer *-Rings

and since D ( e , ) ( z )[O, ~ l l n ) , it results that D ( e ) ( z )F.~ f Incidentally, D(e) is a simple function: if k, is the characteristic function of the set { z : ~ ( e ) ( z ) = v / nthen ) , D(e)=

(v/n)k,.)

I

v=o

Proof o f Theorem I . If u,, ..., uk are orthogonal, nonzero projections in AII, then since DI(ui)2 l / n by Lemma 2 [cf. $40, Def. I], we have

thus k 5 n. This shows that every orthogonal family of nonzero projections in A/1 is finite; since their sum serves as supremum, the discussion in Section 40 shows that A/I is a finite Baer *-factor, with comparability of projections, satisfying LP RP. It remains to show that All is of Type I,. Let el be an abelian projection in A such that ( I : e l )= n, and write

-

with e , -...-en.

Setting ui=Eir we have

with u , -...- u,; in particular, D,(u,)= lln. The proof will be concluded by showing that u , is a minimal (hence trivially abelian) projection. If u is a nonzero projection with u 5 u,, then 0 < D,(u) 5 Dr(ul)=l l n ; but D,(u) 2 l / n by Lemma 2, thus DI(u)= D,(u,), Dr(ul- u) = 0, ul-u=o. I Let us note a slight extension of Theorem 1. With A again a general finite Baer *-ring satisfying LP-RP, suppose h is a nonzero central projection in A such that hA is of Type I,. Let P be the clopen subset of 3 corresponding to h. Fix O E P and let I = I,; thus h(a)= 1, equivalently l - h e I, equivalently, h $ l . We assert that A/I has the properties listed in Theorem 1: this is immediate from [$39, Prop. I ] . and Theorem 1 applied to hA. Exercises 1A. With notation as in Theorem 1 and its proof, identify A with (el Ae,), [$ 16, Prop. I]. (i) I = { x € A : D(RP(x))=O on a ncighborhood of o). (ii) I = J,,, where J = { a t e ,A e , : hu=O for some central projection h with h(o)= I). (iii) .Thus A/I =B,, where B=e, A e l / J has no divisors of zero. 2A. In order that there exist orthogonal projections el,.. .,c, in A with e, +...+en= I and e , -...- r,, it is necessary and sufficient that the order of every homogeneous summand of A be a multiple of n.

b 43 AII Theorem Type l ('ase

5 43.

199

A/Z Theorem: n p e I Case

We assume in this section that A is of Type I. Fix a maximal-restricted ideal I of A, and writc I = I , for a suitable EX [$39, Cor. of Th. I]. Let D, be the dimension function for A/I introduced in Section 40. By the structure theory for Type I rings, there exists an orthogonal sequence (possibly finite) h,, h,, h,, .. . of nonzero central projections, with sup hi= I,such that hiA is homogeneous of Type I,, [fj18, Th. 21. We can suppose n , < n, < n, < ... . Let Pi be the clopen subset of X corresponding to hi and let GY = UPi; since sup hi = 1, C?l is a dense open set in 3. If there are only finitely many hi-say h,, .. . , hr --- then qY =PI u . ..v Pr is clopen, hence C?l =%. Conversely, if @Y = X then since 9" is compact, the disjoint open covering (Pi) must be finite, thus there are only finitely many h,. To put it another way, it is clear that GY is a proper subset of .T iff (hi) is an infinite scqucncc (iff A has homogeneous summands of arbitrarily large order), and in this case n, + c~ as i+ a . Theorem 1. Suppose A is a finite Baer *-ring of Type 1, sati:fying LP RP, and let I be a maximal-restricted ideal of A. Then A/I is a,finite Baer *-factor, .ratisjyiny LP--RP, and any two projections in All are compamble. Adopt the above notations, in particular I = I,. I f ' a € % - CY then A/I is of' Type IT; if o ~ y say , o€Pi, tlzen A/I is of' Type InL. If a ~ q qthen the discussion at the end of the preceding section is applicable. We suppose for the rest of the section that EX -2i (which is possible only if A has homogeneous summands of arbitrarily large order). In particular, as noted above, ni+ a as i+ m. We are to show that A/I is a Baer *-factor of Type TI. Lemma 1. If' 0 < a < I, then tlzere exists a projection f ' A~ such that D ( f ) < a I and D(f')(z)=a ,for all ZEX-"Y. I

-

Proof. {In the application below, we require only D(f ' ) ( a ) =a, b.ut it is no harder to get D(f ) = a on X -CY.} First, a topological remark: if U is any neighborhood of a, then U intersects infinitely many of the P,. {Suppose to the contrary that U nCY c P , u . . . u P , . Since o $ P , u . . . u P m (indeed, o$qY), V=%-(P,u...uPm) is a neighborhood of a ; then U n V is a neighborhood of o with U n V n?4 = @, contrary to the fact that qY is dense in F.} For each i, write Fi = {p/ni:p = 0,1, ..., n,). Since 0 < a < 1, for each i thcrc exists a,€ F, such that 1 (1) 01a-a, I-; since ni + ar, as i k m, we have a, + a.

Hi

Chapter 7. Reduction of Finite Bacr *-Rings

200

Since, for each i, hiA is homogeneous of order n , there exists a projection f , < hi such that D ( f i ) = a i h i (take , f i to be the sum of niai orthogonal equivalent copies of a faithful abelian projection in hiA). Since ai I a by ( I ) , it follows that for every finite set J of positive integers,

Define j = s u p f,. Since the f , are orthogonal and D is completely additive, it results from (2) that m

D(f)=CD(f,)
It remains to show that D( f )=a on Y-GY. Fix ~ ' E Y - T Y ;let us show that D( f ) ( z l )= a. For each i, definc

by the continuity of D( f ) , U i is a neighborhood of z', therefore, by the first part of the proof, U i intersects Pj for infinitely many j. Construct a subsequence of indices i, < i, < i, < ... and a corresponding sequence of points Z,E U , n Pi" ( v = 1,2,3, ...) as follows. Set i, = 1 and let z, be any point of U , nPI. Since U , intersects infinitely many of the Pi, we may choose an index i, > i, and a point z , U~, n Pi2. And so on. As v -t co, we have i, + co hence niy-t cc ; in view of (I), it follows that For all v, la - D ( f ) ( z 1 ) l 5 la - D(.f )(%)I+ l D ( . f ) ( L )- D(.f)(zOI ; since Z,E Pi, and since hi,D( f ) = D(hi,,f ) = D( JiY)= aiYhi" is identically equal to a," on Pi", we have D( f)(z,)=aiv and thus

since

Z,E

U,, citing (3) we obtain Ia-D(.f)(zl)I 5 Ia-ai,,l

+

1

;

ni,, in view of (4), the right side tends to 0 as v -t co, thus a - D ( f ) ( i )= 0.

I

Lemma 2. Every orthogonal sequence of projections in A11 has a supremum.

# 44. Summary of Results

201

Proof. Let u, be an orthogonal sequence of nonzero projections in All. Write u, = F, with en an orthogonal sequence of projections in A [§23, Prop. 21. Let a,= DI(un)=D(e,)(o); by Lemma 1 there exists a projection,f, in A such that D(,f,) 5 a, 1 and D( f,)(o) = a,. The argument proceeds exactly as in the proof of [$41, Th. I]. I Proof of Theorem 1. As noted prior to Lemma 1, we have only to dispose of the case that EX-5Y. By Lemma 2 and the discussion in Section 40, we know that A/I is a finite Baer a-factor, satisfying L P RP, with comparability of projections. It remains only to show that A/I is of Type 11, and this is immediate from the fact that if 0 < a < 1 then Lemma 1 yields a projection u in A/I with DI(u)=a. I

-

Exercise 1A. With notation as in Theorem 1, %-C4 is a closed subset of X with empty interior, therefore is rare (i. e., nowhere dense).

5 44.

Summary of Results

Gathering up the results in Sections 36 and 39-43, we have: Theorem 1. Let A be a finite Baer *-ring such that L P ( x ) - R P ( x ) for all X E A . (1) I f I is a maximal-restricted ideal qj' A, then A/I is a ,finite Baer *-factor, LP(u)-RP(u) for all u € A / I , and any two projections in A/I are comparable. In order that A/I be of Type I, it is necessary and sufjcient that there exist a central projection h with h $ I and hA homogeneous (in which case A/I is also homogeneous, of' the same order); otherwise, A/I is of Type 11. (2) The intersection of all the maximal-restricted ideals of A is ( 0 ) . (3) Let 3 be the Stone representation space of the complete Boolean algebra of central projections of A, let D be the dimension function of A, and, .for each o~ X , write

Then o - I, is a bijection of .% onto the set of all maximal-restricted ideals of A. Proof: (2)This is [$36, Prop. I]. (3) This is [439, Cor. of Th. I]. (1) The first sentence is covered by the theorems in Sections 41-43. If h is a central projection such that lz$I and h A is homogeneous, then, by the discussion at the end of Section 42, A/I is of Type I (and homogeneous, of the same order as hA).

Chapter 7. Reduction of Finite Baer *-Rings

202

Conversely, suppose A / I is of Type I. Let k be the central projection such that kA is of Type I and (1 - k)A is of Type I1 [$15, Th. 21; in view of [441, Th. I], we know that A is not of Type 11, thus kf 0. Let Q be the clopen subset of %corresponding to k. Write I = I,, ae.9". Necessarily ~ E Q that , is, 1 - k ~ l . (If, on the contrary, I -k$l, it would follow from [$39, Prop. I] that A / I is *-isomorphic to a quotient ring of the Type I1 ring (1 - k)A, hence is of Type I1 [$41, Th. I], contrary to hypothesis.) Then k$ I, therefore A/I is *-isomorphic to kA/kl [439, Prop. I]. Let us apply the theory of Section 43 to the Typc I ring kA: let h, be an orthogonal sequence (possibly finite) of nonzero central projections with sup h, = k, such that h,(kA)= h,A is homogeneous of Type I,,, n , < n, < n, <... . Let P, be the clopen subset of !! corresponding to h,, and let C Y = P,; since sup h, = k, CY is a dense open set in Q. We know that ~ E Q Regarding . Q as the Stone representation space associated with the central projections of kA, observe that k l =(kA) n 1 is the maximal-restricted ideal of kA corresponding to ocQ. (For, k l is a restricted ideal of kA, kD(.) is the dimension function of k A , and, for a projection eE A, one has e ~ k iffl e 5 k and e e l iff e ~ k Aand (k D(e))(o)= D(e)(o)= 0.) It follows that aeqY. (For, o e Q -Y would imply, by [543, Th. I]applied to kA, that kA/kl is of Type 11, hence A/I is of Type IT, contrary to hypothesis.) Then oeP, for some i ; thus hi$l, where h, A is homogeneous of Type I,,. I Exercises 1A. Fix o6.T and let J = { x ~ A : x l z = O for some central projection h with h(o) = I} [cf. $ 24, Exer. 53. (i) J = {xEA:D(RP(x))=O on some neighborhood of g). (ii) AIJ is a factorial Rickart *-ring. (iii) A/J is a Baer *-sing iff J = I, iff AJJ is restricted-simple [cf. 36, Exer. 31.

2D. The analogue of Part (2) of Theorem 1 holds for the maximal-strict ideals of a finite Baer *-ring with GC [IS; 36, Exer. 41. Problem: Does the analogue of Part (1) hold (explicitly, is the quotient ring a Baer *-ring)?

5 45.

AIM Theorem for a Finite A W*-Algebra

-

The foregoing results provide a reduction theory for finite Baer a-rings, satisfying L P RP, in terms of finite Baer *-factors. Since an AW*-algebra satisfies L P - R P [$20, Cor. of Th. 31, the results are applicable, in particular, to a finite A W*-algebra A: regarding A as a finite Baer *-ring, we obtain a reduction of A into finite Baer *-factors A/I, where I varies over all the maximal-restricted ideals of A. However, the A / I are in general not AW*-algebras (Exercise 9). To obtain a reduction of A in terms of finite A W*-factors, it is necessary to replace

# 45. AIM Theorem for a Finite

A W*-Algebra

203

the maximal-restricted ideals I by the maximal ideals M (the M's are just the closures of the 1's [$22, Ths. 1, 21); the present section is devoted to the details. For the rest of the section, A denotes a finite A W*-ulyebva. We maintain the notations established in Section 35: Z is the center of A ; % is the Stone representation space of the complete Boolean algebra 2 ; D is the dimension function of A. A major new element in the picture is that Z is an A W*-algebra [$4, Prop. 8, (v)], therefore Z is the closed linear span of its projections [$8, Prop. 31, therefore Z may be identified with C(X); thus the dimension function D is Z-valued. Fix a point OE.%.Let 9={ e ~ dD(e)(o)=O), : I = (xeA: RP(x)eY), M =i;thus 9, I and M are the most general maximal p-ideal, maximalrestricted ideal, and maximal ideal of A, respectively [422, Ths. 1,2] (a pertinent point is that every maximal ideal in a Banach algebra with unity is closed [cf. 75, Cor. 2.1.41). As in Section 40, we write x + + % = + xI for the canonical mapping A H AII, and D, for the dimension function of A/I [$40, Def. I]. We write x*F=x + M for the canonical mapping A + AIM. Our objective is to prove that AIM is an A W*-juctor. The first main step is to construct a dimension function DMfor AIM, largely imitating the techniques used in Section 40 to construct Dr.

Lemma 1. AIM is a simple C*-ulyebru. Proof. As noted above, the closed ideal M is maximal, therefore AIM is a simple Banach algebra with respect to the quotient norm llxll =inf {Ilx+yll: y~ M ) [cf. 75, p. 441. Since I is a *-ideal [$22, Prop. 21, so is its closure M (see also [$22, Th. 21 or [$21, Exer. 5]), thercforc AIM admits the natural involution (F)*=(X*)~.The proof is concluded by noting that llZ*ZI(= 11x112 for all ~ E A ;indeed, it is a standard result that any closed ideal in any C*-algebra is a *-ideal, and the quotient algebra is a C*-algebra with respect to the quotient norm [cf. 24, Prop. 1.8.21. 1 Lemma 2. If u is any projection in AIM, then u = F for a suitable projection e in A. Proof. Say u=E. Then u = u* u = (x* n)- ; replacing x by x*x, we can suppose x 2 0. Since u2 = U, we have x2 - x = c for a suitable c~ M. If c = 0 we are through. Assume c#O. Then also x # 0. Set B = {x,c}"; since c* = c and c commutes with x, B is a commutative A W*-algebra [44, Prop. 81. Wr~teB = C(T), T a compact space. Let N = M nB. Then N is a closed ideal of B; since 1 $ M, N is proper; since ceN, N is nonzero. It follows that there exists a nonempty closed subset S of T such that

N ={beB: b(s)=O for all SES}.

Chapter 7. Reduction of Finite Baer *-Rings

204.

Since x 2 - x=

(1)

C EN,

we have x 2 - x = 0 on S, thus

S E S implies x(s)= 0

or x(s)= I .

Fix a real number E, 0 < e < I, and apply [47, Prop. 31 to B and x : there exist ~ E and B a nonzero projection P E B such that

Citing (2), we have x e = x 2 y = ( x + c ) y = e + c y , thus xe- EN; then

where c y € N B c N ,

x(s)e(s)-e(s)=O forall S E S .

(4)

We assert that x - e ~N (hence x - e ~M , ending the proof); given any it suffices to show that x(s)- e(s)= 0. If e(s)= 0, then Ix(s)l < E < 1 by (3), therefore x(s)= 0 by ( 1 ) ; if e(s)= I then x(s)= I by (4). I

S E S,

Lemma 3. If e, f are projections in A such that e- f E M , then D ( q f f= ) D(,f)(GI.

-

Proof. Write e = e' + e", f =f ' +f" with e' j" and e f " =errf = 0 [$ 13, Th. 1, Prop. 51. Since M contains e -f , it also contains (e-f)e" = e" and ( f - e ) f " =f". Then e", f " E M = l"= ,f=.A (recall that M = 7 contains no projections not already in I [422, Prop. 4]), thus D(eU)(@) = D ( f U ) ( o ) = O . Then D ( e ) ( o ) = D ( f ) ( o )as in the proof of 1440, Lemma]. I The way is prepared for defining a dimension function DM for AIM:

Definition 1. If u is a projection in AIM, write u=F with e a projection in A (Lemma 2) and define D,(u)=D(e)(o) (the definition is D,(Z) [cf. $40, Def. I], Note unambiguous by Lemma 3). Thus D M @ ) = that D,(u)=O iff D,(d)=O iff eEI iff EM iff u=O (recall that ~ = 1 = . % , ) . Lemma 4. If u, 0 are projections in AIM such that u 5 v, and if' v = with f a projection in A, then there exists a projection e in A such that u=i? and e l f .

7

Proof. I f x is any element of A with u =X,then u = v u* u v = (j'x*x f ' ) - ; replacing x by f x * x f , we can suppose x 2 0 and , f x = x. By the proof of Lemma 2, there exists a projection e in A such that u =e and x y = e for a suitable element y. Since f x =x, we have fe = e, thus e 5 J: I

Lemma 5. If u, is an orthogonal sequence ofprojections in AIM, then there exists an orthogonal sequence of projections e, in A such that u, for all n.

=<

?j45. AIM Theorem for a Finite A W*-Algebra

205

Proof. This follows from Lemma 4, by the argument used in the I proof of [§23, Prop. 2, (ii)]. Lemma 6. If'u, v are orthogonal prr!jections in AIM, then DM(u+ u) DM(v).

+

= DM(u)

=y

Proof. By Lemma 5, we may write u = , v with e, f orthogonal projections in A. Then e + f is a projection with ( e+f ) - = u + v, thus We can now use DM to establish orthoseparability:

Lemma 7. AIM is orthoseparable. Proof: Formally the same as [§40, Prop. 21.

1

Since I c M , there is a natural mapping A l l - +AIM, namely 2-X (an epimorphism with kernel MII). If I c M properly (that is, if I is not closed) then the mapping is not injective, but we are able to maneuver around this circumstance. In view of Lemma 2, and the like result for A/I [$23, Prop. 1, (ii)], the correspondence ; + + e ( e ~ Amaps ) the set of all projections of A/I onto the set of all projections of A I M ; it is useful to formalize this:

Definition 2. Define cp: (All)" -t ( A I M ) - as follows. If v ~ ( A / l ) " , write v=P with e ~ and 2 define cp(v)=F. Since I and M contain the same projections, cp(v)= 0 iff v = 0. We know that (All)" is a complete lattice [§44, Th. I]. The general plan is to infer properties of (AIM)" via the mapping cp. The pertinent property of cp is the following:

Lemma 8. If v,, v, are projections in AII such that v, I v,, then cp(v1)6: cp(v2). Proof: Write v, = PI, u,= d , with e l <e2 [$23, Prop. 2, (i)]. Then I e l 5Z,, that is, cp(v,) 5 cp(v,).

-

Lemma 9. I f u , is an orthogonal sequence ofprojections in AIM, then there exists a projection U E A I M such that u, < u ,for all n and DM(4 = C DM(un). Proof. Write u,=Z,, with en an orthogonal sequence of projections in A (Lemma 5).Set v,= d,; thus v, is an orthogonal sequence of projections Say v=P, e a in AII with cp(v,)=u,. Define v=sup v, [444, Th. I]. projection in A. Setting u =F, we have cp(v)= u. Since v, 5 v for all n, we have cp(v,) 6: cp(v) by Lemma 8, that is, u, < u. Since DM(u)=D M @

Chapter 7. Reduction of Finile Baer *-Rings

206

= D,(d)= D,(u) and similarly DM(un) = D,(v,), it results from the countable additivity of D, [$33, Th. I] that

DM(u)= Dr(v)=

1D,(un)= 1D,(u,).

1

The next lemma is of 'EP' type: Lemma 10. If SEAIM, s # 0, then there exist t~ AIM and u nonzero projection U E AIM such that ss* t = u. Proof. Say s=Z, XEA. Since ss*#O, we have xx*$M. Write a = x x * . By [$7, Prop. 31, there exists a sequence b , (a)" ~ such that (1) a b, = en, en a nonzero projection, and (2) /la- u e,,ll < lln. In particular, ae,+ a ; since a $ M and M is closed, there exists an index m such that aem$M, thercforc e,$M. Sct t=zm, u=<. I Lemma 11. Every orthogonal sequence of projections in AIM has a supremum. Proof. Adopt the notation in the proof of Lemma 9. We assert that u = sup u,. It remains only to show that if w is a projection in AIM such that u, 5 w for all n, then u 5 w. Let s = u ( l - w) and assume to the contrary that s f 0. By Lemma 10, there exists t~ AIM such that s t =u', u' a nonzero projection. Since u s = s we have uu1=u', thus u. Moreover, u' is orthogonal to every u,: unu'=unst=u,u(l - w) t u' I = u,(l

-

w) t = 0 (because u, 5 w). Thus, for cvery n, u'

+

n

u, is a 1

subprojection of u; it follows from the additivity of DM (Lcmma 6) that

for all n. Letting n + cx, and citing the formula in Lemma 9, we have DM(u) > DM(u') + D,(u), therefore D,(ul)

= 0,

u' = 0, a contradiction.

I

The principal result, stated in full: Theorem 1. If A is afinite A W*-algebra and M is a nzaximal ideal of A, then AIM is afinite A W*-ficctor. Proof. Let us first show that AIM is a Baer *-ring. Every orthogonal family of nonzero projections in AIM is countable (Lemma 7) hence has a supremum (Lemma 11);thus it will suffice to show that AIM is a Rickart *-ring [$4, Prop. I].

5 45. AIM Theorem

for a Finite A W*-Algebra

207

Let s r AIM. We seek a projection u r AIM such that R ( { s } = ) u(A/M). Consider a maximal orthogonal family of nonzero projections in R((,s}) (if no such projections exist, then a slight rearrangement of the following argument shows that the choice u=O works). We know that the family must be countable, say (u,), and we can form u = sup u,. We assert that su= 0. If, on the contrary, s u f 0, then by Lemma 10 there exists t € A I M such that t(su)*(su)= u', u' a nonzero projection. Clearly u' u= u'. For alln, (su)u,=su,=O, therefore ulu,=O, u,
-

Lemma 12. l f u, v are prqjections in AIM such that u DM(4 = DM(u).

- v,

then

Proof. Write u=; v=,7 with e, f' projections in A. By hypothesis, there exists X E A such that X * X = , XX*=,7. Then Y = , ~ Y T =f(x e ) - ; replacing x by ,fxe, we can suppose f x = x = x e. Thus if jb = LP(x), e,=RP(x), we have f ; 5 f ; e, 5 e and e, - , f , (but the latter equivalence has nothing to do with the hypothesized one). Since x e ,-.- x and eeo=eo, of x * x = T by To - - right-multiplication yields F = q . Similarly f = ,f,. Thus u,= , v = ,fo; since e, i b we have D(e,) = D(,fo), therefore

-

There remains the question of type for AIM, but this is easily referred back to the discussion for AII: Proposition 1. AIM and AII have the same type; i.e., they are both Type I1 or both Type I (with the same order).

Proqf. We know that AIM is afinite A W*-factor, therefore it possesses a unique real-valued dimension function 1633, Th. I ] ; but Lemma 12

208

Chapter 7. Reduction of Finite Baer *-Rings

completes the proof that DMhas the properties required of a dimension function [$25, Def. I], thus DMis the unique dimension function for AIM. In view of the relation D,@= D,(E) for all e ~ d (see Definition I), the functions DMand D, have the same range of values, hence AIM and AII have the same type. I Exercises

1C. There exists a 'reduction theory'-more analytic and less algebraic-valid for any von Neumann algebra on a separable Hilbert space. 2C. If A is a finite von Neumann algebra and M is a maximal ideal of A, then AIM is also a von Neumann algebra (on a suitable Hilbert space). 3A. A finite AW*-algebra A is strongly semisimple (that is, the intersection of all the maximal ideals M is (0)); thus A is *-isomorphic to a subalgebra of the C*-sum of the finite factors AIM. 4A. Let A be a finite A W*-algebra. The following assertions hold (in increasing generality): implies D ( ~ ) E M . (i) If M is a maximal ideal of A, then (ii) If P is a primitive ideal of A, then ~ E implies P D(e)eP. (iii) If I is a closed ideal of A, then e ~ implies l D(e)€I. (iv) If I is a closed ideal of A and if e, f' are projections in A such that e- f c I , then D(e)-D(f)cI.

EM

5A. Let A be a finite AW*-algebra and let I be a closed, restricted ideal of A. (i) e ~ implies l C(e)~l. (ii) If en is a sequence of projections in I, then s u p e , ~ I . (iii) If A is orthoseparable (equivalently, its center Z is orthoseparable [§ 33, Th. 4]), then I = h A for a suitable central projection h .

6A. Let A be an orthoseparable finite AW*-algebra and let I be a maximalrestricted ideal of A. Write I=I, for a suitable a€%. Then I is closed iff a is an isolated point of X. 7A. If A is an orthoseparable finite AW*-algebra of Type I possessing homogeneous summands of arbitrarily large order, and if /=I, with ~ E . % - Y(notation as in [pj 43, Th. I]), then I is not closed. 8A. Let A be a finite AW*-algebra of Type 11, whose center Z is infinitedimensional (cf. Exercise 11). (i) Z contains an element c such that 0 5 c 5 I,c is singular (i. e., not invertible in Z)and RP(c)= 1. (ii) With c as in (i), choose a € % with c(a)=O. Then I, is not closed. 9A. Let A be a finite A W*-algebra and let I be a nonclosed maximal-restricted ideal of A (cf. Exercises 6-8). Then (i) A/I is not semisimple; therefore (ii) A/I is not strongly semisimple; therefore (iii) AII does not satisfy the (VWEP)-axiom. In particular, (iv) A11 is a Baer *-factor that cannot be normed to be a C*-algebra.

5 45. AIM Theorem for a Finite AW*-Algebra

209

10B. Let Z be a commutative AW*-algebra. Write Z=C(X), 9"a Stonian space. Fix O E X and define N = { c e Z : c(a)=O), J = { cZ~: c = 0 on some neighborhood of o) .

(i) J is a maximal-restricted ideal of Z , and I= N. (ii) Z/J is an integral domain, with unique maximal ideal NIJ. (iii) The following conditions are equivalent: (a) J = N (that is, ~ = a (b) ; Z/J is semisimple; (c) Z/J is simple; (d) Z/J is a field; (e) Z/J is the field (C of complex numbers. Let n be a positive integer and let A=Z,. (iv) A is an AW*-algebra of Type I,, whose center may be identified with Z . Let I=I, be the maximal-restricted ideal of A corresponding to a, that is, l = { x ~ A D(RP(x))(o)=O}. : (v) l = { x ~ A D(RP(x))€J}. : (vi) I=J,. (vii) A/I is *-isomorphic to (ZIJ),. (viii)-Z/J is a Priifer ring. (ix) I is the unique maximal ideal of A containing 1(x) The following conditions are equivalent : (a') I = I ; (b') A11 is semisimple; (c') A/I is simple. In this case, A/I = (C,. In particular: For each positive integer n, (ZIJ), is a finite Baer *-factor of Type I,, satisfying LP RP.

-

11C. The following conditions on a finite AW*-algebra A are equivalent: (a) every maximal-restricted ideal of A is closed; (a') every maximal-restricted ideal of A is maximal; (a") every maximal ideal of A is restricted; (b) Z is finitcdimensional; (b') .% is finite; (c) A is the direct sum of finitely many finite AW*factors. In particular, such an algebra is orthoseparable. 12B. Notation as in Exercise 10. Assume, in addition, that Z is infinitedimensional (equivalently, X is infinite) and orthoseparable. (For example, let Z be any infinite-dimensional commutative von Neumann algebra on a separable Hilbert space.) Choose (as we may) o to be a nonisolated point of 37. Then none of the conditions in (iii), (x) of Exercise 10 hold. Application: Z 'the' algebra of diagonal operators on a separable, infinitedimensional Hilbert space _(relative to a fixed orthonormal basis) [cf. 37, p. 291. Then %=PW (the Stone-Cech compactification of the discrete space !N of positive integers), and the nonisolated points a are the points of PW-W.

Chapter 8

The Regular Ring of a Finite Baer *-Ring The present chapter is based on

5 46.

181.

Preliminaries

-

At the outset we assume that A is a finite Buer *-ring sutisfying L P R P (cf. the remarks at the beginning of Chapter 7); at this level of generality, the questions outweigh the answers, but it is surprising how far one can get. Gradually the hypotheses are strengthened, the main results being obtained for a somewhat restricted class of finite Bacr *-rings (including all finite A W*-algebras). As an application, it is shown in the next chapter that the algebra of all n x n matrices over any A W*algebra (finite or not) is also an A W*-algebra [$62, Cor. 1 of Th. I]. Before motivating the chapter informally, we record for convenient reference some consequences of the above assumption:

-

Proposition 1. If A is a .finite Baer *-ring sati.~fying LP RP, then (1) A satisjies the parallelogram law (P), (2) A has GC, (3) A has a unique dimension function D, and D is completely additive, (4) D(eu f )+ D(en,f')= D(e) D(f ) ,for every pair c?f' prqjections e, f in A, and (5) the relations x, y E A, y x = I imply x y = I.

+

Proof: (I), (2) See [914, Cor. 2 of Prop. 71. (3) This is true for any finite Baer *-ring with GC [$33, Th. I]. (4) This follows at once from the parallelogram law and the properties of D. (5) If y x = I it is clear that R({x))= jO), thus RP(x)= 1 ; then LP(x) RP(x)= 1 implies, by finiteness, that LP(x)= 1. Then (1 -xy).x =x-xyx=x-x=O, and 1-xy=O results from LP(x)= 1. I

-

As noted above, y x = 1 implies that RP(x) = 1 (and that x is invertible). The converse is false; RP(x)=I does not imply that x is invertible (cf. Exercise 1). In a von Neumann algebra, RP(x)= 1 means that x is S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

5 46.

Preliminaries

21 1

injective; in general, one can hope to get an inverse for x only if one is willing to accept an unbounded operator. Returning to abstract rings, the 'rings with invertiblity' par excellence are the regular rings: a ring C is said to be regular if, for each X E C , there exists ~ E with C x y x = x . It follows that e =y x and f = x y are idempotents; x e = x = ,f'x shows that they are trying to act like a 'rightidempotent' and a 'left-idempotent' for x ; and y acts like an 'inverse' for x relative to the idempotents e and f . We defer until Section 51 a closer look at regularity; for the present it suffices, by way of motivation, to report the following fact: Each finite von Neumann algebra d may be enlarged, in a canonical and minimal way, to a regular ring %?; the new elements of % are certain unbounded operators 'affiliated' with d, these being assumed densely defined and closed; the algebraic operations in %' are the usual ones followed by the closure operation (e.g., the 'product' of two operators in % is the closure of the usual composition product). Consider now the given finite Baer *-ring A. Can A be enlarged to a regular Baer *-ring C? In general, we do not know. Following the exposition in 181, it is shown in the next section that a *-ring C containing A can be constructed, along the lines of the classical construction for von Neumann algebras; and in Section 48 it is shown that C has no new projections (i.e., none not already in A) and that C is also a finite Baer *-ring satisfying L P RP. Gradually strengthening the hypotheses on A, we reach regularity of C in Section 52; the hypotheses on A are regrettably restrictive, but at least they are inherited by C (Section 53) and they are satisfied by the finite A W*-algebras. {In the wake of all this technology, it should be remembered that some rather unpropertied finite Baer *-rings-for example, the ring of 2 x 2 matrices ovcr the field of three elements-are already regular.) Why consider C at all? The answer lies in the fact that the ring of n x n matrices over a regular ring is known to be regular. Putting aside its intrinsic algebraic interest, the justification for considering C is that it is needed in the proof that the algebra of n x n matrices over an A W*algebra is also an A W*-algebra [ 5 ] ; the latter result is exposed in the next chapter (see Section 62).

-

Remarks. There is a related circle of ideas that should be mentioned here. The projection lattice of A is complemented (with a canonical complementation e - e' = 1 - e) and modular 34, Prop. 11. Suppose, in addition, that for some integer n, n 2 4, there exist n orthogonal equivalent projections el,..., e, with sum 1. (This is equivalent to assuming that A = B, for some *-ring B 16, Prop. I],or that the order of every homogeneous summand of A is divisible by n 1942, Exer. 21.) The e, are pairwise perspective 1917, Exer. 12, (vi)]. Summarizing:

[s

[s

212

Chapter 8. The Regular Ring of a Finite Haer *-King

the projections of A form a complemented modular lattice of 'order' 2 4 in the sense of von Neumann [71, p. 931. It follows that there exists a regular ring R, unique up to isomorphism, such that the projection lattice of A is isomorphic to the lattice 9 of principal right ideals of R (this is von Neumann's coordinatization theorem [71, p. 208, Th. 14.11). The mapping e - r ' = l - e is an order-reversing involution in the projection lattice of A ; it induces a like mapping in the lattice 9, denoted, say, (x R) ++ (x R)'. It follows that the ring R possesses an involution x x* such that (x R)' is the right-annihilator of R x* (order 2 3 is sufficient for this result [71, p. 113, Th. 4.31). Since e ne' = e ( l - e) = 0, it follows that (x R ) n(x R)' = (0) for all X E R, therefore the involution of R is proper 171, p. 114, Th. 4.51. Regular rings with proper involution are called *-regular; we look into such matters in Section 51, but mention here that such a ring is a Rickart *-ring. Thus, R is a regular Rickart *-ring; its projection lattice is identifiable with the lattice 9 in the obvious way, hence is isomorphic with the projection lattice of A , hence is complete; thus R is a regular Baer a-ring. To summarize: Assuming A has an order n, n 2 4, there exists a regular Baer *-ring R such that A and R have isomorphic projection lattices. This is achieved without extra hypotheses on A (other than finiteness, L P -RP, and the order condition). However, the only connection between A and R is their common projection lattice. Whether A can in general be embedded as a subring of R is not known (presumably, one has to be able to 'reconstitute' the elements of A and R in terms of their projections via some sort of 'spectral theory'). How does this compare with what is accomplished in the present chapter? We do reach a regular Baer *-ring C with the samc projection lattice as A . (1) The hypothesis on 'order' is dropped (though we essentially steer clear of abelian summands by assuming partial isometries to be addable); (2) we must impose a number of rather severe axioms on A ; (3) A is visibly a *-subring of C from the start.

-

Exercises

1A. Let Af be a separable, infinite-dimensional Hllbert space, let i;,,i;,, t,, ... be an orthonormal basis of A?,let T be the operator defined by T t n =( l l n )i;,, and let .d= { TI\" be the bicommutant of T, calculated in Y'(2'). Then .d is a commutative (hence finite) von Neumann algebra, RP(T)=I, but T is not invertible. 2A. If e is any projection in A, then eAe is also a finite Baer *-ring satisfying LP-UP. 3A, D. If B is a *-subring of A such that B= B , then B is also a finite Bacr *-ring. Problem: Does B satisfy LP RP? Does it have GC? The answers are affirmative if A satisfies the (EP)-axiom and the (SU)-axiom.

-

5 47. Construction of the Ring C

5 47.

213

Construction of the Ring C

We assume in this section that A is a finite Baer *-ring satisfying LP --RP [cf. 5 46, Prop. 11. This hypothesis is adequate for thc construction of C, and for the development of many of its properties (Section 48), but we are able to show that C is regular only under added hypotheses (Section 52). Definitions and terminology are motivated by the von Neumann algebra case; the construction follows verbatim thc reference [4], to which the reader is referred for heuristic discussions of the operatorial motivation.

Definition 1. A strongly dense domain (SDD) in A is a sequence of projections (e,) such that enf 1 ; that is, el 5 c, 5 e, 5 ... and sup en= 1. {Thinking heuristically in terms of linear operators, the ranges of the en are an increasing sequence of closed linear subspaces whose union is a dense linear subspace. We caution the reader that this picture is adequate for the von Neumann algebra case but may be incorrect in the AW*-case, where the calculation of suprema is not tied down to vectors.) Lemma 1. I f (e,) and ( f,) are SDD, then (ennf,) is un SDD Proof. Setting g, = ennf, and g = sup y, I - g s l -gn

we have y, T g. For all n,

= ( I - e n ) u ( l - f,),

therefore (*)

D(1 - g) 5 D(1- e,) + D(1-

.fn)

[§ 46, Prop. I , (4)]. Sincc D(l -e,)J 0 and D(l - .f,,)J 0 [$ 33, Th. 21, it follows from (a) that D(l -g)=O [cf. $ 27, Lemma I, (ii)], thus g=l. I

Lemma 1 extends at once to finitely many SDD (en),(fn),...,(k,,): the term-by-term infimum (enn,f, n . .. n k,) is also an SDD. Among other things, SDD are a technique for inferring properties 'in the limit'; the following is a sample: Lemma 2. Let (en)be an SDD and let X E A . ( 1 ) If' e,xe,=0 ,fbr all n, then x=O. (2) If' e,xe, is selfladjoint for all n, then x is selfladjoint. Proof. (1) Fix an index m. For all n 2 nz, e,(xem)=(e,xe,)em=O; it follows from supje,: n 2 m ) = I that xe,=O. Sincc m is arbitrary, x=o. (2) For all n, e,(x* - x)en= (e,xe,)* - ( e n xen)= 0 , therefore x* - x = 0 by (1). fi

21 4

Chapter X The Regular Rlng of a Finite Baer *-R~ilg

Definition 2. An operator with closure ( O W C )is a pair of sequences (x,, en) with xnE A and (en)an SDD, such that m < n implies x, e, = x, e, and x,* e, =x:e,,,. {Heuristically, one can think of (x,, en) as a linear operator whose restriction to the range of en is x,e,; the condition (x,e,)e,=x,e,=x,e, for m < n means that the xnen are 'strung together coherently', and similarly for the x,* e,.). Lemma 3. If (x,, en) and (y,, f,) are OWC, then so are (x:, en) and ( x ,+y,, enn f,). I f , in addition, A is a *-algebra over an involutive Jield F, then (Ax,, en) is an OWC for each A E F. Proof. Set g, = enn f,. If m < n then

and similarly ( x , +y,)* g , = ( x , +y,)* g , ; since (y,) is an SDD (Lemma 1) we conclude that (x,+y,, g,) is an OWC. The other two assertions of the lemma are obvious. I The discussion of products requires another concept:

Definition 3. If ~ E and A e is a projection in A, we write x - ' ( e ) for the largest projection g such that ( I - e ) xg =0, that is, e x g = x y ; thus x-'(e)= I -RP [(I - e ) x ] . (Thinking heuristically in terms of linear operators, the range of the projection x - ' ( e ) is the largest subspace mapped by x into the range of e; identifying subspaces and projections, x p l ( e ) is the inverse image of e under x (hence the notation).) Lemma 4. I f X E A and e is any projection, then e 5 x p ' (e). Proof. 1-x-'(e)=RP[(I e ) x ] -LP[(l e ) x ] < 1 e , therefore E L I - L P [ ( 1 - e ) x ] -1 -RP[(1 - e ) x ] by finiteness [$ 17, Prop. 4, (ii)].

=x

' ( ~ )

I

Lemma 5. Suppose (en)is an SDD and (x,) is a sequence in A such that m < n implies x,e, = x,e,. Let ( f,) be any SDD and define g n= enn x i '( f n ) . Then (g,) is an SDD.

'

Proof. Let h, = x i (f,); jection such that

(*I

thus g, = enn h, and h, is the largest pro-

( 1 - fn)xn12,= 0 .

Note that g,T. {Proof: If m < n then

5 47.

Coilstruction of the Ring C

='

hence

( l - f n ) x n g m= (1 - f , ) ( l - . f , ) x m h r n ~ ~ m

by (*), thus g, < h, by the maximality of h,; also g, 5 em5 en, thus g, 5 ennh, = g,.) It is to be shown that g,T I.Since D(h,) 2 D( f,) by Lemma 4, the relation 1 - y, = (1 - en)u (1 - h,) yields

D(l -g,) I D(l -e,)+D(I-h,)

I D(l -en)+D(l- j,),

therefore D(1- y,) 10 (cf. the proof of Lemma 1).

I

This leads to a natural notion of product for OWC

Lemma 6. If(x,, en)und (y,, f,) czre OWC, and k,

=

[ f , ny;'(e,)l

lj

n [enn (x,*)-'(J,)I,

then ( x ,y,, k,) is an OWC. Proof: (k,) is an SDD by Lemmas 5 and 1. Clearly ( I k, = 0 (cf. Definition 3), thus

-

e,)y, k,

= (1 - f,)x;

If m < n then (x,y,) k, = ( x , y,) k, results from the computation (quote the first equation of (*) at the appropriate steps)

(xnyn)k, = xnynfmk,

= xnymfmk, = xn(y,k,)

=xn(e,y,k,)=xme,y,k,=.~,y,k,,

and ( x ,y,)* k, I of (*).

= ( x , y,)*

k, results similarly from the second equation

If ~ E and A (en)is any SDD, clearly ( x ,1) and ( x ,en) deserve to be regarded as 'equal' (they agree, so to speak, on a dense linear subspace); this calls for an equivalence relation:

Definition 4. We say that the OWC (x,,en),(y,, f,) are equii~alent, written (x,, en)=(y,, J,), if there exists an SDD (g,) such that x,g,=y,g, and x,* g, =y:y, for all n. (Cf. [$ 48, Prop. 21.) The equivalence is said to be implemented via the SDD (g,). Lemma 7. The relation in the set of all OWC.

=

dqfined uhoi~eis un equirulence relution

Prooj. Reflexivity and symmetry are obvious. If (x,, en)-(y,, f,) via (h,), and if (y,, 1,)-(z,, g,) via (k,), then (x,, en)-(z,, g,) via (hnn kn). I The following lemma is useful in the manipulation of representatives of equivalence classes:

216

Chapter 8. The Regular Ring of a Finite Baer *-Ring

Lemma 8. (i) If (x,, en) is an OWC and (g,) is any SDD, then (x,, enn y,) is also an OWC and (x,, en)=(x,, enng,). (ii) Suppose (x,, en)r ( y , f,) viu an SDD (g,). Set h, = P , n f , ng,. ?'hen (x,, h,), (y,, h,) are OWC, and (x,, h,) -(y,, h,) via (h,). ProoJ: Routine.

I

Definition 5. We write [x,, en] for the equivalence class of the OWC (x,, en) with respect to the equivalence relation E defined above. The set of all equivalence classes is denoted C, and its elcmcnts are called closed operators (CO); [x,, en] is the C O determined by the OWC (x,, en). {In our heuristic model, the passage from (x,, en) to [x,, en] corresponds to forming the closure of the linear operator (x,,e,), proper care having been taken-via Definition 2-to ensure that the adjoint operator is also densely defined.] We denote the elements of C by boldface letters x, y, z, ... . If X E A we write E = [ x , I] for the CO determined by the pair ( x ,I)of constant sequences; we write 2= j7: ~ E A for ) the image of A in C under the injective (Lemma 2) mapping x H i? and, , more generally, 5 for the image of a subset S c A.

Lemmas 3 and 6 suggest algebraic operations for OWC:

and, when relevant, i ( x n ,en)= (Ax,, en). Algebraic operations in C are defined by passage to quotients modulo thc cquivalcncc rclation =; thc dctails arc as follows. If (x,, en)3 (x:, rk) via (g,), and (y,, f,)-(y:, f,') via (A,), then (xn+yn> en nf n )

3

(x: +Y:, ei nJ,')

via (g, n h,). It is trivial that (x:, en)-(x;*, r:) via (y,), and, when relevant, (Ax,, en)=(Ax:, e:) via (y,). This paves the way for the following definitions: Definition 6. If x , y EC, say x = [x,, en],y = [.v,,, f,], define X+Y

= [xn+~n,en".fnI,

x* = [x:, en]; when A is a *-algebra over an involutive field F, define

$ 47 Colirtruct~onof the R ~ n gC

for all / 1F.~ Define XY =

[xnyn> knl

3

where (k,) is the SDD given in Lemma 6; this is well-defined, by the following lemma: Lemma 9. Suppose (x,, en)-(xk, ek) and (y,, fn) -("v;, f,') und suppose (k,), (k,) are SDD such thut (x,y,, k,) und (xL,yh, k',) arr OWC. Then ( x , ~ , ,k,)-(xkyk,

K).

Pvooj. Let (g,) be an SDD implementing (x,, en)-(xk, rk); wc can suppose, in addition, that (x,, y,) and (xk, g,) are OWC (Lemma 8). Similarly, let (h,) bc an SDD implementing (y,, f n ) r ( y ; , J,'), such that (y,, h,) and (yh, h,) are OWC. Changing notation, we can suppose that (x,, en)-(xi, e,) vla (e,), (y,, J,) =(yb, I,) via (f,), and that (k,), (k',) arc SDD such that (x,y,, k,) and (xLyk, k,) are OWC; we are to show that (x,y,, k,)-(xkyk, ki). Define ~,=f~n~;'(e~); then (y,) is an SDD (Lemma 5) and (1 e,)p,g, y, 5 yil(e,)), that is, e,,y,y,=y,y,; it follows that -

=0

(because

{Proof: xnyngn =xnenyngn =4zenyrzgn=,'kyngn =xi.yi, f n g n =x',,vk fig, = xhyh y,.) Similarly, defining (h,) is an SDD such that (ii)

(Y,* x,*)h, = (yk*xk*) hn.

Setting d,, = g, nh,, (i) and (ii) yield thus (x,y,,k,)-(xkyk,k',)

via(cl,).

I

-

Theorem 1. Let A be u finite Buer *-ring sufisjying L P RP. Define C, and the operations in C, us indicuted ubovc. Then (1) C is u *-rinq with unity 7 (un4 if A is u *-ulgehru ouer an inuoluti~lefield F, then so is C ) , und (2) the mapping x H Y is a *-i.somorplzism of A onto u *-subring 2 of' C. Proof: (1) We illustrate thc routine proof by verifying the associative law (x y) z = x(y z). Say

Chapter 8. The Regular Ring of a Finite Baer *-Ring

21 8

Then x y = [x,y,, h,] for suitable (h,), hence (xy)z= [(x,y,)z,, k,] for suitable (k,). Similarly, x(y z)= [x,(y,z,), kin] for suitable (k,). Since (x,y,~,,k,) ~ ( xy,z,, , ktn)(via any SDD),we conclude that (xy)z = x(y z). (2) For example, to show that (x+y)define Y+y by (x, I), (y, 1) of the equivapplying Definition 6 to the representatives alence classes Y,7:

=~+y,

We write 1 for the unity element of C, that is, we identify 1 with 7. But in general we refrain from identifying x with F;there are conceptual advantages to maintaining the distinction between A and 2 (as we do throughout this chapter) until the properties of C have been fully developed. The question of when A = C is discussed in Sections 51 and 53. In the next section we develop some properties of C that require no further hypotheses on A . Exercises

--

1A. Let A be a finite Baer *-ring satisfying LP R P and let e be a projection in A (thus eAe is also a finite Baer *-ring satisfying LP RP). (i) FCF coincides with thc set of all X E C such that x=[x,,e,] for suitable x,~eAe. (ii) Writing C,,, for the ring accruing to eAe via Theorem 1, we have c,,,=ece in the sense that there exists a natural *-isomorphism between the two rings. 2A. Lemma 2 holds with (en)replaced by any increasingly directed family of projections (e,) such that sup e,= 1. 3A. Let A be a finite Baer *-ring satisfying LP -RP, let A be a fixed limit ordinal, and consider well-directed families of projections (e,),,, with r, T 1. (i) The entire section can be recast in terms of such fam~hes;write C, for the ring given by the analogue of Theorem 1. (In particular, C=C,o, where w is the first infinite ordinal.) (ii) C, +, = C for any ordinal a. (iii) If A= S) (the first uncountable ordinal) and if A is orthoseparable, then the construction collapses: C, = A .

4A. Let A be a finite Baer *-ring satisfying LP -RP and let (en)be an SDD in A. If x,y are elements of A such that, for all n, e,,xe,,commutes with enye,, then xy=yx. In particular, if e,xe, is normal for all n, then x is also normal.

5 48.

-

First Properties of C

As in the preceding section, A is a finite Baer *-ring satisfying LP RP, and C is the ring constructed there. We show in this section that C has no new projections (they are all of the form F,e a projection

# 48. First Properties of C

219

in A) and that C is itself a finite Baer *-ring satisfying LP - R P (and a little more). The first proposition, which confirms the heuristic guide in [$ 47, Def. 21, is useful in lifting properties from A to C :

Proposition 1. If' XEC,x = [x,, en], then -

,for all m.

-

x em= xmem,

-

-

emx = emxm

Proof. Fix an index m. Define an SDD (,f,) by the trivial choices f,=O for n t m and ,f,=l for n 2 m. For all n, one has indeed, for n 2 m these reduce to x,em=xmem, and for n t m they reduce to 0 = 0 . Since x & = [x,em, y,] for suitable SDD (y,), and the foregoing relations show that (x, e,, g,) =(xmem,I)via (J,), we have [x,em, y,] =[xmem,I], that is, ~ c = ( x ~ e , ) It~ .follows that x*& =[x,*,e,]F,,,=(x;e,)-, thus&x=(emxm)-. I In the next proposition, we simplify the task of verifying equivalence of OWC; so to speak, in testing for equivalence, the adjoints take care of themselves:

Proposition 2. If x = [x,, en], y = [y,, f,] and such that xngn = Yny, fbr all n,

if' (g,) is an SDD

then x = y. In fact, it suffices to assume that h, x, g, = h, y, g, ,for a pair of SDD (gn), (An). Proof: We prove the second assertion. Note first that if (k,) is any SDD, then (k,, k,) is an OWC such that (k,, k,)r(l, 1) via (k,), thus [k,, k,] = 1. Then

for a suitable SDD (r,), and similarly y = [h,y,y,, s,] for suitable (s,); since (h,x,g,, r,) =(h,x,g,, s,) = (h,y, y, s,) (via any SDD) we conclude that x = y . I

Lemma. If X E C then there exists a projection ~ E A such that (1) fix=x, and (2) yx=O ifand only if' y , F = ~ . Proof: Say x = [x,, en]. Define .f,= LP (x, en) and let f thus f is the smallest projection in A such that

(*I

fx, en = x, en for all n.

= sup f,;

Chapter 8. The Regular Ring of a Finitc Baer *-Ring

220

Since ,Tx = [f x,, g,] for suitable (g,), Tx = x results from (*) and Proposition 2. Suppose y E C with y x = 0. Say y = [y,, h,]. For all m and n, we have, by Proposition 1, thus (h,y,) (xmem)=Oand therefore (h,y,) fm =O; since m is arbitrary, h,y, f = 0 , and since n is arbitrary, y f = O by Proposition 2. Conversely, y f = O implies ~ X = ~ ( , ~ X ) =IO . Theorem 1. C has no new projections; that is, then e=F with e a (unique) projection in A .

if' ~ E isC a projection

Proof. By the lemma, there exists a projection e in A such that C (I -Z) = L((e)); but L({e)) = C(l - e), therefore I -Z= I e. I -

Corollary 1. C is a Baer *-ring. Proof: Since C has no new projections, its projections form a complete lattice (isomorphic to the projection lattice of A via e - q ; moreover, the lemma to the theorem shows that C is a Rickart *-ring, therefore C is a Baer *-ring [$4, Prop. I]. I We remark that if ~ E and A r=RP(x), then R P ( 9 exists (Corollary I), and it follows from Theorem 1 that RP(Y)=F; thus RP(x) = ( R P ( x ) ) for all x in A . Corollary 2. C has GC und sati.yfies the parallelogram law (P). Proof. Since A has these properties [$ 46, Prop. I] and C has no new projections, C inherits the properties via the embedding x e x. I Corollary 2 will be superseded by the next theorem. Lemma. If (x,) is a sequence in A und (en) is an SDD such that x, em= xmem whenever m t n, and if f , = LP (x, en), then ,f, T. Proqf.

If m < n, then f,(xm em)= ,f,(x, em)= (f, x, en)em= (x, en)rm I

= xmem, therefore fm 5 f,.

Theorem 2. If X E C then LP(x)-RP(x), via a partial isometry of' the form iG, w a partial isometry in A . Proof. Say x = [x,, en]. Adopt the notation of the lcmma, and set f = sup f,; thus f,T f , and we have f = LP(x) by the proof of the lemma to Theorem 1. Write RP(x)=Z, e a projection in A (Theorem I), and let h = I- r . Thus xg=O, and, citing [$47, Def. 61, we have

pi 48. First Properties of C

where =

I n 11-'(e,)

n en n 1 = en n h p l ( e n ) .

In particular,

(2) ( 1 -e,)hk, = 0 by the definition of hpl(e,). Applying Proposition 1 to ( I ) , we infer that x , h k,=0 for all n ; since h k,= enh k, by (2), it follows that (3) x,e,hk, = 0 . Defining g, = RP(x,e,), it results from (3) that g, h k, = 0 , therefore g, LP ( hk,) = 0 ; citing [$ 3, Prop. 71, we have O=g,LP(hk,)=g,[h-hn(l

-k,)],

thus g, I ( 1 - h) + h n (1 - k,) and therefore

(4) Since g,=RP(x,e,)

-

+

D(g,) I 1 - D(h) D(l - k,) . LP(x,e,)

= f,,

it results from (4) that

D(f,) 5 1 -D(h)+D(l -k,), therefore

D(h) I D(l - f,)

(5)

+D(l

-

k,) ;

since D(1- fn)lD ( l - f ) and D ( l - k,)1 0 [$ 33, Th. 21, we infer from (5) that D(h)5 D(l - f ) (cf. [Ej 27, Lemma 1, (ii)]),that is, D ( f ' )5 D ( l - h), therefore f 5 I - h = e. We have shown that f d e , where ,T= L P ( x ) and F=RP(x). Dually, e 5 J; therefore e f by the Schroder-Bernstein theorem (which is trivial in a finite ring). I

-

Corollary 1. C is finite; in fact, y x = I implies x y = 1

Proof. Assuming x * x = l , we are to show that the projection f = x x * is also 1. Write f a projection in A (Theorem 1). Since L P ( x ) = f and R P ( x )= 1 [cf. 5 3, Prop. 41, it results from Theorem 2 that f 1, therefore f = 1 by the finiteness of A . The second assertion then follows from Theorem 2 on applying [$ 46, Prop. 1, ( 9 1 to C. I

-

e-f

f=x

Corollary 2. I f ' e, f are projections in C, su.y e =e and f in C i f a n d o n l y if e-f in A.

-

=

7,then

Proof: If w is a partial isometry in C such that w*w = e and ww* = f , then LP(w)= f and R P ( w ) = e, therefore e f in A by the theorem. The converse is obvious. I

222

Chapter 8. The Regular Ring of a Finite Bacr *-Ring

Note that the central projections of C are the projections E, with h a central projection in A (Theorem 1 and [$ 47, Def. 61); in view of Corollary 2, A and C have the 'same structure' in the sense of Section 15. The final proposition is for application later on (cf. [$ 51, Prop. I ] ) : Proposition 3. If x = [x,, en] and the x , are all invertible in A , then x is invertible in C and x-'= [ x i 1 ,h,] for a suitable SDD (12,).

Proof. Setting f ,= LP(x,e,), we have ,fn f by the lemma to Theorem 2; since f, -RP(x,e,)= en (by the invertibility of x,) and since e n f l , it follows that fnf 1. (Proof: D(sup f,,) = sup D(f,)= sup D(e,) = 1.) Set y n = x i l . If m < n then y n , f , = y m , f , ;for, ( y , -ym)xmem= 0, therefore ( y , -y,) ,fm = 0. Similarly, defining g,=LP(x,*e,), we have g,Tl and y,*gm=y:gm for m < n . Setting h,= f n n g,, it follows that (y,, h,) is an OWC. Let y = [y,, h,]. Since x,y,= y,x,= 1 for all n, it follows that x y = yx = 1. I Exercises It is assumed in the following exercises that A is a finite Baer *-ring satisfying LP -RP, with center Z, and C is the ring constructed in Section 47. 1A. (i) If x = [x,, en] and X,EZ for all n, then x is central in C. (ii) If ~ E A then , Z is central in C iff XEZ. (iii) The central projections in C are the projections z, h a central projection in A. 2A. If e is a projection in C, say e = , then e is abelian relative to C iff e is abelian relative to A. 3A. If (x,,e,)

for all n.

= Cy,,f,),

then x,(e, nL)=y,(e,

nf.1

and x:(e, n f,)=y:(e,

n.1,)

4A. If x = [x,,e,] and if, for each n, x, is invertible in e,Ae,, then x is invertible in C. More precisely, if y , ~ e ,Ar, with x, y,= y,x,= en, then x ' = b,,e,,]. 5A. If (f,) is any orthogonal sequence (finite or infinite)_ofplojections in A and if, for each n, U,E fnAfn, then there exists x e C such that xf,=,f,x =Si, for all n. 6A. Assume, in addition, that A is orthoseparable [cf. $33, Th. 41 and that A satisfies the (EP)-axiom. Suppose U E A with RP(u) = 1. (i) There exists an SDD (en) such that e,~{u*u}" and u*ue, is invertible in e,Ae, for all n. (ii) 5 is invertible in C.

7C. The results of this (and the preceding) section are valid for a finite Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom. 8A. If A satisfies the (EP)-axiom, then C is strongly semisimple; it follows that for every positive integer n, the matrix ring C, is strongly semisimple. 9A. The entire section can be recast in terms of well-directed families (r,,),,,,, A a fixed limit ordinal [cf. $47, Exer. 31.

4 49. C Has no New Partial Isomelries

5 49.

223

C Has no New Partial lsometries

As in the preceding two sections, A is a finite Baer *-ring satisfying LP --RP. To fulfill the promise in the section heading, we assume, in addition, that partial isometries in A are addable (as is automatically the case when A has no abelian summand, or when A is an AW*-algebra 15 20, Th. I]).

Theorem 1. C has no new unitaries. Proqf'. Let U E C be unitary; we are to show that u=ii with U E A unitary. {The following argument uses only the hypothesis u*u=l, hence it shows anew that C is finite [cf. ij 48, Cor. 1 of Th. 21.) In view of [$47, Lemma 81, we can suppose that u = [x,, en], u* u = [x,* x,, en] and (x,*xn, en)= ( I , en) via (en),thus

(*I

x,* xnen = en for all n . Defining wn= xnen, we have w,*wn=en by (*). Setting ,fn = wnw,* = LP(wn), it follows (as in the proof of [ij 48, Prop. 31) that , f n f 1. Note that wnem = wm whenever m < n . Set e, = fb = wO= 0 and define Elementary computations yield v,* vn= en- en-, and vnv,* = f n - f n - , (n = 1,2,3,. ..), thus the v, are partial isometrics with orthogonal initial projections and orthogonal final projections; by the addability hypothesis, there exists U E A such that and u(en-en-l)=un=wn-w,-,

therefore ii=u [ij 48, Prop. 21.

for n=1,2,3 ,.... Then

I

Corollary 1. C has no new partial isometries.

-

-

Proof. Let w be a partial isometry in C , say w* w=e, w w * = f . Since C is finite and has GC [$ 481, it follows from e f that I- e I - f [$17, Prop. 41. Say v*v=l -e, vv* = I - f . Then u= w+v is unitary and ue= w. Say u=ii (Theorem I ) and e =5 [ij 48, Th. I];then w =iiF=(ue)-, where w = u e is a partial isometry in A. I Corollary 2. Partial isometries in C are addable.

224

Chapter 8. The Regular Ring of a Finite Baer *-Ring

Proof: Immediate from Corollary 1 and the assumed addability of partial isometries in A. I Exercises

1A. The results in this section may bc generalized to well-directed families [cf. 6 48, Exer. 91.

2C. The results in this section hold for a finite Baer *-ring satisfying the (EP)axiom and the (SR)-axiom, in which partial isometries are addable (as is the case when there is no abelian summand).

5 50.

Positivity in C

The minimal hypotheses for the rest of the chapter are the following two: (1") A is afinite Baer *-ring sati.~fyingthe (EP)-axiom and the (UPSR)axiom. (2") Partial isometries in A are addable (as is the case when A has no abelian summand, or when A is an A W*-algebra [rj20, Th. I]). It follows that A also satisfies LP -- R P [520, Cor. of Th. 31, therefore the results of Sections 4 6 4 9 are applicable to A. Moreover, a strong form of polar decomposition holds for A [rj 21, Prop. 21. Our objective in this section is to initiate the study of positivity in C (which culminates in Section 53, under much heavier hypotheses on A). For the present we need only the following added assumptions (which will be superseded by the conditions (3") and (4") of the next section): (A) 2 is invertible in A. (B) I f x , y ~ Aand x*x+y*y=O, then, x=y=O. Let us review the notion of positivity in an arbitrary *-ring B. We say for suitable that X E B is positive, written x 2 0, if x=y:y, +...+y;y, elements y,, .. .,y, E B [§ 13, Def. 81. The self-adjoint elements of B may be ordered by writing x I y (or y 2 x) in case y-x 2 0. It is immediate that the relation is reflexive (x I x) and transitive (x I y and y < z imply x < z), but the property of antisymmetry (x I y and y < x imply x=y) may fail in general. In order that antisymmetry hold (and hence that the self-adjoints be partially ordered by this relation), it is sufficient that B have the following property: x: x, +...+x;x, = O implies x , = ...= x, = 0 (for any m) [cf. fj13, Exer. 91. We note, for use later on, that if x and y are self-adjoint elements such that x I y, then t * x t ~ t * y tfor every tgB. In view of the (UPSR)-axiom, the notion of positivity in A is simplified: x 2 0 if and only if x =y* y for some y (indeed, for some y 2 0). It follows at once that (B) extends to finitely many terms: if x:x, +...

5 SO.

Positivity in C

225

+ x: x, = 0, then x , = ...=x, =0. In particular, the ordering of selfadjoints in A is antisymmetric. Though we know nothing of square roots in C, the ordering of its self-adjoints is antisymmetric too: XI

Proposition 1. =...=x,=O.

I f x l , . .. ,X , EC and

xT x , +...+x;x,

= 0,

then

Proof. We illustrate the proof with m = 2. Suppose x* x + y* y = 0. Say x = [x,, en], y = b , , f,]; then [x,*x , y,*y,, gn]= 0 for suitable (g,). For a suitable SDD (h,), (x,*x , +y,*y,) h, = 0, therefore

+

for all n ; in view of ( B )we have x, h, =y, h, = 0 for all n, hence x = y = 0 [$48, Prop. 21. 1 Another application of (B): Lemma. I f x, Y E A then RP(x*x+ y* y)= RP(x)u RPLv).

Proof: Let g

=R

P(x*x+y*y). On the one hand,

[cf. $22, Lemma to Prop. 21. On the other hand

therefore x ( l -g)= y(l - g ) = 0 by (B). Thus x g = x, yg =y, and so R P ( x )5 g, R P b ) I y. {We remark that, in view of Proposition 1, thc same argument yields forany x ,,..., x,EC.}

I

Theorem 1. I f x E A and (en)is an SDD such that enx en2 0 for all n, then x 2 0. Proof. We know that x* = v [ji47, Lemma 21. Let e=RP(x)=LP(x) and set y= x +(1 - e). Since x = e y e = e* ye, it would suffice to show that y 2 0. Since the sum of positives is positive, clearly enye, 2 0 for all n ; moreover, y* y = x* x + I- e, therefore R P b )= e u (I - e)= 1 by the lemma. Changing notation, we can suppose that R P ( x ) = I. Let r. be the unique positive square root of x * x = x 2 , and write x=ur with u*u=uu*=I [$21, Prop. 21; it will suffice to show that u= 1. Note that u* = u. {Proof: r(u* - u)r = (ur)*r - r(ur)=x* r - r.x=xr " Def. l o ] ) ,therefore u* -u=O results -rx=O (because ~ E ( x ~ [$) 13, from RP(r)= 1 .) Thus u is a symmetry [ji 13, Def. 61; in view of (A), the

226

Chapter 8. The Regular Ring of a Finite Haer *-Ring

formula g = (1/2)(l+ u) defines a projection such that u = 2 g - 1. It will suffice to show that g = 1. Write v=s2 with s 2 0 , S E { Y ) " . Since uv=ru (because x = r u with x , r and u self-adjoint), it follows that su = us, thus Since enxe,2 0 we may write e,xe, = t,* t, for suitable t, (e.g., with t, 2 0 ) ;then (*) yields 2(Ssen)*(gsen)=(sen)*(sen)+ t,Y tn . Citing the lemma, it results that but R P(gsen)5 RP(s en) trivially, thus But RP(s)=RP(r)=RP(x)=1 implies that RP(sen)= en [cf. $23, Lemma to Prop. I]; citing (**), we have thus D(en)5 D(g) for all n ; since D(en)f 1 it results that D ( g )= I, g=1. I An important consequence of Theorem 1 is that the notions of positivity in A and C are consistent: Corollary. If x=E, X E A , then x 2 O in C ifandonly i f x 2 O in A.

Proof. Suppose x 2 0 in C, say x = y: y , +...+ y: y,. It is clear from the definition of the operations in C that we can write x = [x,, en] with x, 2 0 for all n. Citing [$48, Prop. 11, we have thus enxen=e,x,e, 2 0 for all n, therefore x 2 0 in A b y Theorem 1. The converse is obvious. I Exercises 1A. The results of this section hold with sequences replaced by well-directed families [cf. $49, Exer. I]. 2A. If A satisfies (lo), (2") and (B), and if, in addition, A is orthoseparable, then, for every U G A ,(1 + a * a ) is invertible in C (in particular, 2 is invertible in C).

3C. Theorem 1 fails when (UPSR) is weakened to (SR). (2"), (A) and (B), and let (e,) be a 4A. Let B be a finite Baer *-ring satisfying (I"), , e,xt x* ex well-directedfamilyof projectionsin B with e, e. Then, for each ~ E Bx* (interpreted in the obvious way relative to the partial ordering of self-adjoints described earlier in the section).

# 51. Cayley Transform

5 51.

227

Cayley Transform

Minimal hypotheses for the rest of the chapter: we assume, in addition to (I"), (2")of the preceding section, that (3")A is symmetric, that is, I + x * x is invertible in A,for ull X E A, and (4") A contains a central element i such that i2 = - 1 and i* = - i. The results of Sections 46-50 are applicable to A by virtue of the following lemma:

Lemma. A satisfies conditions (A)and ( B ) of' Section 50. Proof. 2 = I +I* 1 is invertible in A by (3"). Suppose x, Y E A and x* x +y* y= 0. Write x* x = r2, y* y = s2 with r and s self-adjoint, r~ {r2)",S E {s2}".Since r2 = -s2, obviously r~ {s2}',therefore r s = sr. Then (r + is)* (r + is)= r2 + s2 = 0, therefore r + i s = 0 ; taking adjoint, r-is=O, hence 2r=0, r=O, s=O. Thus x=y=O. I Remarks. 1. In the presence of (47, and the availability of square roots, (3") is clearly equivalent to the invertibility of x + i in A for all self-adjoints x e A . {But (3")can be formulated in any *-ring with unity.) 2. We also write i for the corresponding element of C, that is, we identify i with? clearly i is also central in C. 3. Since 2 is invertible in A, it follows from (4")that every X E A has a unique Cartesian decomposition x =y + iz, y and z self-adjoint ; explicitly, z = (112i)(x-x*). y = (1/2)(x+ x*), Similarly, every X G C has a unique Cartesian decomposition. 4. It follows from (1") and (2") that A may be written as a direct sum A= B @ C, where C satisfies (4") and every element in the center of B is self-adjoint (see Exercise 1). Thus, in assuming that A satisfies (4"), we are setting aside the 'purely real' part B. Property (3")lifts to C :

Proposition 1. For all X E C , I + x * x is invertible in C. Proof. Say x = [x,, en]. Then I + x* x = [ I + x,*x,, J,] for suitable (f,). By (3O), I+ x,* x, is invertible in A for all n, therefore 1 + x* x is invertible in C and ( I + x * x ) - ' = [ ( I +x,*x,)-', h,] for suitable (h,) [§48, Prop. 31.

1

Corollary. If x EC, x* = x, then x + i is invertible in C .

Proof: ( x - i)(x+ i)= ( x + i)(x- i)= I + x 2 is invertible in C by the proposition. I

Chapter 8. Thc Regular Ring of a Finite Haer *-Ring

228

The corollary sets the stage for the Cayley transform; the following proposition (and its proof) is valid in any *-ring B with unity, possessing a central element i with i2 = - 1 and i* = - i, such that x + i is invertible for every self-adjoint element x in B (note that 2 = ( I + i ) ( l - 1) is invertible in such a ring): Proposition 2. The formulas

define mutually inz~ersebijc.ction~between the set of all self-adjoint element^ x and the set of all unztary celements u such that I -u is invertible m C. If x and u are so paired, then jx}' = { u ) ' ,{ x } "= { u)" (the comnzutants are computed in C). W e call u the Cayley tran.sforin of x. Proof. Suppose x* = x. Defining u = ( x - i ) ( x+ i)

', we have

thus u is unitary; moreover,

thus I-u is invertible, and we recover x by the calculation i(l+u)(l- u ) '

= i(l +u)(1/2i)(x+i) = (112)[(x

+ i )+ u(x + i ) ]

=(1/2)[(x+i)+(x-i)]= x . Conversely, assuming U E C is unitary and I x = i ( l + u ) ( I - ~ ) ~ = i ( I u ) '+u). ( l Then

-

u is invertible, define

Also ( I - u ) x = i(1 +u); taking adjoints, x*(l - u*)= - i(l + u*), and right-multiplication by u yields x*(u - I ) = - i(u + I), thus From (1) and (2) we have x ( l - u)= x * ( l - u ) ; since I-u is invertible, we conclude that x = x*. Moreover, (1) may be rewritten ( x + i)u=x - i, therefore ( x i)(x+ i ) ' = u. I -

Remarks. 1. With notations as in Proposition 2, write u=ii with U E A unitary [ji49, Th. I ] . Since I -u is invertible, R P ( l - u ) = I , therefore R P ( l -u)=l (see the remark following [ji48, Cor. 1 of Th. I ] ) . It is shown in the next section (under an extra hypothesis on A ) that,

5 51.

229

Cayley Transforin

conversely, if UEA is any unitary with R P ( l -u)= 1, then Cis the Cayley transform of some self-adjoint element x of C . 2. If X E A then jx)' denotes the commutant oT x in A, whereas {Z)' is the commutant of Z in C ; there can be no confusion as long as we refrain from identifying x with x. 3. Proposition 2 is also valid with A in place of C . Our principal goal (attained in the next section, but only under an additional hypothesis on A) is to prove that C is regular in the sense of the following definition:

Definition 1. A ring B is said to be regular if, for each ~ E Btherc , B x y x = x. A *-regular ring is a regular ring with proper exists ~ E with involution. (Cf. Exercises 5,6.) Proposition 3. If B is a *-ring with unity, the followin~gconditions are equivalent: (a) B is *-regular. (b) fir each X E B, there exists u projectiovz e such t l ~ a t B x = Be; (c) B is regular and is a Rickart *-ring. Proof. (a) implies (b): If XEB, by hypothesis therc cxists y c B with x y x = x . Then g = x y is an idempotent with g x = x , thcrefore x B c g B ; on the other hand, g B = x y B c x B , thus x B = y B . Then

L({x}) = L(x B) = L(g B) = L((g)) = B(l -g), therefore R(L((x}))= R(B(1 -y))

=

R({1 - g ) )

= gB = xB.

Forgetting about g, we have shown that In particular,

R (L({x})) = x B for all x . R(L({xx*)))= x x * B

for all x

Since L(jx))= L({xxs)) by properness of the involution, we infer that (*)

xB=xx*B

for all x .

Let XEB. We scck a projection e such that B x = B e . By (*), we have x=.xx*z for a suitable element z. Setting e=x*z, we have thus e is a projection. Moreover, x = x x Yz = x e shows that B x c Be; since e = e Y= z Y x , we have also Be c Bx, thus B x = Be. . hypothesis, B x = B e , 6. a projection. (b) implies (c): Let ~ E B By Then R({x)) = R(Bx) = R(Be) = R(j6.i) = (1 e ) B .

230

Chapter 8. The Regular Ring of a Finite Baer *-Ring

This shows that B is a Rickart *-ring and that e =RP(x). Since B x = Be, e = y x for a suitable element y; then x y x = x e = x , thus B is regular. (c) implies (a): Every Rickart *-ring has proper involution [$ 3, Prop. 21. 1 Proposition 4. Let B be u *-regulur ring with unity, and let ~ E B , e=RP(x), f =LP(x). (i) There exists u unique element y such thut Y E P Bf ; y x = e , x y = f : (ii) In particular, if e = f (us is the cuse when x is normal), then x is invertible in eBe. (iii) If x is self-adjoint [normal] then y is also self-adjoint /normal]. Proof: As noted in the proof of Proposition 3, B x = B e . Choose y~ B with y x = e . Then also ( e y f ) x = ey(fx) = e y x = e e = e ; replacing y by eyf ; we can suppose y ~ e B j :Then (xy-f')x

=xyx-fx

=xe-

fx

=x-x

= 0,

therefore (xy - j)j =0, thus x y - j =O. If also z ~Bef satisfies z x = e , then y=ey=zxy=zj =z, which proves the asserted uniqueness. If x is normal, that is, if x* x =xx*, then e = j' results from properness of the involution [cf. $ 5, Lemma 51. It is elementary that an invertible normal [self-adjoint] element in a *-ring with unity has I normal [self-adjoint] inverse. In extending A to C, we are reaching for a regular ring; when A is already regular, the extension collapses: Theorem 1 . If A is regular then

2=C

Proof: In view of the Cartesian decomposition, it suffices to show that if X E C , x* = x , then ~ € 2 .Let u = F be the Cayley transform of x. As remarked following Proposition 2, LP (1 - u) = RP(1- u) = 1 ; by Proposition 4, 1 - u has an inverse b in A , hence x = i(l + u) (1 u)- = 2, where x = i ( l +u)h. I -

'

This has some surprising consequences: Corollary 1. If A is regular and (x,, en) is an OWC, then therc exists x~ A suclz thut x e n = x n e n and e n x = enxn for ull n. Proqf. Let x=[xn, en], write x = x by Theorem 1, and quote [$ 48, Prop. I]. I

5 51.

Cayley Transform

231

Corollary 2. If A is regular, ( f,) is a sequence of' ortlzoyonal projections, and an€ fnAfn for all n, then there exists X E A such thut f n x = x fn= a, fbr all n.

Proof: If the sequence (f,) is finite, let x be the sum of the a,. Otherwise, let f = sup f, and define then (x,, en) is an OWC, and Corollary I provides a suitable element x. I Corollary 3. Suppose thut (1) ,for each X E A there exists a positive integer k such that x * x < k 1, and (2) A contains an inlinite sequence (,f,) of nonzero orthogonal projections. Then A is not regulur.

Proof. Assume to the contrary that A is regular. Setting a,=nfn, Corollary 2 provides an element X E A such that .f,x = x fn = n f , for all n. Choose k as in (I); then f,x*xf, k,f,, thus

(*I

n2 f,
for all n.

Fix m with m2 > k ; thus m2 1 - k 1 = 1 +...+ 1 (m2- k summands) is > 0 , - and it is invertible by (1") and (4"). Then m2 ,f, 2 k,f,; in view of (*) this yields (m21- k I ) f, = 0 (cf. the remarks preceding [ij 50, Prop. I ] ) , hence, by the invertibility of m2 1 - k 1 , the contradiction .f, =O. (In connection with (2), see Exercise 11.) I To put it another way, if A is rcgular and satisfies (2), then it can't satisfy (1). We return to such matters in the discussion of boundedness in Section 54. Exercises

1A. Let A be a Baer *-ring satisfying the (EP)-axiom, and assume that partial isometries in A are addable. Then there exists a unique central projection h such that (i) hA satisfies (P),and (ii)every element in the center of (I - h)A is self-adjoint. ( A*-ring of the latter kind is called purely real.)

2D. Problem: How does one proceed in the purely real case (see Exercise I), i. e., in the absence of (4")? 3.4. If (A,) is a family of rings satisfying (1")-(4"), then the complete direct product of the A, also satisfies them [cf. 4 1, Exer. 131. 4A. Let B be a *-ring with unity, possessing a central element i with i2= - 1 and i* = - i, such that x+ i is invertible for each self-adjoint element x. (Briefly, we say that B is a *-ring with Cayley transform.) Suppose that for each x 2 0 there exists a unique r 2 0 with x = r2. Then B satisfies the (UPSR)-axiom.

5A. Let B be a ring with unity, XEB. The following conditions are equivalent: B x y x = x ; (b) there exists an idempotent e € B with (a) there exists ~ E with

232

Chapter 8. The Rcgular Ring of a Finite Baer *-Ring

Bx=Be; (c) there exists an idempotent ~ E with B xB=,fB. Thus the validity of (b) for each XEB is equivalent to regularity. 6A. The following conditions on a *-ring B are equivalent: (a) B is *-regular; (b) the involution of B is proper, and, for each XEB, there exists a projection ~ E B with B x = Be; (c) B is regular and is a weakly Rickart *-ring. 7A. Proposition 4 holds for any *-regular ring. 8A. If B is a *-regular ring and I is any ideal in B, then I is strict [cf. $22, Exer. I] and B/I is *-regular. 9A. If B is a *-regular ring with unity, in which x* x +y* y = 0 implies x = y =0, then B is symmetric. 10A. Assume: (1) C is a *-ring with unity and proper involution, (2) B is a *-subring of C, (3) B contains all the projections of C, and (4) for every XEC, 1 +x*x is invertible and (1 +x*x)-'E B. Then B is a Rickart *-ring iff C is a Rickart *-ring. In particular, B is a Baer *-ring iff C is a Baer *-ring. 11A. (i) Condition (2) of Corollary 3 can fail only if A is the direct sum of finitely many factors of Type I. (ii) An integral domain is regular if and only if it is a field. (iii) A finite Baer *-factor of Type I, satisfying LP RP, need not be regular.

-

12C. Let B be a ring with unity and let n be a positive integer. Then B, is regular iff B is regular. 13C. Let B be a *-regular ring and let n be a positive integer. If the relations x ,,..., x,EB, xTx1+...+x,*xn=O imply x,=...=x,=O, thenB,isalso*-regular. 14A. The following conditions on a *-regular ring B are equivalent: (a) B satisfies LP-RP; (b) if e, f are projections that arc algebraically equivalent [$I, Exer. 61 then e f.

-

-

15C. Let B be a *-regular ring satisfying LP RP (cf. Exercise 14). (i) If x,, ..., x, are elements of B such that xTx, +...+x,*x,=O, then RP(x,), ...,RP(x,) are central abelian projections. (ii) If B has no abelian direct summand, then B, is *-regular for all n. 16A. The results of this section generalize to well-directed families (but the statement of Corollary 2 must be left in sequential form). 17C. In a regular Baer *-ring, (i) y x = 1 implies x y = 1 (in particular, the ring is finite), and (ii) the projection lattice is a continuous geometry. 18A. For any XEC, the element y =(I

+ x*x)- '

satisfies

y2 I

I.

19C. Except for Corollary 3, the results of the present section hold with (UPSR) weakened to (SR).

5 52.

Regularity of C

To proceed further, it is necessary to augment the hypotheses (1")-(4") of the preceding section with some more spectral theory. The appropriate axiom is as follows:

# 52. Regularity of C

233

Definition 1. We say that A satisfies the unitary spectral axiom (briefly, the (US)-axiom) if, for each unitary UEA with R P ( l -u)= 1, there exists a sequence of projections e,E{u)" such that en?I and (I - u) en is invertible in enA en for all n. The following proposition assesses the strength of the (US)-axiom:

Proposition 1. (i) The (US)-axiom is implied by the hypothesis (1 ") when A is orthoseparable (equivalently, the center cfA is orthoseparable). (ii) Every Rickart C*-algebra (in particular, every AW*-alqebra) sati,Cfl'es the (US)-axiom. Proof: (i) We show, more generally, that if A is any orthoseparable Baer *-ring satisfying the (WEP)-axiom, then A satisfies the (US)-axiom. Suppose UEA is unitary with RP(1-u)=I. Set a = l -u. Obviously {a)' = {u)' ; moreover, x u = u x iff u* x = x u* iff x* u = ux*, thus {u)' is a *-subring. It follows that (u$"={u)" is a commutative *-subring of A . Let ( f , ) be a maximal orthogonal family of nonzero projections such that, for each 1 , there exists b , ~ { a ) "with a h , = b , a = f ; (hence a,ji is invertible in j;Aj;, with inverse b, j;).We assert that sup f ; = I.Let g = I-sup f,, note that g s fa}" 154, Prop. 71, and assume to the contrary that y # 0. Since R P ( a ) = I , it follows that ug#O. By the (WEP)axiom, there exists C E {ya* a y)" = {y a* a)" c {a)" with (y a* u)c*c= 1; ,f a nonzero projection. Thus

(*I

,f = (ga*c*c)a = a(gcc*c*c).

Clearly f 5 y, therefore ,f is orthogonal to every ,f,; setting b = q a * C*C E {aj", (*) shows that maximality is contradicted. By orthoseparability, the family (,f,) is countable; write it as a (possibly finite) sequence (.fn). Define e,= ,f; +...+f i (if there are only finitely many n, then e n = l for sufficiently large n). Then e n f 1 and n

e,a= oe,

=x

a f i is invertible in enAen.

1

The foregoing applies, in particular, if A satisfies the hypothesis (1 '); since A then has GC 1646, Prop. I], the finiteness of A yields the parenthetical criterion for orthoseparability [tj 33, Th. 41. (ii) This is easy spectral theory [cf. # 8, Prop. I]. I We assume for the rest of the scction that, in addition to the hypotheses (1")-(4") of the preceding section, (5") A satisfies the (US)-axiom. We can now characterize the unitaries that occur as Cayley transforms (see also Exercise 3):

234

Chapter 8. The Regular Ring of a Finite Baer *-Ring

Proposition 2. I f u c C i.s unitary, then 1 - u is invertible in C if and only if R P ( l - u) = 1 (equivalently, writing u = ii with uc A unitary, R P ( l -u)= 1). Thus, the Cayley transform pairs the selfladjoints x of' C with the unitaries u in A such that R P (1 - u) = I.

Proof. The 'only if part is trivial. Conversely, suppose RP(1- u) = I. Writing u=ii with UEA unitary, we have R P ( l -u)= 1 (remark following [tj 48, Cor. 1 of Th. I]). By the (US)-axiom, there exists an SDD (en) such that en€ {u)" and (I - u)e, has an inverse y, in e,Ae,, thus If m < n, it follows from the uniqueness of inverses that y, em= emy, =ym, - u)y therefore (y,, en) is an OWC; setting y = [y,, en], (*) yields (I = y(1 - u)= I [$48, Prop. 21, thus 1 - u is invertible in C. B At this point, one could develop the spectral theory in C; we defer this until the next section, preferring instead to take the shortest path to regularity. The following formulation of the (US)-axiom will be more convenient : Lemma 1. I f u E A is unitary and e = R P (1 - u), then there exists a sequence of projections enE {u}" such that enf e and (I - u) en is invertible in enAe,. Proof. Note that ec{l -u)" = {u)" [$ 3, Prop. lo]; I-e is the largest projection such that (I -u) (I -e)=O, that is, u(l -e)= 1 -e (hence also u*(l -e)= I -e). So to speak, 'u= I on I -e', and Proposition 2 is not applicable directly when 1 - e # 0; we remedy this by considering the unitary element Obviously vc{u)". We show that RP(l - v ) = I. Let g = 1 -RP(1 - v ) ; thus g ~ {-v}"= l jv)" c {u)", and (I -v)g=O. We have 1- v

=

[e+(l -e)] -[ue-(1

= (1-u)e+2(1-e)

=

-e)] 1-u+2(1 -e),

multiplication by e yields e(1- v) = e(l - u) = 1 - u, and multiplying this by g we have (1 -u)g = e(l -v)g = eO = 0, hence eg = 0 (recall that e = R P ( l - u)). It follows that therefore g=O (recall that 2 is invertible in A). Thus RP(1- v)= I. Note also (from the definition of v) that 11 e = u e, thus (1 - v)e = (I - u)e = 1 - u.

5 52.

Regularity of C

235

By the (US)-axiom, choose a sequence of projections g , ~ { v ) "c {u}" with g,f I and (1 -v)y, invertible in g, Ag,. Then e g , {u)", ~ eg,f e, and (I - u)eg , =(I - v)eg, is invertible in eg, A eg,, thus the sequence en= eg, meets all requirements. I This is the key to constructing the 'relative inverses' needed for regularity. Lemma 2. If X E C ,x* = x , and i f e = R P ( x ) , then x is invertible in e C e .

Proof. Let u = i7 be the Cayley transform of x. Since x = i(1- u)-' ( I + u), it is clear that R P ( x ) = R P ( l +u); thus, writing e=E, e a projection in A, we have e = R P ( l +u). Set u= -u; thus v is unitary and R P ( l -v)=e. By Lemma 1, there exists a sequence of projections en€ (v}"= {u}" such that en e and (1 - v)e, has an inverse z, in e,Ae,. Thus (1 + u)z, = z,(l + u) = en for all n. (*) As argued in the proof of Proposition 2, z,em=zm when m
236

Chapter 8. Thc Regular Ring of a Finite Baer *-Ring

to A, (4) the relations x, y~ D, x* x +y* y = 0 imply x =y =0, und (5) the element i of A is also central in D. Then D is *-isomo~phicto C (via an extension oJ the embedding a cl of A into C).

-

P r o d . It follows from (1) and (4) that D is *-regular [$ 51, Def. I]. By (3), D and A have the same unity element. If e is any projection in D, then the symmetry u = 2e - 1 is in A by (3), thcrcfore e = (1/2)(1 + U)E A ; thus D contains no new projections. Since D is a Rickart *-ring [$ 51, Prop. 31 with complete projection lattice, D is a Baer *-ring [# 4, Prop. I]. For any XED, R P ( l + x*x)= I [$ 50, Lemma], therefore I + x*x is invertible in D by *-regularity [$ 51, Prop. 41. It follows that the Cayley transform is operative in D (remarks preceding [$ 51, Prop. 21). Let XGD. We assert that there exists an SDD (en)such that x e , ~ A and x* en€ A for all n. By the Cartesian decomposition (and [$47, Lemma I]) we can suppose x* =u. Let u be the Cayley transform of x; by (3), ~ E A Since . I -u is invertible in D, we have RP(1 -u)= I (in D or in A-it's the same). Adopt the notations in the proof of Proposition 2, in particular, (I - u)y,= en; writing (I -u)-' for the inverse of 1- u in D, we have y, =(I - u ) ' en, therefore xe, = i(l + v) (1- u ) ' en =i(l+u)y,cA for all n. Each ~ G determines D an element cp(x) of C as follows. By the preceding, we can choose an SDD (en) such that x e , A ~ and x* ~ J , EA for all n. Applying [$ 47, Lemma 51 to the sequence (xe,) in A, we know that g,=e, n(xr,)-'(en) is an SDD; since (I-e,)xe,g,=O, we have (*)

x ~ ~ , = x e , g , = ( e , x e , ) ~ jfor ~ all n.

Similarly, defining h,= enn (x*en)-' (en), (h,) is an SDD such that

(**I

x* h,

= x* enh, = (r,

x* en)h,

for all n.

Setting x,=e,xe, and k,=g,nh,, it is clear from (*) and (**) that (x,, k,) is an OWC. We propose to define cp(x)= [x,,, k,], hence must check that the indicatcd clcmcnt of C is unambiguously determined. Suppose also (ek) is an SDD such that x e k A ~ and X*e : A ~ for all n; applying the foregoing construction, we arrive at an OWC (xk, kn), where x: = eixek. Setting f, = k, n kfn, it is elementary that fnx, f, = fnxf n = f,x; f,, therefore [x,, k,] = [xi, k,] by [$ 48, Prop. 21. It is routine to check that cp: D + C is a *-monomorphism. Finally, cp is surjective. For, suppose XEC,x* =x. If u=G is the Cayley transform of x, then LP(l - u ) = R P ( l -u)= 1; hence I -u is invertible in D, the formula x = i(1 + u) (I - u)-' defines a self-adjoint element of D whose Cayley transform is also u, and a straightforward argument shows that cp(x)=x. I

5 52.

Regularity of C

237

Proposition 1 yields two special cases that are worth noting explicitly:

Corollary. I f A sati.fies (1")-(4") and is orthoseparable, or if A is a finite AW*-algebra, then A has a unique regular ring in the sense of Dejinition 2 and Proposition 3. Proposition 3 yields a remarkable application of regularity:

Theorem 2. If (h,) is un orthogonal family of central projections in A with sup h,= 1, then C is *-isomorphic to the complete direct product of' the Z,C, via the mapping x ~ ( h - , x ) . Proqf. {We remark that it suffices to assume that A satisfies (1")-(4") and that C is regular (cf. Exercise 3).) Write ha=%, C,=h,C, and let D be the complete direct product of the C , [$ 1, Exer. 131. Writc 1 for the unity element (h,) of D. Define a mapping c p : C + D by cp(x)=(h,x). It is clear that cp is a *-monomorphism, with cp(l)=I. It remains to show that cp is surjective. Note that cp(i)=(h,i) is a central, skew-adjoint unitary element of D; that is, writing cp(i)= i, D satisfies (4").If (x,)ED is self-adjoint, then x, is self-adjoint for all a, x,+h,i has an inverse y, in C,, therefore (x,) + i=(x, + h, 1) has inverse (y,) in D. It follows that the Cayley transform is operative in D (remarks preceding [$ 51, Prop. 21). Since 2 = ( l + i) (1- i) is invertiblc, every elcment of D has a Cartesian decomposition; thus, to show that cp is surjective, it will suffice to show that its range contains every self-adjoint element of D. Suppose (x,)ED is self-adjoint. Let (u,) be the Cayley transform of (x,). In particular, (u,) is unitary, therefore u, is unitary in C, for each a. Write u,=iJi, with u , ~ h , A [$49, Cor. I ] , thus uzu,=u,u,*=h,. By A h,u =u, for all a (hence u is unitary), addability, there exists ~ E with thus cp(q=(u,). Observe that RP(l -q=1, that is, RP(l - u)= 1. {Proof: Let g = 1 -RP(1 -u). We have ( I -ti) i)g=O, hence 0 = cp(1 -q cp(3= (1- (u,))cp(Zj); since I- (u,) is invertible in D (by the theory of the Cayley transform), we conclude that cp(Zj)=O, therefore g=0 by the injectivity of cp. Thus g = 0 as asserted.) Since LP(1-Q =RP(I - Q = l and C is *-regular, I -ii is invertible in C [$51, Prop. 41 ; defining x=i(1 + q ( l -ti)-', we have x* = x . Since x ( l -q= i(l +ti), cp(x)(l-cp(q)=i(l +y(ti)), thus

application of cp yiclds

cp(x)(l- (u,))= 4 1 + (u,))

9

therefore cp(x)= i(l + (u,))( I

-

(u,))

' =(x,) .

I

238

Chapter 8. The Regular Ring of a Finite Baer *-Ring

Exercises 1A. Let B be a Rickart *-ring in which 2 is invertible. If B satisfies the (US)axiom and e is any projection in B, then eBe also satisfies the (US)-axiom. 2A. If (A,) is a family of rings satisfying (Io)-(5"), then the complete direct product of the A, also satisfies (Io)-(5").

3C. Suppose that (Io') A is a finite Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom (hence C exists [§48, Exer. 7]), and that (2")-(4") hold. The following conditions are equivalent: (a) C is regular; (b) if U E A is unitary and RP(1 -u)= 1, then I -ti is invertible in C ; (c) if UEA is unitary and e = R P ( l -u), then 1-E is invertible in ece.

4C. Suppose that (Io') A is a finite Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom, and that (2")-(5") hold. Then A has a unique regular ring in the sense of Definition 2 and Proposition 3. {Note: When A has no abelian summand, the hypothesis (2") is redundant; when A is orthoseparable (equivalently, the center of A is orthoseparable), the hypothesis (5") is redundant.) 5C. Let A be a Baer *-factor of Type II,, satisfying the (EP)-axiom and the (SR)-axiom, in which every element of the form 1 +x*x is invertible, and possessing a central element z such that z* # z. Then A has a unique regular ring in the sense of Definition 2 and Proposition 3. 6A. Every *-regular ring with unity trivially satisfies (5"). 7C. Suppose A satisfies (1")-(4") and is regular. If (h,) is an orthogonal family of central projections in A with sup ha= 1, then A is *-isomorphic to the complete direct product of the h,A, via the mapping x-t (hex). The same result holds with (Io) weakened to (Io') (see Exercise 3).

8C. (i) Suppose that (I"') A is a finite Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom, (2"') A has no abelian summand, and that A satisfies (3")-(5"). Then C, is *-regular for all n. (ii) In particular, if A is a nonabelian, finite Baer *-factor satisfying the (EP)axiom and the (SR)-axiom, if A is symmetric, and if A possesses a central element z such that z*# z, then C , is *-regular for all n. 9A. If A is a complete Boolean algebra with the identity involution [cf. 43, Exer. 141, then A satisfies (lo), (2") and (4")trivially and (5") vacuously (but not (3")), and is a commutative regular ring.

10D. Problems: (i) Can the regularity of C be reached with fewer axioms'? (ii) Can orthoseparability be omitted in the Corollary following Proposition 3? (iii) What about the purely real case [cf. ff 51, Exer. I]?

5 53.

Spectral Theory in C

As in the preceding section, we assume that A satisfies the hypotheses (1")-(5"). In this section we exploit (5') to show that all properties hypothesized for A lift to C. A key consequence of the (US)-axiom (5") is that a self-adjoint element of C can be represented in a form suitable for 'spectral theory':

9 53.

Spectral Theory in C

239

Proposition 1. If' x EC, x* = x, and i f ' u = ii is the Cuyley trunsform of x, then one can write x = [x,, en] with x,, enE (u}", x,* = x,, x, en= x,.

Proof. Adopt the notations in the proof of [$52, Prop. 21 (we know that RP(1 -u)= 1 by the trivial half of that proposition). By elementary algebra [ $ I , Exer. 121, y , {(I ~ - u)e,, ((1 - u)e,)*)" c ( u ) " . In e,Ae,, ue, is unitary and y, is the inverse of enPue,=(l -u)e,; defining we have x,*=x, by elementary algebra (see the proof of [$51, Prop. 21). The formula

shows that X,E {u)"; also, m < n implies y,e, =y,, therefore x,e, Since x,* =x,, it follows that (x,, en)is an OWC, and (*) yields

= x,.

The next proposition is a substitute for the assertion that 'functions' of a self-adjoint element x lie in { x ) " : Proposition 2. l f X E C , x* = x, jf u = ii is the Cayley transJi,rm oj ~ jbr all n, then y E { x ] " . X ,and if y = [yn,f',] with y , {u)"

Proof: Assuming Z E { x ) ' , it is to be shown that y z=z y. Since ( x ) ' is a *-subring of C , we can suppose (by the Cartesian decomposition) that z* = z. Let w = i5 be the Cayley transform of z. Since x z = z x , we infer that uw= wu 13.51, Prop. 21, thus uw=wu. Since y , ~ { u ) " , it follows that wy, =y, w for all n, therefore w y = y w; since {w)' = { z ] ' , it follows that z y = y z. I The following proposition well illustrates the power of the (US)axiom and the advantages of unique positive square roots: Proposition 3. If x E C, then x 2 0 !f' and only if' x = y* y ,for some y EC. In fact, C satisfies the (U PSR)-axiom.

Proof: Suppose x 2 0 (that is, x = y: y , +... + y: y, for suitable y .. ., Y,EC). Write x = [x,,,en] as in the statement of Proposition 1. Citing [$48, Prop. I ] we have E,, = (e,x, e n )=<x< 2 0 in C, therefore x, 2 0 in A [$50,Cor. of Th. I ] . Let r, be the unique positive square root of x,; in particular, r,E (x,}" c {u)" [$ 13, Def. 101. Thus, the x,, en,r, all belong to the commutative

,,

240

Chapter 8. The Regular Ring of a Finite Baer *-King

*-subring (u)". It follows from the uniqueness of positive square roots that (1)

r,e,=r,

when m < n ;

= r: e, = x, em= x,. From (2) we see for, rne, =emr, em2 0 and (r, that (r,, en)is an OWC; defining r = [r,, en], it results from r: = xn that

Since r, E (u}", we have by Proposition 2. Next, we note that for, if s, is the unique positive square root of r,, then the above argument shows that sne, = s, when m < n, thus s = [s, en] is a sclf-adjoint element with r =s2. In view of (2), (3), (4), it remains only to prove uniqueness: assuming that t 2 0 and x = t2, it is to be shown that t =r. Obviously t x = x t ; since r E {x)", we have r t = t r. Then

setting y = r - t, it follows that

Since y*ry 2 0 and y* t y 2 0, it results from (5) that yry = y t y = 0 [$50, Prop. I]; then y3=y(r-t)y=0, hence y4=0, and y =O results from the properness of the involution.

I

Definition 1. If XEC, x 2 0, we writc x+ for the unique positive square root of x given by Proposition 3. We know, in addition, that x+E {x)". Lemma. If XEC,x # 0, then tlzrre exists a € A suclz that X Z = , ~ f a nonzero projection. Proof. Say x = [x,, en]. Since x # 0, there exists an index m such 1948, Prop. 21. By the (EP)-axiom in A, there exists that x,e,#O b* = b e {e, x:x, e,)" such that (emx: x, em)b2 = e, e a nonzero projection. Thus br,x~x,e,b=e;

# 53. Spectral Theory in C

241

setting w=xmemb, we have w* w=e. Define f = w w Y . Citing [$48, Prop. I], we have Z=(xmem)-b=x
hence ,f=liiE*=x(e,bw*)-;

let u=embw*.

I

Proposition 4. C .sati,vfies the (EP)-uxiom. In fact, giaen any ZEC, z f 0 , there exi.sts YEA such that y ~ { z * z } " ,y 2 0, and z * z y 2 = z u nonzero projection.

7

Proof. Set x = (z* z)t, let u = G be the Cayley transform of x, and write x=[x,,e,] as in Proposition 1. Adopt the notation in the proof of the lemma. In particular, a = emb w*, where

b~ ( e m x ~ x m e m=){xi)" " c (u}"; then w=x,e,b also belongs to {u}", thus u ~ ( u ) " . Set y=(a*a)i c {u)". Then 2,ye {x)" by Proposition 2, and the relation f'= j *j yields

E {a*a)"

J = z + x * x u = x ~ ( u * u ) -=z*zy2.

I

Proposition 4 completes the proof that C has all the properties hypothesized for A-plus regularity: Theorem 1. Assume A satisjies (1")-(5"). (i) C .satisfies (1")-(5") and is regulur. (ii) A = C (f and only if A is regular. (iii) In particulur, C is its own regular ring, thus the operation A-C is idempotent. Proqf. (i) C is a finite Baer *-ring [$48] satisfying the (EP)-axiom (Proposition 4) and the (UPSR)-axiom (Proposition 3), thus it satisfies (1"); C also satisfies (2")-(4") (see [449, Cor. 2 of Th. 11 and [$51, Prop. I]), it inherits (5") from A since it has no new unitaries [$49, Th. I] or projections [S;48, Th. I], and it is regular [$52, Th. I]. (ii) If A is regular, then 2 = c [fj51, Th. I]; the converse is immediate from the regularity of C. (iii) It follows from (i) that C has a regular ring D in the sense of [$52, Def. 21; but C= D by (ii), thus C is its own regular ring. I It follows that the properties that accrue to A in virtue of (1")-(5") also accrue to C; for example, C admits a strong form of polar decomposition: 1.11a

C can write x = liir with r = (x* x); and Proposition 5. If' ~ E one partial isometry in A such that E* E = RP(x), EM,*= LP(x).

Chapter 8. The Regular Ring of a Finite Baer *-Ring

242

Proof. Since C satisfies (1") and (27, we can apply [ji21, Prop. 21 to it; the only partial isometries available in C are of the form i5 [ji49, Cor. 1 of Th. I]. I Proposition 6. If x E C, then x 2 0 ifand only if'one can write x = [x,, en] with x, 2 0 for all n. Proof. The 'only if part is trivial. Conversely, suppose x = [x,, en] with x,2 0 (or merely e,x,e, 2 0) for all n. Citing [ji48, Prop. 11, we have e,xF,, =(e,x,e,)- 2 0 for all n, therefore x 2 0 by [$50, Th. 11 (which is applicable to C since C also satisfies (1")-(4")). 1 The final result of the section bears on the notion of boundedness introduced in the next section (as do Exercises 3 and 5): Proposition 7. Let x, y E C. Then x* x l y* y if' and only if x = wy with w * w s l . Proof. If x = w y with w*w11, then x*x=y*w*wy
and w* w = (t v*)*x* x(tv*) l (t v*)* y* y(t v*) =vts2tv*=vfv*=vv*=LP(y)~1.

I

Exercises

In the following exercises, it is assumed that A satisfies (1")-(5"). 1 A . If X E C , x 2 0 , R P ( x )= e, and if y is the inverse of x in e C e, then y 2 0 .

2A. If x , ~ E C ,0 5 x 5 y, and if x is invertible, then y is invertible and 05y-'5x-l.

3A. (i) If ~ E C y, 2 0, then y 5 I if and only if y 2 5 1. (ii) In particular, 0 ~ ( 1 + x * x ) - ' ~ 1forall x c C . (iii)If x * = x and y = ( x + i ) l , then y * y < 1 . 4A. If x c C , one can write x = u r with r 2 0 and u = i i unitary.

5A. If X E C and z is in the center of C , then x * x 5 z iff x x * 5 z. 6A. Suppose A has the property that the relations a c A, a a * 5 a * a imply a a * = a * a (as is the case when A is a finite von Neumann algebra). Then the relations X E C , x x * < x * x imply x x * = x * x .

Ji 54. C Has n o New Bounded Elements

243

tj 54. C Has no New Bounded Elements

We assume, as in Sections 52 and 53, that A satisfies the hypotheses (1")-(57, and will shortly add another. The notion of boundedness alluded to in the section heading can be defined in an abstract *-ring:

Definition 1. An element Y. of a *-ring B with unity is said to be bounded if there exists a positive integer k such that x*x I k l (in the sense of the ordering described in Section 50). The set of all such elements of B, in view of the followx i s denoted B,; it is called the hoz~n~lrrlsubring ing proposition : Proposition 1. (i) I f ' B is any *-ring with unity, tlzen thc set B, ~ f ' a l l bounded elemmts in B is a suhring of B. (ii) B, contains all partial isometrics (in particular, all prej~ctions) of' B. (iii) If'n 1 is invertible in B fi)r ruery positive intt.gc.r n, tlien L3, is u *-subring of' B. Proof. (i) Suppose x, ~ E B , . Say x*x < 1111, y*y I n I , m and n positive integers. The boundedness of x-"v and x y result froin the computations ( X -y)* (x -v ) I (x y)* (x y) (x -.v)*(X - y)

+

+ +

= 2 x * x + 2 ~ ~ * y < 2 n z+I2 n 1 = ( 2 m + 2 n ) 1

and

( x y ) * ( x y ) = y * ( x * x ) y < y * ( m l ) y = r n ( ~ ~ ~<(m11)1 *y) ,

where the last inequality is justified by the identity (ii) If w* w = e, e a projection, then 1 - w* w = (I e)*(I - e)2 0, thus wcB,. (iii) We observe first that if ny 2 0, n a positive intcgcr, then v 2 0. (Proof: Obviously n2y=n(ny) 2 0. Say n2y=.yTy , +...+)~:y,,. Write c=(n I ) - ' ; then c is self-adjoint and central. thus y=c(n2y)c = b l c ) * ~ 1 c ) + . . . + b m c ) * ~ , c )0.1 2 Assuming XEB,, we are to show that x*EB,,. Say x*s5 r i l l . Then O < ( ~ x * - r n 1 ) ~ + x ( m-x*x)x* l -

= m2 I

-m(xx*)=m(ml x x * ) ,

therefore m l -xx* 2 0 by the preceding obscrvation. Thus xx* I m I, x*EB~. I

244

Chapter 8. The Regular Ring of a Finite Baer *-Ring

In particular, A, is a *-subring of A, C , is a *-subring of C ; obviously

2, c C,; and A,, C , are themselves Baer *-rings [cf. $4, Exer. 61. The notion of boundedness has occured tacitly before (cf. [$51, Cor. 3 of Th. I ] , [ji 53, Prop. 71). The claim in the section heading is that 2, = C,; to validate it, we require an additional axiom on A :

Definition 2. We say that A satisfies the positive sums axiom (briefly, the (PS)-axiom)provided that if ( f , ) is an orthogonal sequence of projections in A with sup ,fn = 1, and if, for each n, a, E J,A f , with 0 < an< 1, then there exists U E Asuch that a f , = a, for all n. {Informally, 'a= Can7.) In addition to (1")-(5") we assume for the rest of the section that

(6") A satisjies the (PS)-axiom. Remarks. Assume the notations of Definition 2. 1. The conditions 'a, EJ,A f , and 0 5 a, < I' are equivalent to '0 I u, < j,'. 2. The elements a,, fm all commute: a, f,= J,a,=a,, and a,f , =f,an=aman=O when m f n . 3. In an A W*-algebra (or even a Rickart C*-algebra) the construction of u is easy spectral theory; in fact, writing the commutant of the set of all ,fn as a C*-sum [ $ l o , Prop. 31, one need only assume that the a n ~ , f n A Jare , bounded in norm. 4. The element a is unique since supJ, = I. Moreover, 0 5 a 5 1. {Proof: Setting e,=f,+...+f,, we have enT1 and e,ae,=a,+... +a, 2 0 for all n, therefore a 2 0 [§50,Th. I ] ; also e,(l -u)e,=( j , - a l ) +...+(f,- a n ) 2 0 for all n, thus a < 1.) 5. The condition sup f n = l can be dropped, by adjoining I-sup,f, to the sequence (but then a need not be unique). 6. The point of the (PS)-axiom is that U S A ;one can always construct an aEC in the obvious way (cf. the proof of [tj 51, Cor. 2 of Th. I]). Lemma. Let x E C. In order that 0 I x I 1, it is necessary and sujl ficient that x = cl ,for some U E A with 0 5 u < I. Proof. Suppose 0 I x 2 I. Let u = i i be the Cayley transform of x and write x=[x,,e,] as in [$53, Prop. I]. Then O < % X < I & ; sincc <x<=(e,x,e,) =%, wc have 0 I x, < en [$50, Cor. of Th. I ] , therefore (1 Note that

O<xn
foralln.

# 54. C Has n o New Bounded Elements

for, if m < n then, since x$E(x,)"c {u)", we have Set e, = x , =0 and define fn=en-en,,

a,=x,-x,,

.

From (1)and (2) we have 0 5 a, 5 I, and it is easy to see that The (PS)-axiom yields a € A with

~ , , E J ,A h .

u ~ , = ~ , = x , - x , ~ , for all n , therefore

thus Z=x. Moreover, 0 5 u I by Remark 4 above. Conversely, if U E A and O
-

(1 y*y = b . Write b=c2 with c c A, c 2 0 . Then ( 1 ) yields

Write x =Er with r =(x*x)% and w s A a partial isometry [$ 53, Prop. 51. Since sc=s"cs">O, it results from (2), and the uniqueness of positive square roots in C [$ 53, Prop. 31, that r=(sc)-. Then x = Er=Z with a= wsc. Moreover, (a*a)- =x* x 5 k l , therefore a * a s k l by [jj 50, Cor. of Th. I]. Conversely, if a € A and a* a 5 k I, then 3 i*St< k I by the trivial half of [$ 50, Cor. of Th. I ] . I 1

1

Corollary 1. I f xl,. .. ,X ,

E

C and

X T X , +...+x;T,x, = 1 ,

then xj=Zj j;?r .suitable a j € A with ajY a, 5 I.

246

Chapter 8. The Regular Ring of a Finite Bacr *-Ring

Proof: Since 1 - xT xi = C xz x,, we have 0 5 xT lemma. I k+i

5 I ; quote the

Corollary 2. If' every a € A is bounded (for example, i f A is a finite A W*-algebra), then 3 is the bounded subring of C. Proof. Immediate from the theorem.

I

We conclude the chapter with a result on boundedness in quotient rings (see also Exercise 6):

Theorem 2. IJ I i~ a reslricted ideal of A , then the houndc~d.subring of A11 is the canonical image of the hounded subring of A, tlzw AJA, n I is nuturally *-isomorphic to (All),. Proof: Write x H 2 for the canonical mapping A + All. Recall that AII is a Rickart *-ring [tj 23, Prop. I]. If a is bounded in A , it is obvious that c? is bounded in AII. Conversely, suppose 2 is a bounded element of AII; we seek U E A , with 2 = d . Say 2 * 2 5 k 1, k a positive integer. Then

for suitable y,, ...,y ,€A,

thus there exists c t l with

In particular, k I- c 2 0, so we may writc k I - c = r2 with r 2 0. In view of (1) we have x * x l r2, therefore i?*i?
C w*w I I [$ 53, Prop. 71. In view of Theorem 1, for suitable ~ E with w= iG with WEA,, thus ( 2 ) yields

(3) X = wr, and it will suffice to show that 7 =6 with b~ A Let u = F . We know 9' that u*=u and u 2 = k l -c^=kl. Sct s=(k-7r)- (cf. the proof of Theorem 1). Then s* = s and s2 = k-' u2 = 1, thus s is a symmetry in AII. Then ( 1 / 2 ) ( l +s) is a projection in A/I, therefore (1/2)(l+ s ) = & for a suitable projection in A [$23, Prop. I]. Then s = ( 2 e - I)-, thus

writing b = k t ( 2 e - I ) , we have

7=6

with ~ E A , . I

6 54.

C TTas n o New Hounded Elelncnts

247

Exercises 1A. Assume A satisfies (1")-(5"). If every element of A is bounded (that is, A,= A) and if A is not the direct sum of finitely many factors of Type I, then 2 c C properly. 2A. Assume A satisfies (1")-(4"). (i) For every X E C ,(I +x*x)-' is bounded. (ii) If A also satisfies ( 5 3 and if x* = x , then (x+i)-' is bounded. 3A. If B is a *-ring with unity then B, is a *-subring of B under either of the following hypotheses: (i) x*x and xx* are unitarily equivalent, for every XEB; (ii) B is a Rickart *-ring with W P D [$ 21, Def. I]. 4A. (i) If A satisfies (lo)-(@) and is regular, then A satisfies (5") and (6"). (ii) If A satisfies (Io)-(5"), then C satisfies (1")-(6"). 5A. An example of a ring satisfying (1")-(6") that is neither regular nor A W* can be constructed as follows. If (A,) is any family of *-rings satisfying (1 ")-(67, then the complete direct product A of the A, also satisfies (1")-(6"). In particular, let (A,) be a family of finite AW*-algebras: if at least one of the A , is infinitedimensional, then A is not regular; if the family is infinite, then A is not an A W*-algebra. 6A. Assume A satisfies (Io)-(5"). Note that A and C have the same p-ideals, every ideal of C is restricted, and the p-ideals are paired bijectively with the ideals of C . Suppose 9 is a p-ideal of A, I is the ideal of A gencratcd by .a, and I the ideal of C generated by .?. Explicitly,

Thus 1 n 2 = i . (i) I is maximal-restricted in A if and only if I is maximal in C, in which case CjI is a regular Baer *-factor. (ii) The bounded subring of C/I is the canonical image of the bounded subring of C, thus C o j C o n I is naturally *-isomorphic to (Cjl),. (iii) If, in addition, A satisfies (6"), then A,jA, n 1 is naturally *-isomorphic to (C/I)o. (iv) If A satisfies (6") and all elements of A are bounded (as is the case when A is a finite AW*-algebra), then the natural *-monomorphism x + 1tt Y+ I maps AjI onto the bounded subring of CjI, thus AjI is *-isomorphic to (C/I),. {In a sense, we may speak of C/I as 'the regular ring' of A//; the rub is that All need not satisfy (1")-(5"); for instance, the (EP)-axiom may fail in A11 [cC # 45, Exer. 10, (iii)].} 7A. If A is the real AW*-algebra of bounded, complex sequences that are real at infinity [$4, Exer. 141, then A is a commutative Baer *-ring that satisfies (I"), ( 3 3 (5") and (6") but neither (2") nor ( 4 ) . 8A. A complete Boolean algebra trivially satisfies ( 6 ) [cf. $ 52, Exer. 91. 9A. Let A be the *-ring of all complex sequcncos x=(i,) that are ultimately real (that is, An is real for all sufficiently large n); the ring operations of A are the coordinatewise ones, and the involution is x*=(A,*). Then A is a commutative, regular Baer *-ring satisfying the (EP)-axiom and the (UPSR)-axiom; A satisfies (I"), (3"), (5") and (V), but neither (2") nor (4).

Chapter 9

Matrix Rings over Baer *-Rings The chapter is based on [9].

3 55.

Introduction

Let A be a Baer *-ring, n a positive integer, A, the *-ring of all n x n matrices over A (with *-transpose as the involution [$ 161). Is A, a Baer *-ring? In general, the answer is negative (see the exercises). However, if A is properly infinite, then A, is *-isomorphic to A-thereby killing off the question-under rather weak hypotheses (e. g., orthogonal GC is sufficient [$ 17, Cor. of Th. I], hence also PC [§ 14, Prop. 51). We are thus left with the problem of determining conditions on a finite Baer *-ring A that are sufficient to ensure that A, is a Baer *-ring. The problem is largely open. Using the theory of the regular ring developed in the preceding chapter, we show that A, is a Baer *-ring provided that A satisfies conditions (1")-(6") of Section 54, (7") A, satisfies the parallelogram law (P), and (8") every orthogonal sequence of projections in A, has a supremum; the hypotheses (1")-(8") are severe but the case of AW*-algebras is covered (see Section 62). For the rest of the chapter (excluding the exercises, and excluding [$ 62, Cor. 1 of Th. I]) A denotes a finite Baer *-ring, with center Z, satisfying the conditions (1")-(6") of the preceding chapter; for convenient reference, we record them here:

( 1 ") A is a finite Baer *-ring satisjying the (EP)-axiom and the (U PSR)-axiom; (2") partial isometries in A are addable; (3") I+u*a is invertible in A ,for all a € A; (4") A contains a central element i such that i2 = - 1 and i* = - i; (5") if' u EA is unitary and R P (1 - u) = I, then there exists a sequence of' projections e,E {u)" such that e,? I and (I -u)e, is invertible in e, A e, for all k; S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

6 55.

Introduction

249

(67 if f , is an orthogonal sequence of projections in A with sup j, = 1, and i f , ,for each k , ak€ f k A f , with 0 5 a, 5 1, then there exists a € A such that af,= a, ,for all k .

Remarks. 1. All conditions (I0)-(6") hold for A a finite A W*-algebra. 2. When A is orthoseparable, condition (5") is redundant [$ 52, Prop. I, (i)]. 3. When A is regular, conditions (5') and (6") are redundant; for, condition (5") holds trivially (we may take ek = I for all k [$ 51, Prop. 4, (ii)]) and condition (6") is a consequence of (1')-(4") 1951, Cor. 2 of Th. 11. 4. Condition (4") means that A has no 'purely real' part [$ 51, Exer. I]. 5. For a nonregular, non-A W* example of A, see [$ 54, Exer. 51. The following notations are also fixed for the rest of the chapter: C denotes the regular ring of A constructed in the preceding chapter [$ 52, Def. 21 (we drop the boldface, identify 2 with A, and regard A as a *-subring of C). n denotes a jixed positive integer. We write A, and C, for the n x n matrix rings (with *-transpose as involution), and regard A, as a *-subring of C,. For simplicity, we use lower case letters for the elements of all rings in sight; thus x=(x,,) denotes a typical element of C,. It is shown in the next section that C, contains no partial isometrics (projections, unitaries) not already in A,. Exercises 1A. If A={0,1) is the field with two elements, with the identity involution, the *-ring A, is a regular Baer ring but is not *-regular; in particular, A, is not a Baer *-ring.

2A. If A is the real AW*-algebra of bounded, complex sequences that are real at infinity [cf. § 54, Exer. 71, then A, is not a Rickart *-ring. This can be seen in two steps: (i) View the elements of A, as bounded sequences of 2 x 2 matrices (with the norm of a 2 x 2 complex matrix defined as the sum of the absolute values of its entries). (ii) For k= 1,2,3,... let e, be the element of A, whose kth coordinate is the projection / 1 i\

l 22 /

and all other coordinates are the zero matrix. Then e, is an orthogonal sequence of projections in A, with no supremum.

C h a p t e r 9. M a t r i x Rings over Baer *-Rings

250

3A. If A = Z , the ring of integers with the identity involution, then A, is not a Rickart *-ring. 4A. If A is the ring of 2 x 2 matrices over the field of three elements, then A is a Baer *-ring but A, is not a Rickart *-ring. 5A. If A is a Baer *-ring in which partial isometries are not addable, then A, is not a Baer *-ring for any n 2 2.

6A. If A is a Baer *-ring such that A, is a Rickart *-ring, does it follow that A, is a Baer *-ring?

tj 56. Generalities

In this section wc dcrivc some properties of A, and C, using only the hypotheses (1")-(6"). The central result is that A, is a Rickart *-ring (Theorem 1); this is the principal dividend of (6'). The first proposition holds under the weaker hypotheses of Section 50: Proposition 1. If xl,. . . ,.x, E C, and .xTx, + . . .+ x: x, x, =... = x,=O. In particular, the involution of C, is proper.

Proof: Say x,

hcnce

= (x:~) (k = 1,. ..,m).

= 0,

then

The (j,j) coordinate of the given

-eI=O[$ 50, Prop. I]. I

The next proposition requires only (lo)-(5") (see also [$ 52, Exer. 81): Proposition 2. C, is *-regular, hence is a regular Rickurt *-ring.

Proof: Since C is regular [$ 52, Th. I],C, is regular by a general theorem of von Ncumann [$ 51, Exer. 121. Since the involution of C, is proper (Proposition I), the proposition follows from [$ 51, Prop. 31. 1 The full force of the hypotheses (1 ')-(6") is used in the next proposition : Proposition3. If X , ,..., x,EC, and x ~ x , + ~ ~ ~ + x ~ (the x , =idenl tity matrix), then x,, . . . , Y, F A,. In particular, C, has no new projections (unitaries, partial isometrie.~). ProoJ. Say x,=($,).

The (j,j) coordinate of the given equation reads

hence $,t A [$ 54, Cor. 1 of Th. I]. If W E C, is a partial isometry, say w* w=e, then W E A, results from the equation w* w + (1 - e)* (1 - e) = 1. Thus C, has no new partial I isometries (hence no new projcctions or unitaries).

# 56. Generalities

251

Theorem 1. A, is a Rickart *-ring. Proof. Since C, is a Rickart *-ring (Proposition 2), the theorem is immediate from the fact that A, contains all projections of C, (Proposition 3). 1

Corollary. If' A is a Jinite AW*-algebra, then A, is a Rickurt C*-algebra. (It is shown in Section 62 that A, is ulso an AW*-ulqebra.) Proof. The n x n matrix algebra over a C*-algebra is also a C*-algebra (e. g., by the Gel'fand-Naimark theorem); quote Theorem 1. I Note that the center of A, consists of all the 'scalar' matrices diag (r,...,r) =

(a

.,

z

with Z E Z (the center of A), thus A and A , have *-isomorphic centers (as do C and C,). In particular, A and A, have the 'same' central projections, identified via

These are also the central projections of C and C,, since (i) A and C have the same projections [$48, Th. 11, (ii) A, and C, have the same projections (Proposition 3), and (iii) an element UEA is central in A iff it is central in C [cf. 5 47, Def. 61. If h is a central projection in A for) the product of diag (h,. ..,h) and (x,,)EC,, we write h(x,,)= ( h ~ , ~ and (x,,); that is, for purposes of notating products, we identify h with diag (h, ...,h). In view of the coarse structure theory (see [$ 25, Th. 21 and [$ 18, Th. 2]), the effect of the following theorem is to reduce the study of A, and C, to the cases that A is homogeneous or of Type 11:

Theorem 2. If (h,) is an orthogonal .family of central projections in A with sup h,= 1, then C, is *-isomorphic to he complete direct product of the h, C,, via the mapping (xij)++ (h,(xij)). Proof: The indicated mapping is clearly a *-homomorphism. It is injective: if (ha(xij))= 0, that is, if ha(xij)= (h,xij) = 0 for all a, then h,xij = 0 for all a, i, j, hence xij= 0 for all i, j, thus (xij)= 0. It remains to show that the mapping is surjective. Suppose that for C all a, i,j. each a we are given a matrix (x;~)in h,C,; thus X ? ~ E ~ ,for Since C is the complete direct product of thc h, C [$ 52, Th. 21, it follows that for each pair i, j there exists xij€C such that h,xij=x:j for all a. Then h,(xij) = (haxij) = (x:~),

252

Chapter 9. Matrix Rings over Baer *-Rings

thus ( x i j ) e C , maps onto the given element ((x:~)) of the complete direct product of the h, C,. I

Corollary. Notation as in Theorem 2. If, ,for each a , w,~h,A, is a partial isometry [projection, unitary] then there exists a unique partial isometry [projection, unitary] w E A, such that haw = w, for all a. Proof. By the theorem, there exists a unique W E C , such that haw= w, for all a. Since w, w,* wa= w,, that is, h,(w w* w- w)=0 for all a, it follows that ww*w=w, that is, w is a partial isometry. By Proposition 3, WEA,. If, moreover, the w, are projections [unitaries] in h,A,, it is immediate from the theorem that w is also a projection [unitary]. I The center of A,, being isomorphic to the center Z of A, is a Baer *-ring. In addition, there is a smoothly working notion of central cover available in A,:

Definition 1. If x = (aij)€A,, we define C ( x ) = sup{C(aij):i , j = I , ..., n ) , where C(aij)is the central cover of aij relative to A [$ 6, Prop. 31. We regard C ( x ) as a projection in A, as well, and call it the central cover of x ; clearly C ( x )is the smallest central projection h such that hx..=xij for all i and j, that is, h x = x . In particular, the present definition is consistent with [$ 6, Def. I ] . !J

Though we do not know that A, is a Baer *-ring, the characterization of 'very orthogonality' valid for Baer *-rings holds also in A,:

Proposition 4. If x , Y E A,, the followiny conditions are equivalent: (a) C(x)C(y)=O;(b) xA,y=O. Proof. Write B= A,. It is obvious that (a) implies (b). Conversely, suppose x B y = 0. Then also ( B xB ) B ( B y B )=O. Say x = (aij),y = (bij). We assert that aijAb,, = 0 for all indices i,j, r-, s; since pre- and post-multiplication of x by suitable matrices will move aij to the (1,l) position, and similarly for b,,, it clearly suffices to prove that a,, Ab,, =O. Also, pre- and post-multiplying x by the projection diag(l,O, ...,O), we can suppose that aij=O unless i=j= I,and similarly that bii=O unless i=j= 1. It follows that, for all a € A, the condition yields a,, ab,, = 0 ; thus a,, Ab,, = 0. Summarizing, it has been shown that for all indices i, j, r, s we have aijAb,, = 0, hence C(aij)C(b,,) = 0 [$ 6, Cor. 1 of Prop. 31. It is immediate from Definition 1 that C(x)C(y)=O. I

5 56.

Generalities

253

We shall make use of central cover in A, only for projections; the following elementary proposition is a special case of [§ 6, Prop. I]:

-

Proposition 5. For projections in A,, (1) e f implies C(e)= C(f ); (2) e s f implies C(e)< C ( f ) ; (3) C(he)=hC(e) when h is central; (4) if (el) is a family of projections that possesses a supremum e, then C(e)= sup C(e,). Exercises 1C. Let B be any *-ring with unity, n a positive integer, B, the *-ring of all n x n matrices over B [$16, Def. I]. (i) The involution of B, is proper if and only if B satisfies the condition that xfx, +...+x,*x,=O implies x, =...=x,=O . (ii) Let M be the set of all nples x=(x,, ...,x,), xi€ B, regarded as a left B-module in the obvious way; one can identify B, with the ring of all module endomorphisms T: M + M. Assume that B satisfies the condition in (i). For x, Y E M, define [x,y]=x1yf+...+xn.Y,*; thus [x, y] is a B-valued 'inner product' on M such that [x, x] = O implies x = 0. Writing T* for the *-transpose of the matrix T, one has [xT, y]= [x, y T*] identically. For a subset N of M, write

NI

=

{xEM: [x,y]=O for all YEN}.

Prove: B,, is a Baer *-ring if and only if M = N + NL for every submodule N of M satisfying N = NLL. (iii) Assume B is an involutive division ring. Then B, is a Baer *-ring if and only if B satisfies the condition that xTx, +... +x,*x, = O implies x, =...=x,=O . 2A. If A satisfies (1")-(6") then A, and C, are symmetric. In fact, if x is any element of C,,, then I+x*x has an inverse in An.

3A. Assume A satisfies (Io)-(6"). If XEC, then LP(x) and RP(x) are algebraically equivalent in C, [cf. 3 1, Exer. 61. In particular, if e, f are projections in A,,, then e-e n f and e u f - f are algebraically equivalent in C,. 4D. Assume A satisfies (1")-(6"). Problems: (i) Does A,, (equivalently, C,) satisfy the parallelogram law (P)? (ii) Does A,, satisfy LP-RP? Does C,? (iii) Is A,, (equivalently, C,,) a Baer *-ring? 5A. Assume A satisfies (1")-(5"). The following conditions on A are equivalent: (a) the relations XEC,x*x< 1 imply XGA; (b) every partial isometry in C, is in A,; (c) every unitary in C, is in A,; (d) every symmetry in C, is in A,; (e) every projection in C, is in A,. In items (b)-(e), one can replace 2 by any positive integer n 2 2 ; it follows that, assuming (a), A, is a Rickart *-ring for all n.

Chapter 9. Matrix Iiings over Baer *-Rings

254

5 57.

Parallelogram Law and Generalized Comparability

Does the parallelogram law (P) hold in A,? Though the evidence is favorable [$ 56, Exer. 31, the question is open. From the next section on, we assume (P) outright. In the present section, we note that (i) if A, possesses a feeble notion of 'square roots', then A, satisfies (P), and (ii) if A, satisfies (P) then it has GC. Definition 1. A weakly Rickart *-ring is said to satisfy the very weak square root axiorn (VWSR) if, for each element x, there exists an element r such that x * x = r* r (hence RP(x)=RP(r)) and LP(r)=RP(r). It is trivial that a Rickart *-ring satisfying the (WSR)-axiom [$ 13, Def. 51 also satisfies the (VWSR)-axiom; in particular, every Rickart C*-algebra satisfies the (VWSR)-axiom. LP

-

Proposition 1. If' A, satisfies the (VWSR)-axiom, then A, satisfies RP, therefiwe A, sati.sfie,s the paralleloyram law (P).

Proof. The second assertion follows from the first [$ 3, Prop. 71. Thus, if ~ E A , r=RP(x), , j = LP(x), it will suffice to show that e- j. Write x * x = r* r with LP(r) =RP(r) =RP(x) =e. By the *-regularity of C,, we have C,x= C,e= C,,r [Ej 51, Prop. 31; it follows from [Ej 21, Prop. 31 that there exists a partial isometry W E C, such that x = w r, w* w=e, w w* = j . Since W E A, [Ej 56, Prop. 31, this shows that / in A,. I

-

In particular, if A is a finite AW*-algebra, then A, is a Rickart C*-algebra [Ej 56, Cor. of Th. I], therefore A, satisfies (P) by Proposition 1. {But this is not news, since (P) holds in every Rickart C*-algebra [Ej 13, Th. I] by a very different proof.) We denote by u,, . ..,u, the diagonal matrix units in A,: u,

= diag (1,0,. . .,O),

.. . ,u, = diag (0,....O, 1).

Thus, the u, are orthogonal, equivalent projections with sum 1, and A is *-isomorphic to u, A,u, via the correspondence a tt diag (a,0,. . . ,0) [cf. 5 161. It follows that the centers of A, A, and u, A,u, are +-isomorphic via the correspondences z

tt

diag (z, . . .,z) H diag (z,0,. .. ,0)

Lemma 1. Suppose A, ~utisfiesthe parullelogra~nlaw (P). I/ e i~ any projection in A,, there exlsts an orthogonal deconzpo~itione = el + . ..+ e, suclz that e, 5 u, for all i.

5 57. Parallclogra~nLaw and

Generalized Comparability

255

Proof. The proof for n = 3 illustrates the principle. Since A , is a Rickart *-ring [$ 56, Th. I] satisfying (P), we can apply [$ 13, Prop. 51 to the pair r , u,: there exist orthogonal decompositions

such that e l - u and fu,=O; thus e , d u , and f ~ l G u ~ = u , + u , . Repeat the argument on the pair J; u,, obtaining orthogonal decompositions u, = a + h f = e,+e,, with e,--a and e,u,=O. Then e,-a
+

+

e, I (u, u,) (ul u,)

Thus e=e, +e,+e,

= u,

with e i d u l (i=1,2,3).

- u1 1

Lemma 2. If e, f are projections in A, such that e 5 u , and f then e , f are GC.

5 u,,

Proqf. It is no loss of generality to supposc that e 5 u,, f 5 ul. Since A has GC and u, A,u, is *-isomorphic to A , it follows that e , f are GC in ul A n u l ;since the central projections of u, Anul are evidently of the form diag(h,O, ...,O)=hu, with h a central projection in A , it is immediate that e , j are G C in A,. I

Proposition 2. If' A, satisfies the parallelogram law ( P ) , then A, has GC. Proof. {The argument shows, more generally, that if A is any Rickart *-ring with GC, such that A, is a Rickart *-ring satisfying (P), then A, has GC.) Let e, f be any pair of projections in A,; we seek orthogonal decompositions c=ef+e", f=fr+f"

with el- f' and e", f" very orthogonal [$ 14, Prop. I]. By thc finite additivity of equivalence [$ 1, Prop. 81, we are free to discard equivalent subprojections of e , f . The proof for n = 3 illustrates the method. By Lemma 1, we can write e=e1+e2+r,, j=f,+f,+f, with e i 5 u , , J d u , for all i. Applying Lemma 2 to the pair e l , f ; and discarding equivalent subprojections, we can suppose el, 1; are very orthogonal. Then applying Lemma 2 to the pair e l , f ; and discarding equivalent subprojections, we can suppose e l ,f , are very orthogonal too. Thus, by 9 successive applications of Lemma 2, we are reduced to the case that pi, f j are very orthogonal for all i,,j; but then it is clear

Chapter 9. Matrix Rings over Baer *-Rings

256

that e, f are very orthogonal [cf. 3 56, Prop. 5, (4)]. {Explicitly, if hij is a central projection such that e, hij= e, and ,fj hij=0, then, setting

h = inf (sup hij), j

i

we have e, h = ei for all i, and j, h = 0 for all ,j, therefore e h = e and fh=O.) I Combining Propositions 1 and 2, we have:

Corollary 1. If A, sati@es the (VWSR)-axiom, then A, satisfies the parallelogram law (P) and has GC. Another consequence of Proposition 2:

Corollary 2. If A, satisfies the parallelogram law (P), and e, f are projections in A,, then the ,following conditions are equivalent: (a) C(e)C(f ) # 0 ; (b) PA,f # 0 ; (c) e, f are partially comparable, that is, there exist nonzero subprojections e, I e, f , sf such that e, f,.

-

Proof. The equivalence of (a) and (b) is noted in [$56, Prop. 41. It is obvious that (c) implies (a) [cf. $ 56, Prop. 51. Since A, has GC by Proposition 2, it follows that (b) implies (c) [$14, Prop. 21. 1 Exercises

1A. Assume A satisfies (1")-(So). If C, satisfies the (VWSR)-axiom, then C, is *-regular, satisfies LP -RP, satisfies the parallelogram law (P), and has GC. 2.4. Assume A satisfies (Io)-(V), and suppose every element of A is bounded [$54, Def. 11. If C,,satisfies the (VWSR)-axiom, then so does A,.

3A. Consider the following condition on a *-ring B: If X E B and e is a projection in B such that x * x ~ e B e ,then there exists r ~ e B ewith x * x = r * r . Call this condition (C). (i) In a weakly Rickart *-ring, the (VWSR)-axiom implies condition (C). (ii) In a finite Rickart *-ring satisfying LP RP, condition (C) implies the (VWSR)-axiom. (iii) In a *-regular ring, ihe (VWSR)-axiom implies L P RP. (iv) A finite *-regular ring satisfies the (VWSR)-axiom if and only if it satisfies LP R P and condition (C).

-

-

5 58.

-

Finiteness

For the rest of the chapter (excluding Corollary 1 in Section 62) we assume, in addition to (1")-(6") : (7") A, satisfes the parallelogram law (P); (So) every sequence of orthogonal projections in A, has a supremum. On the basis of (I0)-(8") it will be shown that A, is a finite Baer *-ring (see Section 62).

257

§ 59. Simple Projections

If A is a finite A W*-algebra, then A, is a Rickart C*-algebra [§ 56, Cor. of Th. I], therefore A, satisfies (7") (see the remarks following [$57, Prop. I]) and (8") (see [§8, Lemma 3 to Prop. I]). Nevertheless, in the general case (7") and (8") must be regarded as unwelcome guests, since they impose conditions on A, (about which we are trying to prove something) rather than directly on A (about which we are willing to assume practically anything).

Proposition 1. A, is finite. Proof: A, is a Rickart *-ring [$56, Th. 11; it satisfies the parallelogram law (P) by hypothesis, therefore it has GC [$57, Prop. 21. Write u , , . .., u, for the diagonal matrix units (as in Section 57). Since A is finite, so are the isomorphic rings uiA,ui; in other words, the ui are finite projections. Thus u , +...+ u, = I, where the ui are finite; in view of (So), it follows that 1 is a finite projection in A, [Ej 17, Th. 31. 1 In view of (a"), the following variant of finiteness is also valid in A, [$ 17, Prop. 4, (iii)] :

Corollary. A, does not contain an infinite sequence of orthogonal, equivalent nonzero projections. The corollary is important for certain exhaustion arguments that underlie the next section (cf. the proof of [#26, Prop. 161). Exercises

1A. Assume that A satisfies (Io)-(6"), and that A, satisfies the (VWSR)-axiom and (8"). Then the relations x, y e A,, y x = I imply x y = I . (Better yet, see Exercise 3.)

2C. Let B be a regular Baer *-ring satisfying LP-RP and let n be any positive integer. The relations x, y e B,, y x = 1 imply x y = 1; in particular, B, is finite. 3C. Suppose that (I"') A is a finite Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom, and that (2")-(5") hold. Let C be the regular ring of A 52, Exer. 41 and let n be any positive integer. The relations x , y ~ C , y, x = 1 imply x y = 1 ; in particular, C , and A, are finite.

[a

5 59.

Simple Projections

As in the preceding section, we assume (lo)-(8"). For brevity, we write B = A,. As an aid to proving that B is a Baer *-ring, we develop here the rudiments of the dimension theory of projections in B. The essential

258

Chapter 9. Matrix Rings over Bacr *-Rings

idea is to scan certain of the arguments in Chapter 6 with B in mind. The properties of B that are crucial for the following discussion of dimension: (i) B is finite [$58, Prop. I], (ii) B has GC 1457, Prop. 21, (iii) B admits a notion of central cover ([$56, Prop. 51, 1557, Cor. 2 of Prop. 2]), and (iv) the central projections of B form a complete Boolean algebra, whose Stone representation space we denote by 3 (we identify a central projection with the characteristic function of the clopen set in .%"to which it corresponds). Let us scan Section 26 with B in mind. Everything through Proposition 13 holds verbatim for B. In particular, a nonzero projection e is called simple iff there exists a central projection h and an orthogonal decomposition h = e, +...+ em with e e, for k = I, . . ., nz [cf. $ 26, Def. 21; necessarily h=C(e), and the integer m, which is unique by finiteness [cf. 3 26, Prop. I], is called the order ofe; one defines T(e)= (l/m)lz, regarded as a continuous function on .F [cf. $26, Def. 31; if m is a power of 2 then e is called fundamental [cf. $26, Def. 41. If e is simple and e J; then f' is also simple and T(e)= T ( f ) [cf. $26, Prop. 111; conversely, if e and f' are simple projections with T(e)= T(f'), then e -f [cf. $26, Prop. 121.

-

-

Lemma 1. Every nonzero projection in B contains a simple projection. When A is oj Type I, the simple projection can be taken to be ahclian; when A is qf Type TI, it can be taken to he,fundamental.

Pro($ We note first that the lemma is true in A (i.e., for n = l ) [$26, Props. 14 and 161. Write u,, . .., u, for the diagonal matrix units of B. Let e be any nonzero projection in B. Since u , +...+ u, = 1, we can suppose, for example, that eu, # 0. Then C(e)C(u,)#O. In view of [$57, Cor. 2, (c) of Prop. 21 we can suppose, without loss of generality, that e l u, [cf. $15, Prop. 71. Since u, Bu, is *-isomorphic to A, it follows from the first line of the proof that there exists a subprojection e ' l e that is simple in u, Bu,; since the central projections of u, Bu, are the projection hu,, with h a central projection in B (see the discussion in Section 57), and since Iz=hu,+...+hu, with h u , - ~ ~ ~ - l z u , ,it is clear that e' is also simple in B [cf. $26, Prop. 81. If, in addition, A is of Type I [Type TI] then e' can be taken to be abelian [fundamental]. I Although we do not yet know that the projection lattice of B is complete, it is a good omen that the above process can bc continued to exhaustion : Lemma 2. Let e be any nonzero projection in B. There exists an orthogonal family (e,) oj simple projections such that e = sup e,. 1j A is of Type I, the e, can he taken to be abelian; if A is of Type TI, they can he taken to be fundamental.

# 60. Type 11 Case

259

ProoJ: Let (el)be a maximal orthogonal family of projections of the desired kind with el < e for all I . We assert that sup e, exists and is cqual to e. Thus, assuming f is a projection such that el 5 j' for all 1, it is to bc shown that e<,f. Let x=e(I - f ) and assume to the contrary that x f 0. Let y = L P ( x ) ; thusOfg 5 e. By Lemma I, there exists a projection go of the desired kind with go < g. Then for all L one has e,x= e,c(l -f') =e,(l -f ) = O , therefore e,g =0, hence e,go =0, contradicting maxiI mality.

5 60.

'Iype I1 Case

Assuming that (lo)-(8")hold and that A is of Type 11, we show in this section that B = A, is a Baer *-ring. First, we observe that projections in B may be subdivided at will:

Lemma I. I f e is any projection in B and m is ~itzypositi~~r itzteyr~, then e is the sum of m orthoyonal, equivalent projections. Proof. By [$57, Lemma 1 to Prop. 21, we are reduced to the correI sponding result for A 19, Th. I].

[a

We now scan Section 30 with B in mind. Everything through Proposition 4 holds verbatim for B (the needed notion of central covcr, divisibility of projections, and completeness of the lattice of central projections, are available in B); the analogue of 1430, Prop. 41 for B:

Lemma 2. If' (J;),,, is an ortlzogonaljut~iilyqffirndamc~ntulprqjc~ctio~z~s in B, such that the orders of'thr f; are bounded then tlzere cxist.~ an ortlzo~/rloncil family (h,) qf nonzero central projections with sup 11, = I, such th~rtJbr each a, the set ( x K ~: h, f , # 0) is finite. Lemma 3. With notation as in Lemma 2, supf, c>xists. Proof. For each a, write K , = ( X EK : / I , / , # 0) and dcfine

(these are finite sums). By 1556, Cor. of Th. 21, there exists a projection ~ E such B that hag= y, for all a. It will be shown that supj i exists and is equal to g. Given any ~ E K we , assert that /; I g, that is, f i ( l - y)= 0. Given any a, it suffices to show that h , f ; ( l- y)=O [$56, Th. 21. If I?,,f,=O this is trivial. If h, f, f O then h, J; < g, by the definition of g,; since I?,g = (I,, an elementary computation yields h,f;(l -q)=O. Thus g is an upper bound for the J;.

260

Chapter 9. Matrix Rings over Baer *-Icings

Finally, if,f is a projection in B with ,f, < f for all x, it is to be shown that g 5 f. Fixing a, it is enough to show that h,g(l -j') = 0, that is, g, = g, f. For all x, h, J; < h, J; therefore

1 h,.fx 5 h,f>

xrK,

Lemma 4. If(e,),,, is any orthoyonal family of projections in B, then sup el exists.

Proof. For each L we may express e, as the supremum of an orthogonal family of fundamental projections [$59, Lemma 21, say . Let J be the set of all ordered pairs (I,x) with L E 1 and ~ E K , For m =0,1,2,3, .. . let J, denote the set of all ( 1 , x) in J for which el, has order 2". Thus c0 J = Jm.

u

m=O

For fixed m, the projections e,, with (L,x)EJ, have bounded order (=2"), hence by Lemma 3 we can form (ii)

Jm

= SUP

{el,: (1, X)E Jm)

It is routine to check that theJ, are orthogonal. Hence by (8") we can form (iii)

e=sup(,f,:nz=0,1,2,3,

...).

It will be shown that sup el exists and is equal to e. Given any I, let us show that el < e. If x e K, then e,, I e. {Proof: If el, has order 2", then (L,X)E J,, hence e,, <,fm5 e.) Then e, < e results from (i). Finally, if y is any projection in B with el < g for all L,it is to be shown that e 5 g. Given any m, it will suffice by (iii) to show that J, 5 g. For all ( 1 , X)EJ, we have el, < e, < g, therefore ,fmIg by (ii). I Theorem 1. If A satisfies (1")-(6") and is o j Type 11, and $ n is a positive integer such that (7") and (8") hold, then A, is a Baer *-ring.

Proof'. Since B = A, is a Rickart *-ring [$56, Th. I], the theorem is immediate from Lemma 4 [g4, Prop. I]. I

5 61.

Type I Case

We assume in this section that (1")-(8") hold and that A is of Type I. It is to be shown that A, (equivalently, C,) is a Baer *-ring. In view of [$56, Th. 21 and the decomposition theory of Type I rings [518, Th. 21,

5 61.

Type I Case

261

we can suppose that A is homogeneous of Type I,, r a positive integer. (Note that if h is a central projection, then hA,=(hA),, hC,=(hC),, and hC is the regular ring of h A (cf. [§52, Prop. 31 or [§47, Exer. 1, (ii)].} Let e l , ..., e, be orthogonal, equivalent, abelian projections in A with sum 1, and let u,, ..., u, be the diagonal matrix units in B= A,. For i= I , . .., n and v = 1, ..., r, let uiv be the matrix with e, in the (i,i) coordinate and zeros elsewhere:

uiv= diag(0, ..., e,, ..., O ) , with e, occurring in the ith place. By hypothesis, evAe, is an abelian Baer *-ring; thus, if e is a subprojection of e,, then e = h e, for a suitable central projection h of A [$15, Prop. 61. Since uivBu,, is *-isomorphic to e, Ae,, ui, is an abelian projection in B ; better yet, the preceding comment shows that if u is a subprojection of uivthen u = hui, for a suitable central projection h. Summarizing, the uiv are orthogonal, equivalent, abelian projections in B with sum 1, possessing the subprojection property just mentioned.

Lemma 1. B does not contain n r + I orthogonal, equivalent nonzero projections. Proof. Suppose j; ( p = 1,2, . .., n r + I ) are orthogonal, equivalent projections in B ; it is to be shown that f , = 0. Since C uiv= 1 , it suffices i,v

to show that ui,f , = 0 for all i, v. Assume to the contrary, for example, that u,, f , #0. Then [557, Cor. 2 of Prop. 21 provides nonzero subprojections u < u , f
,,

,

,

Lemma 2. If (e,) is any orthogonal family o f projections in B, there exists a nonzero central projection h such that he,=O for all but finitely many 1. Proof. Assuming to the contrary that no such h exists, one shows that for each positive integer k, there exist distinct indices i . .. , l k and nonzero subprojections g,<e,t ( t = l , ..., k ) such that g,-g,... -g, (for k=nr+ I this contradicts Lemma 1 ) ; the argument for this is the same as in [§ 14, Prop. 91. 1

,,

This leads to the analogue of [ $ I S , Prop. 51 for B:

Lemma 3. Let (e,) be any orthogonal familv qf projections in B. There exists an orthogonal family (h,) of nonzero central projections with sup h, = 1, such that for each a, the set ( 1 : h, e, # Of is finite. Proof. Let (h,) be a maximal such orthogonal family, set h, = 1 -sup h,, and assume to the contrary that h, # 0. The direct summand h, A also

Chapter 9. Matrix Rings over Bacr *-Rings

262

satisfies (I0)-(6") and is homogeneous of Type I,, and (11, A ) , = ~ I B , also satisfies ( 7 ' ) and (8");Lemma 2 applied to h,A produces a nonzero central projection h 5 11, that contradicts maximality. I

Lemma 4. With notation us in Lemma 3, sup cJ, c~xists. Proof: Formally the same as [$60, Lemma 31.

1

With Lemma 4 in hand, the argument of [§60,Th. I]shows that B is a Baer *-ring. Putting the homogeneoi~spieces back together via 1656, Th. 21, we have:

Theorem 1. I f A .satisfi.s ( 1 ")-(6") and is of' Type I , and [ f n is a positive integer such that (7")and (8")hold, then A, is a B a ~ r*-ring.

5 62.

Summary of Results

As noted in Section 55, the situation for properly infinite Baer *-rings is essentially trivial. Combining the theorems in Sections 60 and 61, we have :

Theorem 1. If A is a ,Jinite Baer *-ring satisfjing ( 1 ")-(6' ), and ( f n is a positive integer such that (7") and (8") lzold, then A, is a $nite Baer *-ring with GC. Proof. Let 11 be the central projection such that h A is Type I and h)A is Type I1 [$ 15, Th. 21. Since h A,=(h A), and (1 - h ) A , =((I - h ) A ) , are Baer *-rings by the theorems of the preceding two sections, it follows that A,= h A, + ( 1 - h) A, is also a Baer *-ring. I

(I

-

Corollary 1. If A is any AW*-~z/(jehr~~ and n is any positive integer, then A, is an AW*-al~jehra. Proof. At any rate, A, is a C*-algebra (by the Gel'fand-Naimark theorem), thus it is the Baer u-ring property that is at issue. By structure theory [$ 15, Th. I]we are reduced to the properly infinite and finite cases. The properly infinite case is disposed of at once by the observation that every A W*-algebra has GC (see, e. g., [$ 14, Cor. 1 of Prop. 71). If A is a finite AW*-algebra, then A satisfies ( I r ) - ( 6 ) and A, satisfies ( 7 ' ) and (So),as noted in Section 58; quote Theorem 1. I Noting Remark 3 in Section 55, we have:

Corollary 2. If A is a regular Baer *-ring satisfying (1")-(4'), and if n is a positive integer such that (7") and (8') lzold, then A, u a regular Baer u-ring.

5 62. Summary of Results

263

Properties of A, are hard to come by; none is too humble to be noted :

Corollary 3. If A satisfies tlze lzypoth~sesof 71zeort.m I , then partial isometries in A, are addable. Proof. If n = 1 this is the hypothesis (2 ). Supposc n >I.If zl is any central projection then h A,=(hA),, thus it is clear that A, has I no abelian direct summand. Quote [$ 20, Th. 1, (ii)]. Exercises 1C. Assume (1")-(8"). If x , y ~ C , ,and y x = l , then x y = l .

2A. Assume (1")-(8"). If A is of Type I [Type II] then A, is also of Type I [Type 111. 3C. If A=Zr, Z a commutative AW*-algebra (in other words, A is a homogeneous AW*-algebra of Type I,) then every element of A is unitarily equivalent to an upper triangular element of A. 4C. If A is a finite Baer *-ring satisfying the (EP)-axiom and thc (SR)-axiom, then C,, is strongly semisimple for all n. 5A. Let Z be a commutative AW*-algebra, write Z = C ( Y ) , F a Stonian space, fix a point a€%, and let J

=

{ c E Z :c=O on a neighborhood of o).

Then, for every positive integer n, (ZIJ), is a finite Baer *-factor of Type I,.

6C. If A is a commutative ring with unity and descending chain condition on annihilators. then the following conditions arc equivalent: (a) A,, is a Baer ring [cf. 4 4, Exer. 41 for every n 2 2; (b) A, is a Baer ring for some n 2 2; (c) A is the direct sum of finitely many Priifer rings. 7D. Assuming (1")-(8"), does A,, satisfy LP

-

RP? Does C,?

8C. If B is a regular Baer *-factor of Typc I1 satisfying LP-RP, then so is B,,, for every positive integer n. 9C. Let A be a finite Baer *-factor of Type 11, satisfying the (EP)-axiom and the (UPSR)-axiom, in which every element of the form 1+ x * x is invertible, and possessing a central element z such that z*#z. Suppose, in addition, that A satisfies the (PS)-axiom. {Thus A satisfies (1")-(6")) Let n be any positive integer. (i) A, is a finite Bacr *-factor of Type 11. (ii) A, satisfies L P -RP. Thus the conditions (7") and (8") are a consequcnce of (1")-(6') in the factorial Type 11 case.

Hints, Notes and References

DcIfinition 2. The notation is borrowed from [19, p. 1861. Definition 5. Introduced by Murray and von Neumann [67, Def. 6.1.11. In 1541 this is called '*-equivalence', the term 'equivalence' being reserved for the concept in Exercise 6. Proposition 8. The validity of this proposition is the reason for the choice of definition of equivalence. Note that the analogous proposition for unitary equivalence is false. {For example, let &? be a Hilbert space with orthonormal basis ... and let P, Q, R be the projections in 2 ( X )whose ranges are the closed linear subspaces [[I, (3, <5,...Ir 3 5 ] [ < 2 , (4, (6,...], respectively. Then P is unitarily equivalent to Q, but P + R = I is not unitarily equivalent to Q + R.} Cf. [$ 17, Exer. 17, (i)].

el,c2, c3,

Proposition 9. [54,Th. 241. Definition 8. [67, Def. 6.3.11. Theorem I . A theorem of Lebow [57]. Exercise 2. (ii), (iii) follow from (i) or from Exercise 1. Exercise 6. [54,Th. 141. Exercise 7. [54,Th. 261. Exercise 8. Cf. [54, Th. 271. This is worked out in [$ 13, Lemma to Prop. 71. Exercise 9. [52, p. 20, Lemma 21, 154, p. 24, Exer. 41. Hint: The element x = e + ( I - f') has inverse j'+(I-e). Exercise 10. [54,Th. 151. Hint: Cf. Proposition 9. Exercise 12. Used in the proof of [$ 53, Prop. I]. Exercise 14. Masas are the structural elements in the multiplicity analysis of commutative von Neumann algebras 181, Part TI, p. 41, [35, Ch. 1111. Exercises 17-19. [54, p. 39, Exer. 101. Exercise 20. Cf. 137,Solution 1771. Exercise 21. No. See [$ 12, Exer. 31.

Definition 2. The basic rcfcrence for C*-algebras is the treatisc of Dixmier [24]. S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

Hints, Notes and References

265

Exercise 3. Cf. 147, Lemma 4.91. Hint: Let QEA, set z = x x * a y , and calculate z* z. Exercise 5. [40, p. 751.

Definition 2. Cf. [74, Def. 2.11, [64, Def. 6.11 Proposition 2. Cf. 154, Th. 211. Proposition 7. [47, Lemma 5.31, 164, Lemma 6.41 Corollary of Proposition 8. [64, Lemma 6.21. Example 2. [47, p. 2491. Proposition 10. Cf.

[a,Lemma 41,

[54, Th. 201.

Exercise I. [52, p. 201, [54, p. 361. Hint: [$ 1, Exer. 91 Exercise 3. (i) For the Riesz-Schauder theory see, e. g., Banach's book [I, Ch. 10, 5 21. Exercise 6. This is worked out in [$ 51, Prop. 31. Exercise 7. Cf. [54, p. 34, Cor.]. Exercise8. C = B n S ' = ( B i u S ) ' . Exercise 9. Cf. the proof of [$ 59, Lemma 21. Exercise 11. Cf. [Sl, p. 532, proof of Th. 21, [63, Lemma 2.21, 154, p. 43, proof of Th. 281. Exercise 14. Ageneral reference on Boolean algebras is the book of Malmos [36]. Exercise 18. (ii) Cf. [ji 16, Prop. 21.

Definition 1. 1521, [54]. Definition 2. 147, Th. 2.31, [49, p. 8531. Definition 3. When 1 E B this coincides with the concept defined in [54, p. 291. Proposition 6. [54, p. 301. Proposition 7. [54, Th. 201. Definition 4. Consistent with

[a,p. 4621. (Cf. [$

7, Exer. 91.)

Definition 5. The concept is due to von Neumann [68], the terminology to Dixmier [23]. Proposition 9. [47, p. 2361. Exercise 3. (i) The last line alludes to the well-known Wedderburn theorem 142, p. 183, Th. I]. Exercise 4. (i) [54, Th. 31. Exerci.se 5. [54, p. 34, Cor.].

Hints, Notes and References

266

Exercise 9. Cf. [54, Th. 201. Exercise 10. This is

[g

6, Prop. 21.

Exercise 11. Better yet, see Excrciscs 24 and 25. Exercise 13. Cf. [7, Lemmas 2.1, 2.21 Exercise 14. [52, pp. 52-53],

154, pp. 103-1041.

Exercise 17. Cf. [54, Th. 61 Exercise 20. Cf. [54, p. 38, Exer. 81. Exercise 21. (ii) Cf. [$ 7, Exer. 91. Exercise 22. By the hypothesis (iv), there exists a Hilbert space .F, a von Neumann algebra B of operators on .f,and a *-monomorphism H: B+ 9 ( X ) with O(g)=.d. From (iii) it is clear that 0 is completely additive on projections, that is, if (E,) is any orthogonal family of projections in g,then H(sup E,)= sup O(E,) (the latter sup as calculated in Y ( X ) ) . It follows that H is normal [22, p. 16, Footnote 61, therefore .d= H(B) satisfies ..A =d"in Y(x) (cf. [22, Cor. 4 of Th. 31, [23, Ch. I, 4 4, Cor. 2 of Th. 21, [90, Lemma 4.41, [112, Lemma I]). Exercise 23. (a) implies (b): If H : A + 9 ( 2 ) is a *-monomorphism such that O(A)=O(A)" in 9 ( X ) , then 0(A) is an AW*-subalgcbra of Y(e(.X) (Proposition 8, (iv)). (b) implies (c) trivially (Exercise 21). (c) implies (d): Since H(A) is A W*-embedded in 9 ( X ) , it follows that H is completely additive on projections, i. e., if (e,) is any orthogonal family of projections in A, then H(sup e,) = sup O(e,) (the latter sup as calculated in Y(X)). Each vector < E Xdetermines a positive linear form cp: on A that is CAP, via the formula v:(x) = 510 (xEA),

.

thus the family of linear forms ((pi);, is contained in 9'.If rp<(x)= 0 for all ( E X , then O(x)=0, x = 0 . (d) implics (c): If x # 0 then x* x # 0. (e) implies (a): A recent result of G. K. Pedersen [122]. {Historical note: The implication (e) => (b) was proved by J. Feldman 126, Th. 21. Explicitly, Feldman showed, assuming (e), that there exists a Hilbert space .fl and a *-monomorphism 0: A -t Y ( 2 ) such that (i) H(1)= I, and (ii) H(sup e,) =sup O(e,) for every increasingly directed family of projections (e,) in A (the latter sup as calculated in 9(.R)). It follows at once that H is completely additive on projections (consider the family of finite subsums); in particular, if (E,) is an orthogonal family of projections in H(A), then sup E, (as calculated in Y(.#)) is also in H(A). Since O(A) is an AW*-algebra (it is *-isomorphic to A), it follows from a result of Kaplansky [cf. S; 7, Exer. 91 that TtH(A) implies RP(T)EQ(A) (RP as calculated in Y(X)). Thus O(A) is AW*-embedded in Y ( X ) . In particular, RP's in O(A) are unambiguous (they are the same whether calculated in the AW*. algebra H(A) or in Y(X))and therefore H(RP(x))=RP(H(x)) for all ~ E A In view of [S; 3, Prop. 71, it results that N(ev j')=H(e)u H(f') for any pair of projections e, j'in A, thus 0 preserves finite sups; it follows from this, and (ii), that O(sup e,)=sup O(e,) for any family of projections (e,) in A (the latter sup as calculated in Y ( X ) ) . In particular, if (E,) is any family of projections in O(A), then sup E, (as calculated in 9(X)) is also in H(A); thus H(A) is an AW*-subalgebra of Y(X).Cf. Exercise 24.1

Hints. Notes and References

267

Exercise 24. (c) implies (a): Proposition 8, (iv). (a) implies (b): Exercise 21, (i). . Exercise 23, .d is (b) implies (c): Assume .d is AW*-embedded in Y ( 2 ) By *-isomorphic to a von Ncumann algebra. thcrcfore .d=.ce" in Y ( f f )by Exercise 22. {Historical note: For A finite [$ 15, Def. 31, this result is due to H. Widom [PO, Th. 4.41 (see also J. Feldman 126, Th. I]); for ,d semifinite, to K. Sait6 178, Th. 5.21; for general d , to G. Pedersen [122].) Exercise 25. The implications (a) => (b) 3 (c) + (d) are trivial. (d) implies (e): Cf. [49, Exer. 41. (e) implies (a): Assuming (e), d is an AW*-algebra by a result of Kaplansky [cf. 4 7, Exer. 91. By hypothesis, .d is AW*-embedded in 9 ( % ) in the sense of Exercise 21. By Exercise 24, d = d" in Y ( X ) . Exercise 26. The result is due to R. V. Kadison [112, Lemma I], by a proof more direct than the following. (a) implies (b): s u p 9 is adherent to .Y in the strong operator topology. (b) implies (a): It follows from (b) and easy spectral theory that T e d implies R P ( T ) e d [112, p. 1761. If (E,),,, is any orthogonal family of projections in 4 then the sums E,, J a finite subset of I, are also in .sit, therefore sup E,e.d rtJ

by (b). It follows from Exercise 25 that .d= d" in Y ( 2 ) . Exercise 27. (a) implies (b) trivially (see Definition 3). (b) implies (a): The result is due to G. K. Pedersen 11221. Note that if e, f are projections in B, then e u f (as calculated in A) is also in B [$ 5, Prop. 71. Let (f,) be any family of projections in B and let f = sup f, (as calculated in A ) ; it is to be shown that f E B. We can suppose I # 0. Let (e,) be a maximal orthogonal family of nonzero projections in B such that el< f for all I (start with some nonzero f, and expand by Zorn's Lemma). Set e=supe,; by hypothesis, ~ E B . Obviously e l f . To prove that f = e it is enough to show that, for any x, f xe.l If, on the contrary, f,- f,e# 0, then f,u e-e=RP(f,f,e) is a nonzero subprojection of f that belongs to B and is orthogonal to every e,, contradicting maximality.

Definition I. Cf. 174, Def. 2.21. Exerci.se 3. Cf. [$ 3, Prop. 1I]. Exercise5. Hint: Consider a maximal orthogonal family of nonzero projections (e,), show that supe, is a unity element for A, and quote 194, Prop. I]. Exercise 8. Cf. [3, Lemma 3.21. Exercise 9. The desired norm on A, can be introduced via either the Gel'fandNaimark theorem 175, Th. 4.8.1 I] or the regular representation 175, Lemma 4.1.131. Exercise 15. Hint: Exercise 14.

Definition 1. Cf. [71, p. 242, Def. 1.11, [18, Lemme 3.11, 119, p. 1861, [35, 551, [SO, Def. 3.31, 154, p. 9, Def.]. Proposition 2. [47, p. 237, Cor. I], 154, Th. 221.

Hints, Notes and References

268

Proposition 3. [54, Th. 91. Corollary I . Cf. [80, Lemma 3.31, [54, Th. 131. Definition 2. The term was introduced by Halmos in the context of multiplicity theory [35, p. 931. Corollary 2. An early special case (in the context of orthoseparable AW*algebras) was proved by Rickart [74, Th. 5.181. Definition 3. Cf. 167, Def. 3.1.21, [54, p. 18, Exer. 71. Proposition 4. [54, Th. 22; p. 18, Exer. 101. Corollary 2. [SO, Th. 3.11. Definition 4. Cf. 154, p. 9, Def.]. Proposition 5 . Cf. [18, Lemme 6.61. Exercise6. Cf. [18, Lemme 3.11. Hint: In a C*-algebra with unity, every element is a linear combination of unitary elements [18, p. 2501. Exercise 7 . (iii) [18, Lemme 3.11, [80, Cor. 3.61. Exercise 8. Hint: Exercise 1. Exercise 10. Cf. [74, Lemma 5.61, 121, Lemme 3.101, 180, Lemma 5.21. Exercise 11. Same as [§ 14, Exer. 171. Exercise 12. Cf. [54, p. 18, Exer. 71. Exercise 14. Cf. [67, Ch. IV, Cor. of Th. 1111. For a matricial generalization see [Ej 16, Exer. 41. Exercise 15. Cf. [23, Ch. I, 2, Prop. 21. Exercise 17. In the case of von Neumann algebras, the result is dramatically sharper [23, Ch. I, § 2, Prop. I]. Exercise 18. Cf. [§ 56, Prop. 41. Exercise 19. Cf. [$4, Exer. 31.

Theorem 1. 147, p. 2361. Proposition 3. [47, Lemma 2.11. Definition 1. [52, p. 301, [54, p. 891. Exercise 1. See 147, Th. 2.31 and [49, p. 8531. Exercise2. Cf. [20, Th. 21, [22, p. 16, footnote (6)], [23, Ch. I, 9 4, Exer. 91, [94, p. 214, Cor.]. The word "commutative" can be omitted [g 4, Exer. 231. Exercise 3. [20, Th. 21. Exercised. Hint: Let T be a Stonian space that is not hyperstonian, and represent d = C ( T ) as operators on a Hilbert space 2;it follows from Exercises 2 and 3 that the intrinsic projection lattice operations in d must differ from the operatorial ones.

Hints, Notes and References

269

Exercises 5,6. Hint: [$ 3, Exer. 151. Exercise 8. The equivalence of (a) and (b) follows readily from Exercise 7 [cf. 4, Exer. 271.

48, Lemma 21. In view of (I), (a) and (c) are equivalent by

[a

Exercise 9. Immediate from Exercise 8. Exercise 11. [IS,Cor. 2.31.

$8 Theorem I . Cf. 174,Th. 2.101. Lemma 3. Cf. [74,Th. 2.71. Proposition 4. [47, Lemma 2.11. Exercise 1. Cf. [47,p. 2491. $9 Proposition 4 . [47, Lemma 2.11. Exercise 1. Hint: [§ 5, Exer. 51. Exercise 2. 147,Lemma 2.11. Exercise 3. [47,Lemma 2.21. Exercise 4. Essentially the same argument as for [§ 7, Exer. 91

Definition 1. [45,p. 4111. Proposition I.[47,p. 2381. Proposition 2. [47,Lemma 2.51. Exercise 1. (i) [44]. (ii) In fact, A,, = (C already provides an example. Exercise 2. [45,p. 4111. Exercise 3. (ii) Cf. [75,Lemma 4.4.61. (iii) For the case of general C*-algebras, see [45, Th. 6.41, [75, Th. 4.8.51, [24,Props. 1.3.7, 1.8.11.

Theorem 1. [47,Lemma 3.11, [54,Th. 301. Lemma 2. [52,p. 25, Lemma 41, [54,p. 39, Exer. 121. Proposition I. Cf. [52,p. 27, Lemma 71, [54,Th. 381. Exercise 1. [54,Th. 291. The proof is written out in [§ 16, Prop. 21. Exercise5. Hint: A, contains two 'orthogonal' equivalent copies of A. Cf. [§20, Prop. 21.

Hints. Notes and References

270

Proposition 1. Cf. 147, Th. 3.21, [52, p. 28, Th. 21, [54, Th. 411. This is proved for von Neumann algebras in [67, Lemma 6.1.31. Exercise 3. No. Hint: Look at matrices.

Definition I. [54, p. 811. Proposition I . Cf. 118, proof of Lemme 3.31 Proposition 2. 147, proof of Th. 5.41. Corollary. The fact that LP(T)-RP(T) in a von Neumann algebra is proved in 167, Lemma 6.2.11. The parallelogram law is proved in 167, proof of Lemma 7.3.41. Definition 2. 116, p. 3881, 117, p. 71. Proposition 3. [86, Th. I]. Proposition 5. Cf. [18, proof of Th. 61, 12, Appendix IT, Lemma 11.41 Definition 3. Same references as for Definition 2 Proposition6. Cf. 116, p. 391, No. 31. For Kickart C*-algebras, the result is due to Kaplansky [cf. 2, Appendix 11, Lemma 11.7 and p. 861. Definition 4. [52, p. 371, [54, p. 901. Lemma. Cf. [54, Th. 271. Proposition 7. Cf. [52, p. 15, Th. 2; p. 371, [54, Th. 601 Theorem I. The result is due to Kaplansky [cf. 2, Appendix I, Th. 1.191. Proposition 8. Better yet, see Exercise 5. Lemrrlu

to

Theorem 2. Cf. [54, proof of Th. 621.

Theorem 3. For von Neumann algebras, see [16, p. 3901; for AW*-algebras, see [2, Appendix 11, Th. 11.121. See also Exercise 6. Theorem 4. Better yet, see Exercise 7. Exercise 5. Cf. [54, Th. 601. Exercise 6. Hint: Revise the proof of Theorem 3 in the light of Exercise 5. Exercise 7. [54, Th. 621. Exercise 8. Hint: Revise Theorem 5 in the light oC Exercises 6 and 7. Exercise 10. Cf. the proof of [jj 53, Prop. 31.

Exr~rcise11. Cf. [a 51, Lemma].

Exrvcisc~14. t l i n l : Dropping down to (e u f ) A(e v ,f), the parallclogram law yields I -e-1'; cf. 14 20, Prop. 2) (in particular, we get addability of the partial isometries here too). Exercise 15. Consider w+w*, where w is a partial isometry implementing e - j'.

Hints, Notes and References

Exercise 18. Cf. [S; 2, Exer. 181. Exercise 19. A theorem of A. Brown [12].

Definition I . Cf. [71, p. 268, Th. 2.71. Proposition I. Cf. [71, p. 265, Th. 2.11, [59, p. 87, Satz 1.11. Dejinition 3. PC corresponds to Axiom E of [54, p. 411. Proposition 3. Cf. [74, Th. 5.21, [47, Lemma 3.31, [54, Th. 581. Proposition 4. Cf. [67, proof of Lemma 6.2.31, [18, proof of Th. 61, [47, proof of Th. 5.61, [54, p. 871. Proposition 5. Cf. [71, p. 264, Lemma 2.11, [47, Lemma 3.41, [54, Th. 351. Proposition 6. Cf. [18, proof of Th. 61. Corollary 1. [47, Th. 5.61. Comparability in von Neumann factors goes back 10 [67, Lemma 6.2.31. For generalized comparability in arbitrary von Neumann algebras, see [18, Th. 61. For generalizcd comparability in continuous gcometrics, see [71, p. 265, Th. 2.1; p. 268, Th. 2.71, [59, p. 87, Satz 1.1; p. 89, Satz 1.21. Theorem 1. Better yet, see Exercise 5. Proposition 8. Cf. [47, Lemma 6.11, [54, Th. 551. Proposition 9. Cf. 147, proof of Lemma 4.111, [54, proof of Th. 481. Exercise I. [54, pp. 43-44]. Exercise 2. This is [S;20, Th. 21. Exercise 3. Hint: Proposition 4. Exercise 4. Hint: Proposition 5, [# 20, Prop. 31, Proposition 4. Exercise 5. [54, Ths. 57, 621. Exercise 6. Hint: [# 6, Prop. 41. Exercise 8. Hint: [# 13, Prop. I]. Exercise 9. Hint: It is enough to show PC; cf. Exercise 8. Exercise 11. Cf. [71, p. 269, Dcf. 2.4],[59, p. 97, Satz 4.11, [61, p. 225, Remark]. Exercise 12. [lo, Th. 21. Hint: Exercise 5 and [# 13, Exers. 8, 151. Exercise 13. Same as [# 20, Exer. 21. Exercise 14. Hint: Exercise 2. For a simplcr proof in the finite case, cf. 159, p. 100, Satz 4.41. Exercise 15. Same as [#6, Exer. 101. Exercise 16. Same proof as for [#57, Prop. 21 Exercise 17. Cf. [47, Th. 6.6, (a)], [54, Th. 701. Hint: [fi 17, Exer. 12, (vi)].

Hints, Notes and References

272

Exercise 18. (i) Same as 156, Exer. 81. (ii) See Exercise 1 and [$3, Exer. 18, (iv)]. Exercise 19. Cf. 1141. The parenthetical result on extremal points is due to Kadison [43] (cf. [23, Ch. I, 4 1, Exer. 21). Exercise 20. If there exists a counterexample A, then equivalence must fail to be additive in A (Proposition 4), the parallelogram law must fail to hold in A (Proposition 7), and A must have a nonzero finite direct summand (Exercise 4); in particular, if there exists a counterexample, then there exists a finite counterexample. Exercise 21. If the answer is yes, then every Baer *-ring satisfying the parallelogram law has GC (Proposition 7). If the answer is no, a counterexample must fail to satisfy any of the conditions (1)-(3) of Exercise 9. Exercise 22. If the answer is yes, then PC and GC are equivalent conditions on a Baer *-ring (Proposition 7).

The principal references for structure theory ($$15-19) are 1671, [18], [19], [47] and [54]. Definition 1. [67, Def. 7.1.1]. Proposition 3. [67, Lemma 7.1.11. Dejinition 2. [51, p. 5331, [54, pp. 10, 361. For von Neumann algebras, see [18, Def. 4.41 (the term used there is 'irreductible'); for A W*-algebras, see [47, p. 2411. Proposition 6. Cf. [19, Lemme 3.31, [47, Lemma 4.71, [54, Th. 131. Proposition 8. Cf. 119, Lemme 1.11, [54, pp. 12-14]. For a general formulation of such phenomena in lattices, see [60, Def. 1.31. Theorem I . [19], [47], [51, p. 5331, [54, Ths. 10, 111. Definition 4. For factorial von Neumann algebras, the nomenclature goes back to [67, Ch. VIII]. Exercise 1. (iii) In detail, see [$ 18, Exer. 131. Exercise 2. Cf. [52, p. 61, [54, p. 37, Exer. 21 Exercise 9. [81, Part 11, Cor. 10.21, L23, Ch. I, $8, Th. 1; Ch. III,$2, No. 4, Cor. 31. Exercise 10. 160, Th. 1.11. Exercise 11. [18, Ths. 10, 141, [23, Ch. 111, $8, Th. I]. For finite A W*-algebras, the question of the existence of trace is open. Exercise 12. (i) Hint: [$17, Prop. I] and [$3, Exer. 18, ($1. (ii) Hint: [§19, Lemma I]. Exercise 13. (i), (ii) [lo]. (iii) [29]. Exercise 14. Cf. Proposition 6 and [$4, Exer. 31.

Hints, Notes and References

Proposition 2. 147, proof of Lemma 3.11, [54, Th. 291. Exercise 1. 154, p. 40, Exer. 131 Exercise 2. Cf. [42, p. 40, Prop. I]. Exercise 4 . Cf. [67, Ch. IV, Th. 1111, 123, Ch. I, $2, Prop. 71. For the case n = 1, see [$6, Exer. 141. Exercise.5. Cf. [29, p. 3331, [lo, Lemma I].

Proposition I . Cf. 167, Lemma 7.1.31, [18, Lemme 4.21, 147, Lemma 4.31, [54, Th. 371. Definition 1. [18, DCf. 3.21. Proposition 2. [47, Lemma 3.51, [52, p. 411, 154, Th. 431. Proposition 3. [47, Lemma 4.41, [52, p. 40, Th. 41, [54, Th. 441. Theorem I . Cf. [67, Lemma 7.2.31, [19, Lemme 1.31, [47, Lemma 4.51, 152, p. 41, Cor.], [54, p. 65, Exer. I]. Corollary. Cf. [54, p. 66, Exer. 21. Theorem 2. Cf. [67, Lemma 7.3.51, [19, Lemme 1.51, [47, Th. 6.21, [52, p. 43, Th. 61, [54, Th. 561. Proposition 5 . Cf. [19, Lemme 1.61, [47, Th. 5.71, [54, p. 88, Exer. 21. Exercise 2. Used in [$30, Lemma 51. Exercise4. Hint: f = x y is an idempotent, algebraically equivalent to 1 [cf. $ I, Exer. 61. Write f A = eA, e a projection [$I, Exer. 71; then e is algebraically equivalent to 1. Cf. [rj 1, Exer. 81. Exercise.5. Hint: Writing f = x y , e = 1 - R P ( l f), one has fA=R((l Cf. Exercise 4.

-f

})=eA.

Exercise6. Worked out in [$46, Prop. 1, (5)]. Cf. [23, Ch. 111, $1, Exer. 31. Exercise 7. Cf. [54, p. 71, Exer. 21. Hint: [$16, Exer. I]. Exercise 9. Hint: Drop down to (e u f )A(e u f ). Exercise 10. (i) Same proof as for [$34, Prop. I]. Cf. [47, Th. 6.31, [54, Th. 671. (ii) Cf. Exercise 9. Exercise 11. [51, p. 524, Th.]. Exercise 12. (iv) Cf. [47, Th. 6.6, (b)], [54, pp. 119-1201. (v) Hint: Proposition 4, criterion (2a). (vi) Cf. [63, Lemma 2.41. (vii) Cf. [47, Th. 6.6, (c)]. Hint: One can suppose that A is finite and e u f = I. Write e= e' + e", f =f' +f" as in [$13, Prop. 51. Setting h = e' u f ', apply (iii) to e', f ' in hAh; since e" f" by Exercise 3, part (vi) is applicable to e", f" in (1 - h)A(1- h).

-

Hints. Notes and References

274

(viii) Under hypothesis (2) or (3),cf. L54, Th. 711 (but note that the above sketch of (vii) avoids continuous geometry). Under hypothesis (4), see [47, Th. 6.6, (c)]. (ix) Hint: [jj 14, Exer. 121. (x) A theorem of Fillmore 1271. (xi) The question is asked in 154, p. 2201, [55, p. 111. Exercise 13. (1) Hint: [#14, Exer. 161 and Theorem 3. Cf. the proof of [$58, Prop. I]. (2) Hint: [S; 13, Th. I], [$8, Lemma 31 and (1). Exercise 14. Cf. 118, Lemme 6.3./3]. For the proof, cf. 1526, Prop. 51. Exercise 15. Cf. 163, Th. 2.1, (ii), (2,[)], [54, p. 87, Excr. I]. Exercise 16. Cf. [67, Lemma 7.2.11, [19, Lemme 1.81, 180, Lemma 3.51. Exercise 17. (i) Cf. [19, Lemme 1.71, 182, p. 404, Remark 1.11, [80, Lemma 3.41. Hint: Decompose according to [$IS, Exer. 71, and apply Proposition 4 and Theorem 1 (with m = 2) to the pieces. (ii) See Exercise 12, (x). Exercise 18. Cf. [80, Lemma 3.71. Hint: Expand (8) to a maximal homogeneous partition; cf. the proof of Proposition 2. Exercise 19. (i) Hint: [Cj 14, Exer. IS]. (iii) Hint: Cf. the proof of [jj18, Lemma 2 to Prop. 31. Exercise 20. (i) [lo, Th. 31. Hint: [Cj 14, Exer. 121 and Exercise 3. (ii) See [#15, Exer. 131. Exercise 21. (i) Hint: [S;14, Prop. 51, [S;20, Prop. 31. (ii) Hint: [$14, Prop. 41. Exercise 22. (i) 1281. Exercise 23. Hint: Using criterion (2 b) of Proposition 4, construct inductively an infinite sequence of orthogonal projections equivalent to e.

Definition I . Cf. 119, Def. 3.31, [ a , p. 4691, [52, p. 32, Def. I]. Proposition I. Cf. [19, Lemme 3.41, [ a , Lemma 191, [54, Th. 531. Proposition 3. Cf. [19, Lemme 3.91, [47, Lemma 4.81, [54, Th. 461. Lemmu 2. Cf. [35, $61, Th. 31, [52, p. 34, Lemma 51. Theorem I. 152, p. 34, Th. I]. Theorem 4. 1.18, Lemma 181. For the case of von Neumann algebras, see 181, Part 11, Th. 101, [19, Th. 21; in this case, all the h, are uniquely determined (see Exercise 10). Proposition 4. Cf. [47, Lemma 4.101, [54, Th. 471. In the presence of GC, thc proposition is also immediate from the dimension theory of projections 1542, Lemma 21; in particular, the von Neumann algebra case is already covered by [18, Th. 81. Proposition 5. Cf. [47, Lemma 4.111, [51, p. 5331, [54, Th. 481.

Hints, Notes and References

Proposition 6 . Cf. 152, p. 501 Exercise 2. (i) Hint: [$14, Exer. 151 or Lemma 1 to Proposition 3. (ii) Hint: Lemma 2 to Proposition 3 and 136, Cor. 3, (i) of Prop. 41. Exercise S . Cf. [54, p. 1051. Hint: Let h = C(e) and write h = h , + h 2 , where h , = h C (f ) and h, = h - h,. Note that C(h, e ) = C(lz,l)= h , and apply Propositions 1 and 6 to get h , e d h , ( l - f). On the other hand, h, f =O. Finally, e = h , c ~ + l z 2 e . Exercise 7. A theorem of Kaplansky 148, Th. 21 (for an alternative exposition, see 149, Ths. 8 and 41). Exercise 8. A theorem of Wiciom [9O, Th. 4.11. K. Saiti, has extended the result to the case that B is a semifinite A W*-subalgebra with center Z [79, Th. 5.21. Exercise 9. [49, Th. 81. Exercise 10. (i) Cf. 181, Part 11, Lemma 2.71, 148, p. 471, footnotc]. (ii) [48, Th. 41. (iii) [19, Lemma 3.51. (iv) Kaplansky has conjectured that the answer is negative [49, p. 843, footnote]. Exercise 11. Cf. 129, p. 3341. Exercise 14. (i) Cf. [44, Exer. 31. Exercise 16. (ii) Cf. [$20, Exer. 121.

Theorem I . Cf. [18, Lemma 6.21, [47, Lemma 4.121, [54, Th. 491. Exercise I . Hint: [S;26, Prop. 51. Exercise 2. (i) Hint: By structure theory, reduce to thc case that A is properly infinite [$17, Cor. of Th. I] or continuous (Corollary of Thcorem 1). (ii) Hint: [I§16, Exer. 51. § 20 Theorem I . Cf. [47, Th. 5.51, 148, Lemma 201, 151, p. 5341, L54. Ths. 54, 641. Exercise I. Same as [S; 11, Exer. 51. Exercise 2. (a) implies (b): See Theorem 3. (b) implies (c): [$13, Prop. 21. (c) implies (a): A has PC [$14, Prop. 31 hcnce also GC

[s 14, Prop. 71.

Exercise 3. Hint: [Cj 14, Prop. 51. See also 1514, Exer. 41. Exercise 4 . [54, pp. 103-1 04, 1091. Exercise 6. See [S;14, Exer. 51. Thc more general version appears in [54, Ths. 54, 64 and 631. Exercise 7 . Better yet, see 1821, Exer. 111. Exerclse 8. Cf. [62, Lemma I]. Hint: [$14, Exer. IS], [Cj17, Th. I] and Thcorem 1. Exercise 9. Cf. 134, Lemma 2.11. (i) Hint: Write e = sup en with en an orthogonal sequence such that e, - e for all n. Expand (e,) to a (necessarily countable) maximal homogeneous partition (g,).

Hints, Notes and References

276

-

Invoke [ij17, Prop. 21 to get a central projection h and a partition (1,) oflz equivalent to (Izg,). Note that he, f , and apply Proposition 3. (ii) Hint: Exhaustion, via (i) and Proposition 3. Exercise 11. (i) The question is raised in [47, p. 2491. (ii) An affirmative answer to (ii) would imply an affirmative answer to (I), as follows. Write x* x= r2, r 2 0, let C = {x*x)", let en be an orthogonal sequence of projections in C such that supe,=RP(r)=RP(x) and re, has an inverse yn in e,C [$8, Prop. 1, (4)], and let w, = xy,; continuing as in the proof of [$21, Prop. I], one would arrive at a factorization x = w r with w* w = R P(x), w w* = LP(w). (iii) An affirmative answer to (iii) would also imply an affirmative answer to (i); this is clear from the remarks for (ii) and the proof of Thcorcm 3. (iv) Even assuming GC, the prognosis for (iii) seems unfavorablc; note, for example, the lack of an analogue of structure theory (in the sense of Section 15) for Rickart C*-algebras. Exercise 12. (i) The equivalence of (a) and (b) is Theorem 2. The proof that (c) implies (b) is easy if A is properly infinite (Exercise 3) or if A is Typc I,,, [cf. ij 18, Prop. 51; the residual case of Type IIfi, is stubborn [54, Th. 521. (ii) Cf. [$11, Exer. 31.

Proposition 2. See also Exercise 1. The von Neumann algebra casc appcars in [67, Lemma 4.4.11; for the casc of A W*-algebras, see 180, Lemma 4.21, 194, Lemma 2.11. Exercise 1. [54, Th. 651. Hint: Partial isometrics are addable in A [$20, Exer. 61. Exercise 3. [54, p. 1091. Exercise 4. Hint: Observe that all elements of A , have countable spectrum, and construct an element x of A , for which the element w of Proposition 3 could only have uncountable spectrum. For example, let t , , t,,t,, ... be an orthonormal basis of X and let x be the compact operator defined by x<,=(l/n)(,+ ,. Thus x = u r , where r(,=(l/n)(, and u t n = tn+ If A , had WPD, there would exist a factorization x = ws with s in the bicommutant of r2 in A,, w in A , , and s*s=r2. Argue that w is a weighted shift w~,=p,(,+, with weights p, of absolute value 1, therefore w has uncountable spectrum (equal to the closed unit disc [cf. 37, Problem 78]), contrary to W E A , . Exercise 7. Better yet, see [ij 17, Exer. 61. Exercise 8. Cf. [14, Th. I]. Exercise 9. Cf. [14, Th. 31. Exercise 11. 1391. § 22

Proposition 1. 192, Lemma 2.11. Definition 1. [%, Def. 21, [92, Def. 2.11. Proposition 2. Cf. [92, Lemma 2.31, [3, Lemma 3.61. Definition 2. [21, Def. 3.31.

Hints, Notes and References

Theorem 1. Cf. [21, Lemme 3.91, [3, p. 5021, [63, Lemma 3.31. Theorem 2. Proved for A W*-algebras by F. B. Wright [92, Th. 2.41. Exercise 1. 164, Th. 6.41. Exercise 3. Hint: I is the left ideal generated by p (Proposition 2) and is closed [S; 8, Lemma 31; cf. Proposition 4. Exercise 4. (i) Cf. [47, proof of Th. 2.31. (ii) [23, Ch. I, # 3, Cor. 3 of Th. 21. Hint: If (e,) is any orthogonal family of projections in L, then the net of finite sums converges to supe, in each of the indicated topologies. Exercise 5 . (i) Example: A = S?(.fl), 2 infinite-dimensional, 1 the ideal of compact operators. Cf. [S; 15, Exer. 13, (i)]. (iii) For the application to A W*-algebras, see [$15, Bxer. 13, ($1. Exercise 6. Hint: Let e be a nonzero idempotent in 7; since eAe is a Ranach algebra with unity, it can have no dense proper ideal 175, Cor. 2.1.41. Exercise 7. Proved for AW*-algebras by Yen [95, Lemma I]. Exercise 8. Cf. [80, Cor. 4.21, [93, Th. 4.11. Exercise 9. (i) Hint: If g such that g 5 u(e, u e,)u*.

-

e 5 e, u e,, then there exists a unitary element u

Proposition I . Cf. [3, Th. 3.71, [63, Lemma 3.41. Proposition 2. Cf. [92, Lemma 3.41, [3, Lemma 3.101. Exercise 2. Cf. [92, Th. 3.21. The argument for (ii) is given in [$ 45, Lemma 21. Exercise 3. [3, Lemma 2.11 Exercise 4. See Proposition 5. Exercise 8. (i) Cf. [S; 7, Exer. 81. (ii) Cf. [# 22, Exer. 4, (i)]. (iii) Follows easily from (i) and (ii). Exercise 9. (ii) and (iii) are covered by Excrc~se8. (i), (iv) See 123, Ch. I, 9 4, Excr. 9 and Cor. 2 of Th. 21. (v) Consider the family of finite subsups [cf. # 3, Prop. 71.

Proposition 3. Cf. [59, Ch. V, Satz 3.11, [63, p. 861. Exercise I. Hint: [# 23, Exer. 41. Exercise 2. This is the usual form of 'weak centrality', proved for finite von Neumann algebras by Godement [32, Lemme 151, for general von Neumann algebras by Misonou [66, Th. 31 and for AW*-algebras by F. B. Wright [92, Th. 2.51. Incidentally, Dixmier's central approximation theorem [la, Th. 71, on which Misonou's proof is based, is also valid for AW*-algebras, and somewhat more generally [33, Th. I].

278

Hints. Notes and Refere~lccs

Exercise 3. (i) Cf. [75, Cor. 2.1.41. (ii) For the one-dimensionality of the center of a primitive Banach algebra with unity, see [50, Lemma 91, [75, Cor. 2.4.51. (iii) Cf. 142, p. 4, Th. I]. (iv) [24, p. 49, Th. 2.9.71, [75, Th. 4.9.61. Exercise 4. Cf. [92, Th. 2.61, [3, Lemma 1.51. Exercise 5. (iii) See [$ 23, Exer. 1, (viii)]. Exerci.se 6. (i) Cf.

[a 6, Exer. 201.

Definition 1. Dimension functions may also bc defined in certain infinite situations; without going into details, the penalty for dropping finiteness of the ring is to admit infinite-valued functions in the definition of a dimension function. For factorial von Neumann algebras, dimension goes back to Murray and von Neumann, who used it as the basis for classifying factors according to type [67, Ch. VIII]. Dimension was defined for an irreducible (i. e., 'ccnterless') continuous geometry [$ 34, Def. I]by von Neumann [71, p. 52, Th. 6.91; for any continuous geometry by Iwamura [41]; for a finite von Neumann algebra by Dixmier [18]; for a finite AW*-algebra by Kaplansky [47, pp. 247-2481; for any von Neumann algebra by Segal [82, Th. I]; for any AW*-algebra by Sasaki [80, Th. 5.11 and Feldman [106]; for any Baer *-ring satisfying the parallelogram law, by S. Maeda (163, Th. 2.11, [65, $ 101). Without attempting to cover the literature of dimension in lattices, we mention also the memoir of Loomis [58], the paper of Ramsay [73] and the survey of Holland [110]. {Warning: The term 'center' is used in a variety of ways in lattice theory.) The exposition in the present chapter (especially for Type 11) leans most heavily on the construction of dimension in [18], as transcribed for finite AW*algebras in [2, Appendix 1111.

Propositions 1-13. Cf. [18, Dkf. 4.3, Lemme 6.31 Proposition 14. Cf. [18, Lemmcs 4.10, 4.111 Proposition 16. Cf. [18, Lemme 6.41. Exercise 1. Hint: By hypothesis, there exists a simple abelian projection with (I: e ) = n ; cf. [(i 18, Cor. of Prop. I], [$ 15, Prop. 61.

Exercise 2. Same as [(i 20, Exer. 91.

Exercise 1. This is worked out in [(i 42, Lemma 21.

Hints, Notes and References

§ 30

Proposition 4. Cf. 118, Lemme 6.101. Proposition 5. Cf. [18, Lemme 6.131.

§ 33

Theorem I . See the notes for [# 25, Def. I]. Theorem 2. Cf. [59, Ch. V, p. 115, Hilfsatz 1.51. Theorem 3. Cf. [41, Th. 41, [59, Ch. V, Satz 1.61. Theorem 4. For the case of a finite von Neumann algebra, see [82, Lemma 1.11. Exercise I. This is an easy application of Theorem 2. For a direct (dimensionfree) argument, see [47, Lemma 6.41, [54, Th. 681. Exercise 4. Cf. [59, Appendix IT].

Proposition I . [47, Th. 6.31, /54, Th. 671. Definition I . The concept is due to von Neumann 1691, [71, pp. 1-2, Axioms I-V]. See also the treatise of F. Maeda [59, Ch. V, Def. 2.1; p. 237, Appendix 111 and the brief survey in Birkhoff's book [ll, Ch. XI, ## 8-1 I]. Theorem 1. Cf. [63, Th. 2.21, 154, Th. 691. Exercise 2. [51, p. 524, Th.]. Exercise 3. The result is due to U. Sasaki (see Apendix I1 of F, Maeda's book ~591).

Proposition I . Since 2 is a continuous geometry [$ 34, Th. I], this is a special case of a theorem of Kawada, Higuti and Matusima [56, Satz 41. Corollary. Cf. [54, Th. 661. Exercise 1. The result, which is immediate from the Corollary, is due to F. B. Wright [92, Th. 2.71. For finite von Neumann algebras, it was proved by Godement [32, Lemme 15, Cor.]. Exercise 2. Cf. [75, Th. 2.3.1 I].

Proposition I . Cf. [71, pp. 273-274, Ths. 2.14, 2.151, [59, Ch. IV, Satz 4.51. Proposition 2. Cf. [71, p. 274, Th. 2.161, [59, Ch. IV, Satz 4.61.

Hints, Notes and Keferences

280

§ 39

Theorem 1. Cf. [56, Satz 31, [59, Ch. V, Satz 3.11. Corollary. 163, p. 861. Exercise 2. (iii) Hint: See the proof of [ij 40, Lemma]. Exercise 3. Cf. [3, Lemma 1.21. § 40

Lemma. Cf. [3, Cor. 1.71. § 41

Theorem I. See the references for [$44, Th. I].

5 43 Theorem 1. See the references for [$44, Th. I].

Theorem I. Since the projection lattice of A is a continuous geometry [$ 34, Th. I], the present theorem is essentially an application of the reduction theory of continuous geometries. The latter is due to Iwamura 1411 (see also [56], [59, Ch. V, Satz 3.21). For the case that A is a finite AW*-algebra, see [3, Ths. 5.3, 5.71; for A a finite Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom, see [63, p. 87, Cor. I]; for A a regular Baer *-ring, see [63, p. 87, Cor. 21 (also [87, Th. 61). The key proofs in Sections 41-43 are modeled on arguments in Wright's paper on the reduction of finite AW*-algebras into finite AW*-factors [92, Ths. 4.1, 4.2, 5.11. The latter theory (as generalized by Yen [95]) is exposed in Section 45.

Lemma 2. 192, Th. 3.21. See also [S; 23, Exer. 21. Theorem 1. For a finite A W*-algebra with trace [cf. $ 15, Exer. 1 I], the theorem is due to F. B. Wright [92]; the assumption of a trace (whose existence is an open problem) was removed by Yen [95]. The exposition in this section is strongly influenced by [3]. {But the order of events here is the reverse of that in 131, where the AII theorem is deduced from the AIM theorem.) Exercise 1. Cf. [70], [81, Part I], [19, Section IV], [23, Ch. 111. Exercise 2. This is a special case of a result of Feldman 126, Th. I]. Sketch of argument: If x-xh is the center-valued trace function of A [cL 1$ 25, Exer. 111, then X E M implies X ~ E M [l8, Th. 121, therefore a positive linear form 1' on AIM may be defined by the formula f(7)=xh(o)( x t A). Since x t M iff ( ~ * xE) M ~ [32, Lemme 151, f is faithful. By uniqueness of dimension, ~ ( e ) = e(PEA), ~ therefore 2 H f @= D(e) (o)is the dimension function of AIM; in particular, f is completely additive on projections. By the proof of the cited theorem of Feldman, AIM is represented as a von Neumann algebra on the Hilbert space derived canonically from j'.

Hints, Notes and Rcfcrcnccs

28 1

Exercise 3. Strong semisimplicity is noted in [$ 36, Exer. I]. Exercise#. (i) Sketch: M is a maximal p-ideal of A ; write M=.JJ,={~EA~: : = 0) is a maximal ideal of D(e) (a)=O] for suitable EX. Then N = ( ~ E Zc(n) . f i c ~ , n ~M =~ Z ;sincc N is the Z with M = { ~ E AD: ( ~ ) E N ) Evidently closed linear span of its projections, N c M n Z (hence N = M n Z by the maximality of N). Then eEM implies D(e)€N c M n Z. (ii) See [$ 24, Exer. 4, (iv)] and (i). (iii) See [$ 24, Exer. 3, (iv)]. (iv) Cf. the proof of Lemma 3. The result is worked out in [3, Cor. 1.71. Exercise 5. Hints: (i) [$ 27, Exer. 31 and Exercise 4. (ii) [$ 8, proof of Lemma 21. (iii) Exhaustion on (ii). Cf. [$ 22, Excr. 4, (i)]. Exercise 6. Hint: Exercise 5, (iii). Exercise 7. Hint: Exercise 6. Exercise 8. Hints: (i) [jj 8, proof of Lemma 21. (ii) [jj 33, Th. 31 and Exercise 5, (i). Exercise 9. Hints: (i) [$ 24, Exer. 3, (iii)]. (ii) [$ 36, Exer. 21. (iii) [$ 36, Cor. of Prop. I]. (iv) A C*-algebra-or any *-algebra of operators on a Hilbert space--is semisimple [cf. 75, Th. 4.1.191. Exercise 10. Hints: (i) Note that J = {cEZ:RP(c) (a)=O). In the notation of the book of Gillman and Jerison [31, p. 621, J = 0,. (ii) [$ 4, Exer. 31. (iv) [jj 61, Th. I]. (v) [jj 42, Exer. 1, (i)]. (vi) Let e, be a faithful abelian projection in A . Then el Ae, is *-isomorphic with Z [$6, Cor. 2 of Prop. 41, thus J coincidcs with the ideal dcscribcd in [$42, Exer. I, (ii)]. (vii) [$ 16, Exer. 21. (viii) [54, p. 17, Exer. 31. (x) If B is a ring with radical R , then B, has radical R, [42, Ch. 1. 4 7, Th. 31. Exercise 11. Sketch: The crux of the mattcr is to provc that (a) implics (b). Assuming Z infinite-dimensional, it is to be shown that there exists a nonclosed maximal-restricted ideal I. One easily reduces to thc case of Type I or Type 11. The latter is covered by Exercise 8. Assuming A is of Type I, write A = A , @ A , @ A , @.-. with A, of Type I,,, n , < n, < n , < ... . If every Ai has finite-dimensional center, then there are infinitely many Ai and we are in the situation of Exercise 7 (cf. [jj 33, Th. 41 for orthoseparability). Otherwise, some Ai has infinite-dimensional center; dropping down, we can suppose A = Z,, with Z = C(%) infinitedimensional. Then Z is not regular (in the sense of von Neumann [$ 51, Def. I]), therefore there exists 0 t . T such that, in the notation of Exercise 10, J # N [31, P 63, (2) -=(8)1. Exercise 12. For the stone-tech compactification, see [31]. § 46 Exercise 3. Cf. [$ 7, Exer. 51, [$ 13, Exer. 21, [$ 20, Excr. 61.

Hints, Notes and References

282

Dtlfinition I . Cf. 167, Def. 16.2.11, [82, Def. 2.11, [4, Def. 1.I]. Defnition 5 . Cf. 1767, Def. 4.2.1, Lemma 16.2.31, [82, Def. 2.11, [4, Def. 2.31. Theorem I . For the case of a finite von Neumann factor, the theorem (slightly reformulated) is due to Murray and von Neumann [67, Ch. XVI, Th. XV]; their construction was subsequently generalized to arbitrary von Neumann algebras by Segal [82, Cor. 5.21. The case of a finite AW*-algebra is treatcd in [4] (scc also the paper of Roos [76, Cor. of Th. 2]), subsequently generalized to arbitrary AW*-algebras by K. SaitG [77], [78]. The present result appears in [8]. Exercise I . (ii) The only nontrivial point is that en 1 implies e n e, T e [$ 34, Prop. 21. Exercise 4. See the proof of [jj 48, Prop. 21.

Theorem 2. A considerable improvement on [8], where this was proved only under heavy additional hypotheses [8, Prop. 8.71. Exercise 3. Hint: Proposition 1. Exercise 7. See [#20, Exer. 61. Exercise8. Hint: [# 53, Lemma to Prop. 41, [$ 36, Cor. of Prop. I], [# 16, Exer. 21.

Exercise I . The main change is that in the proof of Theorem 1 one defines Exercise 2. Cf. [#48, Exer. 71.

Exercise 2. Hint: [$48, Exer. 61. Exercise 3. A counterexample may be constructed as follows. (i) Let F be a field, equipped with the identity involution. It is trivial that F is a Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom. An element u of F is positive in the sense of [jj 13, Def. 81 if and only if it is a sum of squares; we say that a has length n if it is the sum of n squares but no fewer. If F has an element of length 2 2, it cannot satisfy the (UPSR)-axiom. (ii) For every positive integer n, let F, be a field having an element a, of length 2 n. {It is a theorem of J. W. S. Cassels [99] that if t , , . .., t , are indeterminate~,then t: +...+t: is not the sum of n - I squares in the rational function

field IR (t,, ...,t,).) Let A =

n F, m

n= 1

be the complete direct product of the F, [jj 1,

Exer. 131. Then A is a commutative, regular Baer *-ring satisfying the (El')-axiom and the (SR)-axiom; since the partial isometries in A are the elements w=(w,) with w,=O, 1 or -1, it is clear that partial isometries in A are addable.

Hints. Note? and References

283

(iii) With notation as in (ii), let x=(u,). For 1?1=1,2,3,... lct 11, =(b,,,), e,,,=h,+...+h ,,,. Then e,TI, p,,,x=(u,,...,a,,,0,0.o ,...) is 2 0 for ail rn, but x is not 2 0 . Exercise 4. The problem is easily reduced to the case that rl= I. From e , < 1 it is immediate that x*e,x<x*x, thus x*x is an uppcr bound for thc family (x* r,,x). On the other hand, if y is a self-adjoint element of B such that x*r,,x < y for all p, it is to be shown that x * x < y. By Theorem 1 (generalized to welldirected families), it suffices to find a well-directed family (f,) of projections with . f , t I and j,(y-x*x) f ; 2 0 for all p. Set f , = x '(e,,). Then f,T 1 [cL 9 47, Lemma 51; moreover, (I -e,)x fi,=O, that is, x f;,=opxf;,. thus f,x*.u f ; = .fix* el,X) .f, .f,,y f;.

Regrettably, the numbering of the cond~t~ons (3 ). (4 ). etc dcv~atcsfrom that In r8i Lemma. Note that the (SR)-axiom may bc substituted for the (UPSR)-axiom here. Dcyinition I. The concept of regularity is due to von Neumann [cf. 71. p. 70, Def. 2.2; p. 114, Th. 4.51. Proposition 3. Cf. [71, p. 114, Th. 4.51. Corolluries 1, 2. The possibility of combining sequences of clcmcnts along an orthogonal sequence of projections in a regular Raer *-ring is a themc in [SI, Section 31. Corollary 3. In particular, if A is regular and satisfies (I), then A is the dlrect sum of finitely many factors of Type I (Exercise 11, (i)). A more general result of Vidav [89] : If A is a regular Baer *-ring in which

n

1x: xi = 0 implies x,

(n any positive intcgcr), and if A satisfies (I), thcn A is of Type I Exercise 1. Cf. [54, p. 130, Th. A]. Exercise 5. See [71, p. 70, Th. 2.21 Exercise 6. Hint: Exerci.ce 8. Hint: Exercise Y. Hint:

[a 5, Prop. 31. [a 23, Exer. I] 15 50,

Lemma].

Exercise 10. Cf. [7 1. Exercise 11. (i) See the notes for Corollary 3. Exrrcise 12. Sce [71, p. 81, Th. 2.131, 1113, esp. p. 1671 Exercise 14. Hint: Exercise 7. Exercise 15. See [72, Th. 21. Exercise 16. Cf. [/3 50, Exer. 21. Exercise 17. Theorems of Kaplansky [SI, Ths. 2, 31. Exercise 18. Cf. [89, Lemma 61. Exercise 19. Cf. [ji 49, Exer. 21.

=

.. . = t-,, = O

284

Hints, Notes and References

Theorem I . For A a finite von Neumann factor, the theorem (slightly reformulated) is due to von Neumann [cf. 71, p. 89, (VI)]; the case of a finite von Neumann algebra is covered by the construction of Segal [82, Cor. 5.21 (cf. [105, Th. 21). The case of a finite A W*-algebra is treated in [4]; the present result appears in [8]. Roos [76] has observed that any Baer ring A may be enlarged to a regular ring R--the maximal ring of right quotients of A in the sensc of Uturni [88]; when A is a finite Baer *-ring satisfying the (EP)-axiom and the (SR)-axiom, R coincides with the regular Baer *-ring that accrues to the projection geometry of A (which is a continuous geometry; cf. the remarks at the end of Section 46) [76, Th. 21; R coincides with C when A is a finite AW*-algebra 176, Cor. of Th. 21, but in general the connection between R and C (for the various extensions of [$47, Th. I]) remains to be explored. Theorem 2. Proved in [9]. For the case that A is a finite AW*-algebra, see 16, p. 177, Lemma]. Cf. the discussion in 151, Section 31. Exercise 4. Cf. [$ 51, Exer. 191. Exercise 5. Hint: Partial isometries in A arc addable [$ 20, Exer. 61; see [$49, Exer. 21, [$ 51, Exer. I] and Exercise 4. Exercise 7. Hint: [$ 51, Th. I] (as generalized in [$ 51, Excr. 191); see the proof of Theorem 2.

-

ExerciseH. (i) Sketch: A also satisfies (2") [$ 20, Exer. 61, thus C is regular by Exercise 4. Since C satisfies LP R P [$48, Exer. 71 and has no abelian central projections, C, is *-regular for all n [$ 51, Exer. 151. {In effect, [$ 51, Exer. 151 is a substitute for [§ 50, Prop. 11-at the cost of excluding abelian summands.) Incidentally, if, in place of (2"'), A is assumed only to satisfy (2") then at lcast C, is *-regular. {Proof: The relations x , y s A, x* x+ y* y = 0 imply x = y = O [$ 51, proof of Lemma], therefore the relations x, ~ E C x*x+y*y=0 , imply x = y = 0 [$ 50, proof of Prop. I]. Since C is regular (Exercise 4), so is C, [#51, Exer. 131.)

Theorem 1. [8, Prop. 8.61. There is an intriguing analogue for the maximal ring of right quotients [88, l'h. 41 (cf. the notes for [# 52, Th. I]). Exercise I . [8, Lemma 8.101. Exercise 2. [8, Prop. 8.121. Exercise 3. (i), (ii) [8, Prop. 8.131. (iii) [8, Prop. 9.31. Exercise 5 . Hint: Exercise 4. Exercise 6 . Same proof as for the casc that A is a finite AW*-algebra [6, Th. 81.

Definition I . The concept appears in an unpublished manuscript of von Neumann 153, L4]. See also 13, p. 5011. For a discussion of boundedness in *-regular rings, see [89].

Hints, Notes and References

Proposition 1. Cf. [3, Lemma 3.111, 1891 Theorem 1. [8, Cor. 9.51. Theorcw 2. An analogous result is proved in 13, Th. 3.121 Tor certain Rickart *-rings. Exercise I. Hint: [# 51, Exer. 11, (i)]. Exercise 2. (i) Cf. La 51, Exer. 181. (ii) Cf. 13 53, Exer. 31. Exercise 4. (i) Hint: [ij 52, Exer. 61, [# 51, Cor. 2 of Th. I]. (ii) Hint: [$ 53, Th. 1, (i)]. Exercise 6. (ii) Hint: Excrcise 4, (ii). (iv) For the case that A is a finite AW*-algebra, see [3, Th. 4.21.

Exercise 2. Hint: [# 8, Lcmma 21. Exercise 5. Same as [ij 11, Exer. 51. Exerci.se6. No. Sketch of example: Let A be the *-ring of all ultimately real 5 54, Exer. 91. Since the relations x,, ...,x,EA, sequences of complex numbers 1 51, xTx, +...+x,*x,=O imply .x, =-..=x,=O, A, is a Rickart *-ring for all n Exer. 131. But partial isometrics in A are not addable [cf. 3 20, Excr. 41. thus A,, is not a Baer *-ring for any n 2 2 (Exercise 5).

[a

Exercise I. (ii) Cf. [5, p. 441, [91, Th. 91. (iii) Cf. 154, p. 38, Exer. 81, [46, p. 31. Exercise 2. Hint: [# 51, Exer. 9, 181 and Proposition 3 Exercise 3. See [# 51, Prop. 41. Exercise 5. (a) implies (b): See the proof of Proposition 3. The implications (b)=.(c)3 (d)* (e) are trivial. (e) implies (a): Suppose XEC, x * x < l . Write x = w r with X E A and r>() [# 53, Prop. 51. Since 01 r < I 1953, Exer. 3, (i)], one can form s = (r(1-r))2 [453, Def. 11; the matrix

(T

is a projection in C, [cf. 37. Solution 177, p. 3271, therefore YEAby hypothesis.

Proposition 2. Cf. [5, Lemma 3.31. Exercise I. Hint: [# 54, Exer. 4, ($1. Exercise 2. Hint:

54, Th. I].

Exercise 3. Condition (C) is discussed in a papcr of Prijatelj and Vidav 172, esp. Th. I]. (iii) Hint: [S; 51, Exer. 63, [5; 21, Prop. 31.

286

Flints. Notes and References

Proposition I. Cf. [S, Lemma 4.11. Exercise 1. Hint: [$ 17, Exes. 61. Exercise 2. Sketch: By a theorem of Kaplansky 151, Th. 21, every regular Baer *-ringpin particular B-has the desired property (yx= 1 implies xy = 1) and is therefore finite. Let I be a maximal ideal of B. Since I is strict [$St, Exer. 81, by reduction theory B/I is a regular Baer *-factor satisfying LP-RP [544, Th. I]. (BII),. If B/I is of Type 11, then (BJI),, is also a regular Baer Note that BJI, *-factor of Type I1 [$62, Exer. 81, therefore BJI, has the desired property by Kaplansky's theorem. If B/I is of Type I,, then (BJI), is the ring of linear mappings on an nr-dimensional vector space over a division ring ([$ 18. Exer. 171, [$56, Exer. 1, (ii)]), therefore BJI, has the desired property by linear algebra. Since the (0) [$36, Prop. I]. it follows that B, has intersection of the I -hence the I,-is the desired property too. (Incidentally, if B has no abelian summand, then B, is *-regular [$51, Exer. 151.)

-

Exercise 3. Since C is a regular Bacr +-ring satislying LP R P [$48, Exer. 71, Exercise 2 is applicable. (Incidentally, C, is *-regular when A has no abelian summand or when A satisfies (1')) § 60

L e m m 3. Cf. [5, Lemma 5.41. Lenzrna 4. Cf. [S, Lemma 5.51.

Theorem I. The AW* case, as proved in [S, p. 371, was based on the theory of A W*-modules [49].

Corollary 1. This is proved in [S]. Exercise 1. See [$51, Exer. 171. Better yet, see L$58. Exer. 31 Exercise 3. A theorem of Deckard and Pearcy [IS,Th. 21 Exercise 4. See [$ 48, Exer. 7, 81 Exercise 5. See [§ 45, Exer. 101. Exercise 6 . A theorem of Yohe [96]. Exercise 8. For the proof that B, is a regular Baer +-factor, view the results of von Neumann [71, p. 230, Th. 17.4; p. 236, Lemma 18.61, Kaplansky [SI, Th. 31, and Halperin [135, Th. I ] in the light of [$51, Exer. IS]. The fact that B, also satisfies L P - R P is a recent result of J.L. Burke [136, Th. I]. Exercise 9. Cf. [$52, Exer. 51, Exercise 8, [$56, Prop. 31, and [$17, Excr. 171. A substitute for the (PS)-axiom is criterion (a) of [$ 56. Exer. 51.

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Index abelian projection 90,95, 113 *-ring 90. 95 addabiliFy ofpartial isometries 55, 129, 131,263 N-addability 55 additivity of equivalence, complete 55, 122,129,130,183 finite 6, 264 orthogonal 55 N-additivity of equivalence 55 adjunction of a unity element 11, 30, 31 *-algebra 3, 9, 101, 121 algebraically equivalent idempotents 9 projections 9, 18, 66, 230, 232, 253 ALP 28 annihilating left (right) projection 28 annihilator 12 ARP 28 A W*-algebra 21,24,25,43,262 commutative 40,44, 209,263 equivalence in 97 gencrated by projections 97 real 26 A W*-embedded 26,27 AM/*-factor 36, 206, 209 AW*-subalgebra 23,25,27,44 Baer C*-algebra 21 *-factor 36, 201 ring 25 *-ring 20 *-subring 22,27,44, 145 B;-algebra 13 bicommutant 16, 19 Boolean algebra 19 ring 19 bounded element 243 subring 243

(C) 256 C 216,218

C*-algebra 11 Baer 21 Rickart 13 CAP 27 Cartesian decomposition 227 category of Baer *-rings 145 Cayley transform 228, 231, 234 center 17,23 central additivity 132 central cover 34,252 central projection 17, 18 clopen set 40 closed operator 216 CO 216 coarse structure theory 87, 93-96 commutant 16, 19 commutative A W*-algebra 40,44, 209,263 C*-algebra 11 projection 95 Rickart C*-algebra 44 weakly Rickart C*-algebra 48 compact operators 15, 135 comparability, generalized 77 comparable projections 77 complement 39,108 complementary projections 39, 108 complementation 185,211,212 complete additivity, of equivalence 55, 122,129,130,183 of dimension 163,181 complete Boolean algebra 19 complete direct product of *-rings 9 complete lattice 7 of projections 20,21 complete *-regular ring 25 completely additive on projections 27, 43 continuous geometry 185,232 continuous *-ring 92 C'*-sum 52 C ( T ) ,C , ( T ) 11

S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

D(e) 153, 181 diagonal operators 209 dimension function 153, 181 direct product, complete, of *-rings 9 of Baer *-rings 25 of regular Baer *-rings 237 of Rickart *-rings 19 direct summand 18,25 discrete 92, 93 divisibility of projections 119, 155 domination of projections 6

involution 3 proper 10 involutive ring 3

(EP)-axiom 43,241 equivalence of idempotents, algebraic 9 equivalence of projcctions 4, 6, 109,110 in an A W*-algebra 97 in a von Neumann algebra 109,110 *-equivalence of projections 264 equivalent partitions 102 exchange by a symmetry 69 extremal partial isometry 83, 135 extremal point, of unit ball 83 extremally disconnected space 40

masa 9,43 matrix rings 97, 248, 253, 262 maximal abclian sclf-adjoint subalgebra 9 maximal-restricted ideal 146, 186, 192,201,208,209,247 maximal rings of quotients 284 maximal-strict ideal 187, 202 minimal projection 25, 96, 97, 118, 1I 9 modular lattice 108, 184, 185, 211 morphisms 145 multiplicity sequcncc 190

factor 36 factorial ideal 144, 146 p-ideal 144, 149 *-ring 36 faithful projection 38, 96 final projection 5 fine structure theory 87 finite projection 89, 101, 104, 106, 107 *-ring 89 fixed-point lemma 7 fundamental projection 158, 159 GC 77,130,132,181 orthogonal 79 Gcl'fand-Naimark theorem 11, 251, 262 generalized comparability 77, 130, 132, 181

homogeneous Baer *-ring 111 homogencous partition 102 hyperstonla11 spacc 44 ideal, restricted 138, 142 factorial 144 strict 141, 144, 149, 232 idempotent 3, 8,9, 11, 18. 96 infinite *-ring 89 initial projection 5

lattice of project~ons14, 29 complete 20, 21 left-annihilator 12 left projection 13, 28 locally orthoseparablc 118 L P 13,28 LP-RP 131,136,186,220,232,263

normal element 9 normal measure 44 operator with closurc 214 operators with separable range 3 6 order, of a ho~nogencous*-ring 111 of a simple projection 157 ordering, of projections 4 of self-adjoints 224, 245 orthogonal addability of partial isometrics 55 orthogonal additivity of cquivalcnce 55 orthogonal GC 79 orthogonal prqjections 4 very 36,77 orthoseparable 118, 131, 182 OWC 214 (PI 62 p, position 65 pi, position 64, 77 parallelogram law 62 partial isometry 5, 10, 55. 129, 223. 250 extremal 83, 135 partially comparable 78, 256 partition 102 homogeneous 102 maximal homogeneous 102

PC 78,130,132 PD 134 perspective projections 108-1 10 p-ideal 137, 138, 247 factorial 144, 149 maximal 148, 192 strict 141, 149 polar decomposition 134 position p 65 position p' 64, 77 positive element in a *-ring 70, 224, 242 positivc square roots 70,240 primitive ideal 149, 208 projection 3 abelian 90, 95 central 17 finite 89, 101, 104, 106, 107 proper involution 10 properly infinite 91,103, 110 properly nonabelian 91 Priifer ring 209, 263 (PS)-axiom 244 (PSR)-axiom 70 P*-sum 9, 19, 25, 54, 59 purely infinite 92 purely real *-ring 231 quotient rings 142, 186, 201, 246 quotients, rings of 284 real at infinity, ring of sequences 26, 131,135,249 real A W*-algcbra 26, 249 reduced ring 19,97, 101, 121, 141 reduction, of finite Baer *-rings 186, 20 1 of finite AW*-algebras 206 of finite von Neumann algebras 208 of von Ncumann algebras 208 regular Baer *-ring 25, 21 1, 230, 232, 235,241,257, 262,263 regular ring 229,235, 241 *-regular ring 229 complete 25 restricted ideal 138, 142, 246, 247 restricted-simple 187, 192, 202 Rickart C*-algcbra 13 commutative 44 Rickart ring 18 Rickart *-ring 12 right-annihilator 12 right projection 13, 28

ring 3 *-ring 3 R P 13,28 Schriider-Bernstein theorem 7, 59, 60 failure of 62 SDD 213 self-adjoint element 3 semifinite 91 von Neumann algebra 97 semisimple, strongly 186, 208, 222, 263 separable range. operators with 16 similarity 9, 18 simple projection 157, 257 square roots 66,70,240 (SIC)-axiom 66, 131 (SSR)-a-xiom 76 Stone-Cech compactification 209 Stone representation space 19, 148. 149 Stonian space 40,44 strict ideal 141, 144, 149, 232 p-ideal 141,149 strong semisimplicity 186, 208, 222, 263 strongly dense domain 21 3 structure theory 87 *-subring, Baer 22, 27,44, 145 *-subset 9, 17 summand, direct 18 sums of squares 282 symmetric *-ring 9, 18, 25, 108, 227, 232,253 symmetry 19, 56, 69, 110 T(e) 157 trace 97,280 Type 1 93,95 A W*-algebra 88,118, 119 Baer *-ring 110, 199,201 von Neumann algebra 88,97, 118 Type If,, 94, 170, 199 Type I,,, 94 Type 1, 112,116,119,160,170,197, 209,261,263 factor 119, 197, 199, 201,209,232, 263 Type I, 116, 119 Type I1 93 Type 11, 182 factor 195,199,201,238,263 Type IIf," 94

296

Index

Type II,,, 94 Type 111 93, 165 ubiquitous set of projections 18 ultimately real sequences, ring of 247, 285 unitary equivalence of projections 109, 264 unitification 30, 31 unity element, adjunction of 11, 30, 31 (UPSR)-axiom 70,239 (US)-axiom 233 very orthogonal 36, 77 von Neumann algebra 24,100,109, 110, 118, 141, 145 commutative 43,209 embedding in 25, 27, 145 finite 97, 208,211, 242

von Neumann algebra generated by project~ons97 projections in position p' 77 reduction of 208 semifinite 97 Type 1 88,97,118,119 (VWEP)-axiom 43 (VWSR)-axiom 254 weak centrality 147, 149 weakly Rickart C*-algebra 45 commutative 48 weakly Rickart *-ring 28 weiehted shift 276

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete Eine Auswahl ByrdJFriedman: Handbook of Elliptic Integrals for Engineers and Scientists Aumann: Reelle Funktionen Boerner: Darstcllungen von Gruppcn Tricomi: Vorlesungen iiber Orthogonalreihen BehnkeISommer: Theorie der. analytischen Funktionen einer komplcxen Verandcrlichen 78. Lorenzen: Einfiihrung in die operative L.ogik und Mathematik 86. Richter: Wahrschcinlichkeitstheorie 87. van der Waerden: Mathematische Statistik 94. Funk: Variationsrechnung und ihre Anwendung in Physik und Technik 97. Greub: Linear Algebra 99. Cassels: An Introduction to the Geometry of Numbers 104. Chung: Markov Chains with Stationary Transition Probabilities 107. Kothe: Topologische lineare Raume 114. MacLane: Homology 116. Hormander: Linear Partial Differential Operators 117. O'Meara: Introduction to Quadratic Forms 120. Collatz: Funktionalanalysis und numerische Mathematik 121./122. Dynkin: Markov Processes 123. Yosida: Functional Analysis 124. Morgenstern: Einfiihrung in die Wahrscheinlichkeirsrcchnung und mathematische Statistik 125. ItBIMcKean jr.: Diffusion Processes and their sample Paths 126. LehtoIVirtanen: Quasikonforme Abbildungen 127. Hermcs: Enumerability, Dccidability, Computability 128. Braun/Koecher: Jordan-Algebren 129. Nik0dj.m: The Mathematical Apparatus for Quantum Theorics 130. Morrey jr.: Multiple Integrals in the Calculus of Variations 131. Hirzebruch: Topological Methods in Algebraic Geometry 132. Kato: Perturbation Theory for Linear Operators 133. Haupt/Kiinneth: Geometrischc Ordnungen 134. Huppert: Endliche Gruppen I 135. Handbook for Automatic Computation. Vol. IIPart a : Rutishauser: Description of ALGOL60 136. Greub: Multilinear Algcbra 137. Handbook for Automatic Computation. Vol. l/Part b: Grau/Hill/Langmaack: Translation of ALGOL60 138. Hahn: Stability of Motion 139. Mathematische Hilfsmittel des Ingenieurs. 1. Teil 140. Mathematischc Hilfsmittel des Ingenieurs. 2. Teil 141. Mathematische Hilfsmittel des Ingenieurs. 3. Teil 142. Mathematischc Hilfsmittel dcs lngenieurs. 4. Teil 143. Schur/Grunsky: Vorlesungen iiber Invariantentheorie 144. Weil: Basic Number Theory 145. Butzer/Berens: Semi-Groups of Operators and Approximation 67. 68. 74. 76. 77.

'l'reves: 1.ocally Convex Spaccs and Linear Partial DilTerential bquations Lamotke: Semisimpli7iale algebraischc Topologic. Chandrasekharan: Introduction to Analytic Number Theory SariojOikawa: Capacity Functions losifcscu/Theodorcscu: Random Processes and Learning Mandl: Analytical Treatment of One-dimensional Markov Processes Hewitt/Ross: Abstract Harmonic Analysis. Vol. 2: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups Federer: Geometric Measure Thcory Singcr: Bases in Banach Spaces I Miiller: Foundations of thc Mathematical Theory of 1:lectromagnetic Wavcs van der Wacrden: Mathematical Statistics Prohorov/Rozanov: lJrobability Theory KBthc: Topological Vcctor Spaces 1 Agrest/Maksimov: Thcory of lncompletc Cylindrical Functions and Thcir Applications BhatiaJS7cgii: Stability Thcory of Dyna~nicalSystems Nevanlinna: Analytic Functions Stoer/Witzgall: Convexity and Optimizatioll in Finite Dimensions I SariojNakai: Classification Theory of liicmann Surfaces Mitrinovii.,'Vasii.: Analytic Ii,cqualitics GrothcndicckjDieudonnC: Elemcnts dc Gt-omctric Algi-brique I C'handrasekharan: Arithmetical Functions P;~lamodov:Lincar Diffcrcntial Operators with Constant Coefficients Lions: Optimal Control Systems Govcrncd by Partial Differential Equations Singcr: Best Approximation in Normed Linear Spaces by tlements of Linear Subspaces Biihlmann: Mathematical Methods in Risk Thcory T;: Maeda/S. Maeda: Theovy of Symmetric Lattices StiefeIJScheifele: Linear and Regular Celestial Mechanics. Perturbed Two-body Motion-Numerical Mcthods-Canonical Thcory Larscn: An Introduction of the Theory of Multipliers GrauertjRemmert: Analytischc Stellenalgebren Fliiggc: Practical Quantum Mechanics I Fliiggc: Practical Quantum Mechanics 11 Giraud: Cohomologie non abelienne Landkoff: Foundations of Modern Potential Theory LionsIMagenes: Non-13omogeneous Boundary Value Problems and Applications I LionsjMagencs: Non-Homogeneous Boundary Value Problems and Applications I 1 LionsJMagencs: Non-Homogeneous Boundary Value Problems and Applications 111. In preparation Koscnblatt: Markov Proccsscs. Structure and Asymptotic Bchavior Kubinowicz: Sornmcrfeldsche Polynommctl~odc Wilkinson/Rcinsch: Handbook for Automatic Computation 11, Linear Algebra SiegeljMoser: Lectures on Celestial Mechanics Warner: Harmonic Analysis on Semi-Simple Lie Groups 1 Warner: Harmonic: Analysis on Semi-Simple Lie Groups 11 Faith: Algcbra: Rings, Modules, and Catcgorics I. In prcyaration Faith: Algebra: Rings, Modules, and Categories 11. In preparation Maltsev: Algebraic Systems. 111 preparation I'olya/Szcgo: lJroblems and l'hcorcms in Analysis. Vol. I. In preparation lgusa: Theta Functions

Errata and Comments for Baer ∗-Rings Errata p. 36, . 17. For “import” read “important”. p. 42, . 15. Read “For example:” p. 100, . −21. In (i) of Exer. 3, read B instead of A . p. 109, . −2. For GC read (P). p. 119, . −20. In Exer. 14, read “involutory automorphism” in place of “automorphism”. p. 141, . 8. In (a) of Exer. 1, read “A strict ideal of A is. . .” (the initial capital letter should not be italicized). p. 160, . −12. In (D3) of Prop. 1, for D(h) read D(h) = h . p. 242, . −3 to −1. Exer. 6A should have been placed in the next section, where the additional assumption 6◦ ensures that C has the property x*x ≤ 1 ⇒ x ∈ A (§54, Th. 1); granted this property, the proof given in [6, Th. 8] for finite AW*-algebras can be adapted to the present situation [the author, “Note on a theorem of Fuglede and Putnam”, Proc. Amer. Math. Soc. 10 (1959), 175–182; the material between Th. 6 and Th. 8 is irrelevant here]. In §53, the exercise is an open question (the answer is not known to me in 2009); it should have been phrased as a question—“Do the relations . . . ?”— and it should have been labeled 6D instead of 6A. p. 274, . 13. Assuming GC has been replaced by (P) in . −2 of p. 109, Theorem 3 should be added to Proposition 1 and Theorem 1 in the Hint. Note that (P) ⇒ GC (referenced in the comments below). p. 285, . −12. In the hint for “(e) implies (a)” of §56, Exer. 5, read w ∈ A in place of x ∈ A . p. 286, . −2. In the hint for §62, Exer. 9, in place of [§17, Exer. 17] read [§51, Exer. 17]. Comments p. 75, Ths. 3, 4, 5. It suffices that A be a Rickart ∗-ring satisfying (SR) [S. Maeda, “On ∗-rings satisfying the square root axiom”, Proc. Amer. Math. S.K. Berberian, Baer ∗-Rings, Grundlehren der mathematischen Wissenschaften 195, c Springer-Verlag Berlin Heidelberg 2011 

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Soc. 52 (1975), 188–190; MR 51#8158]; cf. Th. 12.13 on pp. 56–57 of [B&B*]1 p. 76, Exer. 16. See the preceding comment. p. 76, Exer. 17. In a Rickart ∗-ring, the following conditions are equivalent: (a) every pair of projections in position p  can be exchanged by a symmetry; (b) for every pair of projections e and f , u(ef )u = f e for a suitable symmetry u of the form u = 2g − 1 with g a projection [S. Maeda, op. cit.]. p. 80, Prop. 7, its Cor. 2, and Th. 1. By a theorem of S. Maeda, every Baer ∗-ring satisfying the parallelogram law (P) also satisfies GC [S. Maeda and S.S. Holland, Jr., “Equivalence of projections in Baer ∗-rings”, J. Algebra 39 (1976), 150–159; MR 53#8121]; cf. Cor. 13.10 on p. 61 of [B&B*]. Thus, in any proposition about a Baer ∗-ring that assumes (P) and GC, the assumption of GC is redundant (in particular, Maeda’s theorem vaporizes Prop. 7). The examples noted below are not exhaustive. p. 82, Exer. 5 and p. 83, Exer. 12. It suffices that A satisfy (SR), since (SR) ⇒ (P) ⇒ GC [Maeda and Holland, op. cit.]; cf. [B&B*], Th. 12.13 and Cor. 13.10. p. 83, Exer. 17. The conditions (a), (b), (c) are equivalent in every Rickart ∗-ring; i.e., the assumption of orthogonal GC can be omitted [S. Maeda, letter to the author, October 8, 1974]. p. 83, Exer. 21. Yes; in fact, (P) ⇒ GC in a Baer ∗-ring (referenced in the comment for p. 80). p. 104, Th. 2. (P) ⇒ GC. p. 106, Prop. 5. Since (P) ⇒ GC, the hypothesis (P) suffices. p. 109, Exer. 12, (xi). Yes; in fact, the answer is yes for any Baer ∗-ring satisfying (SR) [Maeda and Holland, op. cit.]. p. 109, Exer. 15. (P) ⇒ GC. p. 110, Exer. 18. (P) ⇒ GC. p. 111, Remark 4. (P) ⇒ GC. p. 115, Th. 3. (P) ⇒ GC ⇒ PC. p. 117, Prop. 6. (P) ⇒ GC. p. 132, Exer. 11, (i) and (iii). The answers are “yes” for A a finite Rickart C*-algebra [D. Handelman, “Finite Rickart C*-algebras and their properties”, Studies in analysis, pp. 171–196, Adv. in Math. Suppl. Stud., 4, Academic Press, 1979; MR 81a:46073]. Baer and Baer ∗-rings (briefly, B&B*), a 1992 update of Baer ∗-rings, posted (as baerings.pdf) on the University of Texas web site for archiving mathematical publications (www.ma.utexas.edu/mp− arc) as item 03-179 in the folder for 2003.

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p. 142, Exer. 10. Yes, if the algebra is finite (see the comment for p. 132, Exer. 11). p. 144, . −13. In (viii) of Exer. 1, the notion of GC must be extended to accomodate ‘formal projections’ 1 − e when A has no unity element. p. 185, Th. 1. (P) ⇒ GC. p. 206, Th. 1. More generally, D. Handelman has shown that if A is a finite Rickart C*-algebra and M is a maximal ideal of A , then A/M is a finite AW*-factor (referenced in the comment for p. 132, Exer. 11). p. 208, Exer. 3. More generally, every finite Rickart C*-algebra is strongly semisimple [D. Handelman, D. Higgs and J. Lawrence, “Directed abelian groups, countably continuous rings, and Rickart C*-algebras”, J. London. Math. Soc. (2) 21 (1980), 193–202; MR 81g:46100]. p. 253, Exer. 2. For A a complex algebra with an involution (but no C in the picture), there is a far-reaching generalization by J. Wichmann [Proc. Amer. Math. Soc. 54 (1976), 237–240; MR 52#8947]. S.K. Berberian 27 August 2009