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 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?( 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 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, 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;',+ 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 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 .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 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,) (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 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 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,)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, () [# 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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Supplementary Bibliography 1971 Akemann, C.A.: Le$ ideal structure of' C*-algebras. J. Functional Analysis 6, 305-317 (1970). [98] Rirkhoff, G.: What can lattices do jbr vou?. Published in Trendy in lattice theory (Abbott, J.C., Editor), pp. 1-40. New York: Van Nostrand Reinhold 1970. [99] Cassels, J. W. S.: 011the representation qf'rrrtionulfunctions as sunzs qfsquarrs. Acta Arith. 9, 79-82 (1964). [I001 Clark, W.E.: Baer rings which arise ,fiorn certain transitive gruplzs. Duke Math. .I. 33, 647-656 (1966). [I011 Davies, E. R.: The structure qf'C*-ulgehra,s. Quart. .I.Math. Oxford Ser. (2) 20, 351-366 (1969). [I021 Dye, H.A. : On the geometry of projections in certain operator algebras. Ann. of Math. (2) 61, 73-89 (1955). [I031 Dianscitov. K. K.: Some properties of' special *-regular rings (liussian). Izv. Akad. Nauk SSSR Ser. Math. 27, 279-286 (1963). [I041 Feldman, J.: Non.separability of cerlain finite factors. Proc. Amcr. Math. SOC.7, 23-26 (1 956). [I051 - Isomorphi,sms of ftnite tjpr 11 rings ( f operators. Ann. of Math. (2) 63, 565-571 (1956). [I061 -- Some connections between topological and algebraic properties in rincgs of qerutor.~.Duke Math. .I.23, 365-370 (3956). [I071 - Fell, J.M.G.: Separable representation.^ of' rings of' operators. Ann. of Math. (2) 65, 241 -249 (1 957). [I081 Fell, J.M.G.: Representations of weakly closed u1~jehra.s.Math. Ann. 133, 118-126 (1957). [I091 Halpcrn, H.: The maximal GCR ideal in an A W*-alcgebra.Proc. Amer. Math. Soc. 17,906-9 14 (1966). [I101 Holland, S.S., Jr.: The current interest in orthonzodular httices. Published in Trends in luttice theory (Abbott, J.C., Editor), pp. 41-126. New York: Van Nostrand Reinhold 1970. [Ill]Johnson, B.E.: A W*-algehra,~(Ire QW*-a1gehva.s. Pacific J. Math. 23, 97-99 (1967). [I 121 Kadison, R.V.: Operator algebras with a,faitl?ful weakly-closed representation. Ann. of Math. (2) 64, 175-1 81 (1956). [I131 - Pedersen, G.K.: Equivalence in operator algebras. Math. Scand. 27, 205-222 (1970). [I141 Kehlet, E.T.: On the rnonofone sequrntiul closure ( ? f a C * - ~ / / g ~ h rMath. ~r. Scand. 25, 59-70 (1969). [I151 Maeda, F., Maeda, S.: Theory of symmetric lattices. Berlin-HeidelbcrgNew York: Springer 1970. [I161 Maeda, S.: On relativelj~senzi-ortlzoconzplenzentt~(1 1uttice.s. J. Sci. tliroshima Univ. Ser. A Math. 24, 155-1 61 (1960). S.K. Berberian, Baer *-Rings, Grundlehren der mathematischen Wissenschaften 195, © Springer-Verlag Berlin Heidelberg 2011

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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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301

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