C*-AL(JIzBRAb
VOLUME 3" GENERAL THEORY OF C*-ALGEBRAS
North-Holland Mathematical Library Board of Honorary Editors: M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. H6rmander, J.H.B. Kemperman, W.A.J.Luxemburg, F. Peterson, I.M. Singer and A.C. Zaanen
Board of Advisory Editors: A. Bj6mer, R.H. Dijkgraaf, A. Dimca, A.S. Dow, J.J. Duistermaat, E. Looijenga, J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sj6strand, J.H.M. Steenbrink, F. Takens and J. van Mill
VOLUME 60
ELSEVIER Amsterdam- London- New York- Oxford- Paris - Shannon- Tokyo
C*-Algebras Volume 3: General Theory of C*-Algebras Corneliu Constantinescu Departement Mathematik, ETH Ziirich CH-8092 Ziirich Switzerland
2001 ELSEVIER A m s t e r d a m - L o n d o n - New Y o r k - Oxford- Paris - Shannon- Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
9 2001 Elsevier Science B.V. All rights reserved.
This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail:
[email protected]. You may also contact Global Rights directly through Eisevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WI P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.
First edition 2001 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.
ISBN (this volume): ISBN (5 volume set): Series ISSN:
0 444 50751 5 0 444 50758 2 0924 6509
The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.
Preface Functional analysis plays an important role in the program of studies at the Swiss Federal Institute of Technology. At present~ courses entitled Functional Analysis I and II are taken during the fifth and sixth semester, respectively. I have taught these courses several times and after a while typewritten lecture notes resulted that were distributed to the students. During the academic year 1987/88~ I was fortunate enough to have an eager enthusiastic group of students that I had already encountered previously in other lecture courses. These students wanted to learn more in the area and asked me to design a continuation of the courses. Accordingly, I proceeded during the academic year, following, with a series of special lectures, Functional Analysis III and IV, for which I again distributed typewritten lecture notes. At the end I found that there had accumulated a mass of textual material, and I asked myself if I should not publish it in the form of a book. Unfortunately, I realized that the two special lecture series (they had been given only once) had been badly organized and contaiued material that should have been included in the first two portions. And so I came to the conclusion that I should write everything anew - and if at a l l - then preferably in English. Little did I realize what I was letting myself in for! The number of pages grew almost imperceptibly and at the end it had more than doubled. Also, the English language turned out to be a stumbling block for me; I would like to take this opportunity to thank Prof. Imre Bokor and Prof. Edgar Reich for their help in this regard. Above all I must thank Mrs. Barbara Aquilino, who wrote, first a WordMARC TM, and then a I ~ T ~ TM version with great competence, angelic patience, and utter devotion, in spite of illness. My thanks also go to the Swiss Federal Institute of Technology t h a t generously provided the infrastructure for this extensive enterprise and to my colleagues who showed their understanding for it.
Corneliu Constantinescu
This Page Intentionally Left Blank
vii
Table of Contents of Volume 3
Introduction 4
...............................
xix
C'-Algebras ............................... 3 4.1 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . 3 4.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 4 4.1.2 The Symmetry of C*-Algebra . . . . . . . . . . . . . . . 30 4.1.3 Functional calculus in C*-Algebras . . . . . . . . . . . . 56 4.1.4 The Theorem of Fuglede-Putnam . . . . . . . . . . . . . 75 4.2 The Order Relation . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.1 Definition and General Properties . . . . . . . . . . . . . 92 4.2.2 More about the Order Relation. . . . . . . . . . . . . . . 101 4.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Powers of Positlive Elements . . . . . . . . . . . . . . . . 4.2.5 The Modulus . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Ideals and Quotients of C*-Algebras . . . . . . . . . . . 4.2.7 The Ordered Set of Orthogonal Projections . . . . . . . 4.2.8 Approximate Unit . . . . . . . . . . . . . . . . . . . . . 4.3 Supplementary Results on P-Algebras . . . . . . . . . . . . . . 4.3.1 The Exterior Multiplication . . . . . . . . . . . . . . . . 4.3.2 Order Complete C*-Algebras . . . . . . . . . . . . . . . 4.3.3 The Carrier . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Hereditary C*-Subalgebras . . . . . . . . . . . . . . . . 4.3.5 Simple C*-algebras . . . . . . . . . . . . . . . . . . . . . 4.3.6 Supplementary Results Concerning Complexification . . 4.4 W*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 General Properties . . . . . . . . . . . . . . . . . . . . . 4.4.2 F as an E-submodule of E' . . . . . . . . . . . . . . . 4.4.3 Polar Representation . . . . . . . . . . . . . . . . . . . . 4.4.4 W*-.Homomorphisms . . . . . . . . . . . . . . . . . . . . Name Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116 123 143 150 162 178 208 208 215 243 263 276 286 297 297 309 335 361 385
viii
Table of Contents
Subject Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
388
Symbol Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
411
Contents of All Volumes
Table of Contents of Volume 1
Introduction
...............................
xix
....................
1
Some Notation and Terminology
1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 NormedSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Some Standard Examples . . . . . . . . . . . . . . . . . 1.1.3 Minkowski’s Theorem . ; . . . . . . . . . . . . . . . . . . 1.1.4 Locally Compact Normed Spaces . . . . . . . . . . . . . 1.1.5 Products of Normed Spaces . . . . . . . . . . . . . . . . 1.1.6 Summable Families . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 General Results . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Standard Examples . . . . . . . . . . . . . . . . . . . . . 1.2.3 Infinite Matrices . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Complemented Subspaces . . . . . . . . . . . . . . . . . 1.2.6 The Topology of Pointwise Convergence . . . . . . . . . 1.2.7 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 The Alaoglu-Bourbaki Theorem . . . . . . . . . . . . . . 1.2.9 Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . 1.3.1 The Banach Theorem . . . . . . . . . . . . . . . . . . . . 1.3.2 Examples in Measure Theory . . . . . . . . . . . . . . . 1.3.3 The Hahn-Banach Theorem . . . . . . . . . . . . . . . . 1.3.4 The Transpose of an Operator . . . . . . . . . . . . . . .
7 7 7 12 31 35 37 40 58 61 61 74 92 113 123 134 138 148 150 153 159 159 171 180 191
Table of Contents
X
Polar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . The Bidual . . . . . . . . . . . . . . . . . . . . . . . . . The Krein-Smulian Theorem . . . . . . . . . . . . . . . Reflexive Spaces . . . . . . . . . . . . . . . . . . . . . . . Completion of Normed Spaces . . . . . . . . . . . . . . . Analytic Functions . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Baire's Theorem . . . . . . . . . . . . . . . . . . 1.4.1 The Banach-Steinhaus Theorem . . . . . . . . . . . . . . 1.4.2 Open Mapping Principle . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Banach Categories . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . Ordered Banach spaces . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Ordered normed spaces . . . . . . . . . . . . . . . . . . . 1.7.2 Order Continuity . . . . . . . . . . . . . . . . . . . . . .
1.3.5 1.3.6 1.3.7 1.3.8 1.3.9 1.3.10 1.4
1.5
1.6
1.7
199 211 228 240 245 246 254 256 256 264 280 281 281 288 308 308 322 334 334 340
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
357
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
359
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
371
Name Index Subject Index
xi
Table of Contents of Volume 2
Introduction
...............................
2 Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Invertible Elements . . . . . . . . . . . . . . . . . . . . . 2.1.3 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Standard Examples . . . . . . . . . . . . . . . . . . . . . 2.1.5 Complexification of Algebras . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Normed Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Standard Examples . . . . . . . . . . . . . . . . . . 2.2.3 The Exponential Function and the Neumann Series . . . 2.2.4 Invertible Elements of Unital Banach Algebras . . . . . . 2.2.5 The Theorems of Riesz and Gelfand . . . . . . . . . . . . 2.2.6 Poles of Resolvents . . . . . . . . . . . . . . . . . . . . . 2.2.7 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Involutive Banach Algebras . . . . . . . . . . . . . . . . . . . . 2.3.1 Involutive Algebras . . . . . . . . . . . . . . . . . . . . . 2.3.2 Involutive Banach Algebras . . . . . . . . . . . . . . . . 2.3.3 Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . 2.3.4 Positive Linear Forms . . . . . . . . . . . . . . . . . . . 2.3.5 The State Space . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Involutive Modules . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Gelfand Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The Gelfand Transform . . . . . . . . . . . . . . . . . . . 2.4.2 Involutive Gelfand Algebras . . . . . . . . . . . . . . . .
xix
3 3 3 13 17 32 51 65 69 69 82 114 125 153 161 174 197 201 201 241 275 287 305 322 328 331 331 343
xii
Table of Contents
2.4.3 2.4.4 2.4.5 2.4.6
Examples . . . . . . . . . . . . . . Locally Compact Additive Groups Examples . . . . . . . . . . . . . . The Fourier Transform . . . . . . Exercises . . . . . . . . . . . . . . .
............ 358 . . . . . . . . . . . . . 365 ............ 378 . . . . . . . . . . . . . 390 ............ 396
3 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Fredholm Operators . . . . . . . . . . . . . . . . . . . . 3.1.4 Point Spectrum . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Spectrum of a Compact Operator . . . . . . . . . . . . 3.1.6 Integral Operators . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Differential Equations . . . . . . . . . . . . . . . . . . . . 3.2.1 Boundary Value Problems for Differential Equations . . 3.2.2 Supplementary R.esults . . . . . . . . . . . . . . . . . . 3.2.3 Linear Partial Differential Equations . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . .
399 399 399 419 437 468 477 489 517 518 518 530 549 563
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
565
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
568
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
588
Name Index Subject Index
xiii
T a b l e o f C o n t e n t s of V o l u m e 3
4
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C*-Algebras
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
The General Theory 4.1.1
4.2
4.3
4.4
xix
3
. . . . . . . . . . . . . . . . . . . . . . . .
3
General Results . . . . . . . . . . . . . . . . . . . . . . .
4
4.1.2
T h e S y m m e t r y of C * - A l g e b r a . . . . . . . . . . . . . . .
30
4.1.3
F u n c t i o n a l calculus in C * - A l g e b r a s . . . . . . . . . . . .
56
4.1.4
T h e T h e o r e m of F u g l e d e - P u t n a m
. . . . . . . . . . . . .
75
The Order Relation . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.2.1
Definition and General Properties . . . . . . . . . . . . .
4.2.3
Examples
4.2.4
P o w e r s of Positive E l e m e n t s
92
. . . . . . . . . . . . . . . . . . . . . . . . . .
116
. . . . . . . . . . . . . . . .
123
. . . . . . . . . . . . . . . . . . . . . . . .
143
4.2.5
The Modulus
4.2.6
I d e a l s a n d Q u o t i e n t s of C * - A l g e b r a s
4.2.7
T h e O r d e r e d Set of O r t h o g o n a l P r o j e c t i o n s
4.2.8
Approximate Unit
...........
150
.......
162
. . . . . . . . . . . . . . . . . . . . .
178
S u p p l e m e n t a r y Results on C * - A l g e b r a s . . . . . . . . . . . . . .
208
4.3.1
208
The Exterior Multiplication
. . . . . . . . . . . . . . . .
4.3.2
Order Complete C*-Algebras
4.3.3
The Carrier
4.3.4
Hereditary C*-Subalgebras
4.3.5
Simple C*-algebras . . . . . . . . . . . . . . . . . . . . .
4.3.6
Supplementary Results Concerning Complexification
W*-Algebras
. . . . . . . . . . . . . . .
215
. . . . . . . . . . . . . . . . . . . . . . . . .
243
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
263 276 . .
286 297
4.4.1
General Properties
4.4.2
F
4.4.3
Polar Representation
. . . . . . . . . . . . . . . . . . . .
335
4.4.4
W*-Homomorphisms
. . . . . . . . . . . . . . . . . . . .
361
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
385
Name Index
. . . . . . . . . . . . . . . . . . . . .
297
as an E - s u b m o d u l e of E ~ . . . . . . . . . . . . . . .
309
xiv
Table of Contents
Subject Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
388
Symbol Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
411
XV
T a b l e o f C o n t e n t s of V o l u m e 4
Introduction
5
Hilbert Spaces 5.1
5.2
5.3
5.4
5.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
P r e - H i l b e r t Spaces
. . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1
General Results . . . . . . . . . . . . . . . . . . . . . . .
5.1.2
Examples
5.1.3
Hilbert sums
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
O r t h o g o n a l P r o j e c t i o n s of H i l b e r t Space
. . . . . . . . . . . . .
3 3 14 19 24
5.2.1
P r o j e c t i o n s o n t o C o n v e x Sets
. . . . . . . . . . . . . . .
24
5.2.2
Orthogonality . . . . . . . . . . . . . . . . . . . . . . . .
29
5.2.3
Orthogonal Projections . . . . . . . . . . . . . . . . . . .
33
5.2.4
Mean Ergodic Theorems
5.2.5
The Fr~chet-Riesz Theorem
. . . . . . . . . . . . . . . . . .
54
. . . . . . . . . . . . . . . .
63
Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
72
5.3.1
General Results . . . . . . . . . . . . . . . . . . . . . . .
72
5.3.2
S u p p l e m e n t a r y Results . . . . . . . . . . . . . . . . . . .
86
5.3.3
Selfadjoint O p e r a t o r s . . . . . . . . . . . . . . . . . . . .
108
5.3.4
Normal Operators . . . . . . . . . . . . . . . . . . . . . .
123
Representations
. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1
Cyclic R e p r e s e n t a t i o n
5.4.2
General Representations
5.4.3
E x a m p l e of R e p r e s e n t a t i o n s
130
. . . . . . . . . . . . . . . . . . .
130
. . . . . . . . . . . . . . . . . .
146
. . . . . . . . . . . . . . . .
156
. . . . . . . . . . . . . . . . . . . . . . . . .
166
5.5.1
General Results . . . . . . . . . . . . . . . . . . . . . . .
166
5.5.2
Hilbert Dimension
. . . . . . . . . . . . . . . . . . . . .
191
5.5.3
Standard Examples . . . . . . . . . . . . . . . . . . . . .
206
5.5.4
The Fourier-Plancherel Operator
. . . . . . . . . . . . .
218
5.5.5
O p e r a t o r s a n d O r t h o n o r m a l Bases
. . . . . . . . . . . .
223
5.5.6
Self-normal Compact Operators . . . . . . . . . . . . . .
243
5.5.7
E x a m p l e s of Real C * - A l g e b r a s
258
O r t h o n o r m a l Bases
. . . . . . . . . . . . . .
Table of Contents
xvi
5.6
Hilbert right C*-Modules
. . . . . . . . . . . . . . . . . . . . .
5.6.1
Some General Results
5.6.2
Self-duality
5.6.3
Von Neumann right W*-modules
5.6.4
Examples
. . . . . . . . . . . . . . . . . . . . . . . . . .
373
5.6.5
ICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
430
5.6.6
M a t r i c e s over C * - a l g e b r a s
. . . . . . . . . . . . . . . . .
477
5.6.7
Type I W*-algebras
. . . . . . . . . . . . . . . . . . . .
515
Name Index
. . . . . . . . . . . . . . . . . . .
286
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
286 310 341
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
535
Subject Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
539
Symbol Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
567
xvii
T a b l e o f C o n t e n t s of V o l u m e 5
Introduction
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Selected C h a p t e r s of C * - A l g e b r a s 6.1
6.2
6.3
/:P-Spaces
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix
3 3
6.1.1
C h a r a c t e r i s t i c Families of Eigenvalues . . . . . . . . . . .
6.1.2
C h a r a c t e r i s t i c Sequences . . . . . . . . . . . . . . . . . .
3
6.1.3
P r o p e r t i e s of the s
6.1.4
Hilbert-Schmidt Operators .................
46
6.1.5
T h e Trace . . . . . . . . . . . . . . . . . . . . . . . . . .
56
6.1.6
D u a l s of s
72
10
................
21
. . . . . . . . . . . . . . . . . . . . .
6.1.7
Exterior Multiplication and s
6.1.8
T h e C a n o n i c a l P r o j e c t i o n of t h e T r i d u a l of K: . . . . . .
..........
102
6.1.9
I n t e g r a l O p e r a t o r s on H i l b e r t Spaces
116
Selfadjoint Linear Differential E q u a t i o n s
79
...........
.............
124
6.2.1
Selfadjoint B o u n d a r y Value P r o b l e m s . . . . . . . . . . .
6.2.2
The Regular Sturm-Liouville Theory
125
6.2.3
Selfadjoint Linear Differential E q u a t i o n s on T
6.2.4
A s s o c i a t e d P a r a b o l i c a n d H y p e r b o l i c E v o l u t i o n E q u a t i o n s 153
...........
139 ......
6.2.5
Selfadjoint Linear P a r t i a l Differential E q u a t i o n s
6.2.6
A s s o c i a t e d P a r a b o l i c a n d H y p e r b o l i c E v o l u t i o n E q u a t i o n s 192
Von N e u m a n n Algebras
.....
150
. . . . . . . . . . . . . . . . . . . . . .
184
202
6.3.1
T h e S t r o n g Topology . . . . . . . . . . . . . . . . . . . .
203
6.3.2
B i d u a l of a C * - a l g e b r a . . . . . . . . . . . . . . . . . . .
218
6.3.3
E x t e n s i o n of the F u n c t i o n a l Calculus
263
6.3.4
Von N e u m a n n - A l g e b r a s
6.3.5
The Commutants
6.3.6
Irreducible Representations
...........
. . . . . . . . . . . . . . . . . .
283
. . . . . . . . . . . . . . . . . . . . . .
293
................
299
6.3.7
C o m m u t a t i v e von N e u m a n n A l g e b r a s . . . . . . . . . . .
320
6.3.8
R e p r e s e n t a t i o n s of W * - A l g e b r a s
325
6.3.9
Finite-dimensional C*-algebras ..............
.............
334
Table o] Contents
xviii
6.3.10 A generalization . . . . . . . . . . . . . . . . . . . . . . . 7
C * - a l g e b r a s G e n e r a t e d by Groups 7.1
7.2
...................
Projective Representations of G r o u p s
...............
369 369
7.1.1
Schur functions
7.1.2
Projective Representations . . . . . . . . . . . . . . . . .
404
7.1.3
S u p p l e m e n t a r y Results . . . . . . . . . . . . . . . . . . .
431
7.1.4
Examples
. . . . . . . . . . . . . . . . . . . . . . . . . .
466
Clifford Algebras
. . . . . . . . . . . . . . . . . . . . . . . . . .
492
7.2.1
. . . . . . . . . . . . . . . . . . . . . . .
355
General Clifford Algebras
.................
369
492
7.2.2
Cgp,q . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
518
7.2.3
Ct(lN)
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
538
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
559
Subject Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
563
S y m b o l Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
592
N a m e Index
xix
Introduction This book has evolved from the lecture course on Functional Analysis I had given several times at the ETH. The text has a strict logical order, in the style of "Definiton- Theorem - P r o o f - E x a m p l e - Exercises". The proofs are rather thorough and there are many examples. The first part of the book (the first three chapters, resp. the first two volumes) is devoted to the theory of Banach spaces in the most general sense of the term. The purpose of the first chapter (resp. first volume) is to introduce those results on Banach spaces which are used later or which are closely connected with the book. It therefore only contains a small part of the theory, and several results are stated (and proved) in a diluted form. The second chapter (which together with Chapter 3 makes the second volume) deals with Banach algebras (and involutive Banach algebras), which constitute the main topic of the first part of the book. The third chapter deals with compact operators on Banach spaces and linear (ordinary and partial) differential equations - applications of the theory of Banach algebras. The second part of the book (the last four chapters, resp. the last three volumes) is devoted to the theory of Hilbert spaces, once again in the general sense of the term. It begins with a chapter (Chapter 4, resp. Volume 3) on the theory of C*-algebras and W*-algebras which are essentially the focus of the book. Chapter 5 (resp. Volume 4) treats Hilbert spaces for which we had no need earlier. It contains the representation theorems, i.e. the theorems on isometries between abstract C*-algebras and the concrete C*-algebras of operators on Hilbert spaces. Chapter 6 (which together with Chapter 7 makes Volume 5) presents the theory of s of operators, its application to the self-adjoint linear (ordinary and partial) differential equations, and the von Neumann algebras. Finally, Chapter 7 presents examples of C*-algebras defined with the aid of groups, in particular the Clifford algebras. Many important domains of C*-algebras are ignored in the present book. It should be emphasized that the whole theory is constructed in parallel for the real and for the complex numbers, i.e. the C*-algebras are real or complex. In addition to the above (vertical) structure of the book, there is also a second (horizontal) division. It consists of a main strand, eight branches, and additional material. The results belonging to the main strand are marked with (0). Logically speaking, a reader could restrict himself/herself to these and ignore the rest. Results on the eight subsidiary branches are marked with (1), (2), (3), (4), (5), (6), (7), and (8). The key is
XX
1. 2. 3.
Infinite Matrices Banach Categories Nuclear Maps
4. 5. 6. 7. 8.
Locally Compact Groups Differential Equations Laurent Series Clifford Algebras Hilbert C*-Modules
These are (logically) independent of each other, but all depend on the main strand. Finally, the results which belong to the additional material have no marking and - from a logical perspective - may be ignored. So the reader can shorten for himself/herself this very long book using the above marks. Also, since the proofs are given with almost all references, it is possible to get into the book at any level and not to read it linearly. We assume that the reader is familiar with classical analysis and has rudimentary knowledge of set theory, linear algebra, point-set topology, and integration theory. The book addresses itself mainly to mathematicians, or to physicists interested in C*-algebras. I would like to apologize for any omissions in citations occasioned by the fact that my acquaintance with the history of functional analysis is, unfortunately, very restricted. For this history we recommend the following texts. BIRKHOFF. G. and KREYSZIG, E., The Establishment of Functional Analysis, Historia Mathematica 11 (1984), 258-321. 2. BOURBAKI, N., Elements of the History of Mathematics~ (21. Topological Vector Spaces), Springer-Verlag (1994). 3. DIEUDONNI~. J., History of Functional Analysis, North-Holland (1981). 4. DIEUDONNt~ J., A Panorama of Pure Mathematics (Chapter C III: Spectral Theory of Operators), Academic Press (1982). HEUSER, H., Funktionalanalysis, 2. Auflage (Kapitel XIX: Ein Blick auf die werdende Functionalanalysis), Teubner (1986)~ 3. Auflage (1992). KADISON, R.V., Operator Algebras, the First Forty Years~ in: Proceedings of Symposia in Pure Mathematics 38 I (1982). 1-18. MONNA, A.F., Functional Analysis in Historical Perspective, John Whiley & Sons (1973).
xxi
8. STEEN, L.A., Highlights in the History of Spectral Theory, Amer. Math. Monthly 80 (1973), 359-382. There is no shortage of excellent books on C*-algebras. Nevertheless, we hope that this book will be also of some utility to the mathematics community.
This Page Intentionally Left Blank
VOLUME 3" GENERAL THEORY OF C*-ALGEBRAS
This Page Intentionally Left Blank
4. C*-Algebras
In this chapter we leave the realm of operators on Banach spaces and hence that of Banach algebras for the realm of operators on Hilbert spaces. The C*algebras play the same role with respect to these operators as did the Banach algebras with respect to the operators on Banach spaces. We therefore formulate abstractly the algebra of operators on Hilbert spaces. But in contrast to Banach algebras, every C*-algebra is isometric to a closed involutive subalgebra of the algebra of operators on a Hilbert space. The principal difficulty constantly confronted in the study of C*-algebras is the lack of commutativity. It would not be unfair to say that to certain e):tent all that has been done in the theory of C*-algebras consists of exploiting a small commutative toe-hold, namely selfadjointness and normality, in this non-commutative world to find locally commutative C*-algebras which are embedded in the hostile non-commutative C*-algebras. Once this toe-hold has been consolidated, the commutative theory can be applied in the noncommutative case as well, for example by using the absolute value.
4.1
The
General
Theory
Three central results on C*-algebras are proved in this section. First we show that these algebras are symmetric, which implies that the Gelfand transform is an isometry. Next we use this result to develop a functional calculus for selfnormal operators, which is the principal tool in the theory. Finally the FugledeP u t n a m Theorem enables us to enlarge the toe-hold of commutativity provided by self-normal operators to commutative sets of self-normal operators.
4
4. C*-Algebras
4.1.1 G e n e r a l R e s u l t s ( 0 ) (Gelfand-Naimark, 1943). A complex (unitaO C*-algebra is an involutive (unital) complex Banach algebra E such that
D e f i n i t i o n 4.1.1.1
IIx*xll = Ilxli 2 for every x E E . A real (unitaO C*-algebra is an involutive (unital) real Banach algebra E such that its complexification E (Proposition 2.3.1.40) admits a norm extending the norm of E 5dentified with E • {0}) and making a complex C*-algebra (by Corollary 4.1.1.21 this norm on E is unique). A C*-algebra is a complex or a real C*-algebra. Let E be a (unital) C*-algebra. A (unital) C*-subalgebra of E is a closed involutive (unital) subalgebra of E . It is a (unital) C*-algebra with respect to the structure induced from E (Proposition 2.3.1.4 0). The smallest (unital) C*subalgebra of E containing a given subset A of E is called the (unital) C*subalgebra of E generated by A . A Gelfand C*-algebra is a C*-algebra which is an involutive Gelfand algebra. Every closed involutive ideal of a C*-algebra E is a C*-subalgebra of E . If E is a C*-algebra then
iizii ~ < I1~*~ + y*yll
(*)
for all x, y C E (Corollary 4.2.1.18, Theorem 4.2.2.1 c :=v a). Conversely, it can be shown, that if E is an involutive real algebra such that (.) holds, then E is a real C*-algebra (T.W. Palmer, Pacific Journal of Math. 35 (1970) 195-204). If E is a unital C*-algebra, then 1 + x*x is invertible for every x C E (Theorem 4.2.2.1 c =:v a). Conversely, it can be shown, that if E is an involutive unital real Banach algebra such that
IIx*xll = ilxli 2 and 1 + x*x is invertible for every x C E , then E is a real C*-algebra (K.R. Goodearl, Notes on Real and Complex C*-Algebras, Shiva Publishing Limited, 1982). If E is a real C*-algebra, then the identity map E • E --+/~ is an isomorphism of Banach spaces. Indeed
I1~ + ~yll_ Ilxli + I[yil
4.1 The General Theory
5
for every (x, y) C E • E , so that the above map is continuous. By the Principle of Inverse Operator, this map is an isomorphism. The real Banach algebra ~ endowed with the involution ~ ---+~,
xJ
>x
is not a real C*-algebra although llx*xtl)ll.ll =
holds for all x E ~ . Remark.
If E is a C*-algebra and x , y C E then:
a) x * x y -- O => x y -- O ; b) y x x * - O => y x - O ; c) x * x y -
x*x ~
xy--
x ;
d) y x x * - x x * ~ y x - x . a) Since x* x y - O ,
y* x* x y = O , i.e.
II~yll ~ -
II(xy)*(~y)ll = Ily*~*~yll = o.
Hence x y = O. b) Since y x x * = 0, x x * y * = 0. Hence x ' y * - 0 by a). It follows y x = O.
r
II~y-
xll ~ - l l ( ~ y -
~)*(xy-
~)ll = II(y*x* - ~ * ) ( ~ y -
x)11-
= II~*x*~y - ~ * x y - y * x * x + ~*xll = tly*(x*~y - ~*~) - ( x * x ~ - ~*~)ii - 0 , xy --x.
d) We have x x * y * = x x * . By c), x ' y * = x * . Hence y x - x . \
[
P r o p o s i t i o n 4.1.1.2 real f u n c t i o n
on E
(
0
) L e t E be an i n v o l u t i v e algebra a n d f a p o s i t i v e
s u c h t h a t f o r e v e r y x, y C E ,
f ( x y ) <_ f (x) f (y) ,
a)
f(x) 2 ~_ f ( x * x ) .
F o r all x E E , f (x) = f (x*) ,
f(x)2 = f ( x * X ) - f ( x x * ) .
6
4. C*-Algebras
b)
For all x E No E and n E iN , f (x ~) - f (x) '~ "
c)
If E possesses a unit then
f(1) e {0, 1}. a) We have successively that f (x) 2 ~ f (x*x) ~ f (x*) f (x) ,
f(x*) ~_ f(x**) : f ( x ) ,
f(x)=
f(x*),
f(x*x) ~ f ( x * ) f ( x ) =
f(x) 2 ,
f (x)2 = f (x* x) ,
f ( x x * ) = f ( x * * x * ) = f(x*)2 = f ( x ) 2
b) By a) f(x)4 = f ( x * x ) 2 = f ( x * x x * x ) = f((x*)2x 2) = f((x2)*x 2) = f(x2) 2, such that
f (x ~) = f (Z) Let n E iN and assume f (x 2") = f (x) 2"
Since x 2~ is normal, we get f(x2 "+1) __ f((x2") 2) -- f(x2n)2 - f(x) 2n+1 by the above considerations. Hence, by complete induction,
4.1 The General Theory
7
f ( x 2'~) = f(x) 2" for all n E IN. By Proposition 2.2.3.1. inf f ( x n ) l = lim f(xn) 1 nEIN
n---> o o
lim f(x2n) 2 - n - f ( x ) n---~oo
f (x) < f (x n) 88<__f (x), f (x n) - f (x) n . c) If E possesses a unit, then by a),
f ( 1 ) 2 = f ( 1 * 1 ) = f(1) so that f(1) C {0, 1}.
I
P r o p o s i t i o n 4.1.1.3 ( 0 ) Let E be an involutive (unital) algebra which is a Banach algebra with respect to the norm ]l" II- Suppose further that I1~11~ < IIx*xll
for every x E E . Then E is a (unital) involutive Banach algebra in which Ilxll 2 =
IIx*xll = Ilxx*ll
for every x G E . The assertion follows from Proposition 4.1.1.2 a).
Remark.
m
a) The relation IIx*xll = Ilxx*ll
is an important drop of commutativity in the uncommutative world of involutive Banach algebras. b) Let E be as in the proposition. The proposition shows that the relation e E ~
l]~*xli = Ilxll ~
implies the relation
The reverse implication holds too for lK = r
but the proof is long.
8
4. C*-Algebras
P r o p o s i t i o n 4.1.1.4
( 0 )
The unitary elements of a unital C*-algebra
have norm 1. Let x be a unitary element of a unital C * - a l g e b r a . Then llxll 2 -
IIx*xtl = Illll -
I
1
4.1.1.5 ( 0 ) If T is a set then e~176 is a unital C*-algebra, c(T) is a unital C*-subalgebra of g~ and co(T) is a C*-subalgebra of c(T). If T is a topological space, then C(T) is a unital C*-subalgebra of ~ ( T ) ; if T is locally compact then Co(T) is a C*-subalgebra of C(T). If 9~ is a a-algebra on T , then
Example
{x E e~ (T) I x is !R-measurable} is a unital C*-subalgebra of t~ If ( T , ' ~ , # ) is a measure space, then the involutive commutative unital Banach algebra L~176 (Example 2.3.2.3) is a C* -algebra. I Remark.
If T is an infinite set and p E [1, c~[ , then there is no norm on the
involutive subalgebra t~P(T) of t ~ ( T )
which renders it a C*-algebra. Indeed,
assume we had such a norm. Let (A,,)ne~ be a disjoint sequence in ~ s ( T ) such that Card An = 2 n for every n E IN. For n E IN, eAn i]2 -
]]e*A~eA~ I = ]]eAn il,
so that I1~.11 = 1 .
Thus the sequence (~-~eAn)ne~ ] is absolutely summable. Let x be its sum. Then
1 x(t) = n2 for every n E IN and t E An and we obtain the contradiction 2n OO -
~ nEIN
~Tt2P < Z
tET
I~(t)l ~ <
~
4.1 The General Theory
Example 4.1.1.6 let
9
( 0 ) Let (Et)t~T be a family of (unital) C*-algebras and
tET
be the involutive normed (unital) algebra defined in Example 2.3.2.4). Then E is a (unital) C*-algebra (called the C*-direct product of the family (Et)teT) and {x e E l([]xtl])tcT e
co(T)}
is an involutive closed ideal of E (and therefore a C*-algebra), called the C*direct sum of the family (Et)teT . If T is a topological space and all JEt (t C T) are equal, then {x E E Ix is continuous} is a (unital) C*-subalgebra of E .
Example 4.1.1.7
( 0 ) For every n E IN the involutive unital Banach algebra L(]K n) (Example 2.3.2.6) is a unital C*-algebra. For every set T , the involutive unital C*-algebra /:(t~2(T)) (Example 2.3.2.2"/c)) is a unital C*algebra. For every measure space (T, ~, #), the involutive unital Banach algebra 1:(L2(#)) (Example 2.3.2.27 c)) is a unital C*-algebra.
The last two assertions follow from Example 2.3.2.27 c), d). The first assettion is a special case of the second one. 9 Remark. In general, neither of the involutive Banach algebras g2'2(T, T) (Example 2.3.2.5) and L2(p | p) (Proposition 3.1.6.18 b)) is a C*-algebra.
T h e o r e m 4.1.1.8
( 0 ) Let E be a complex (unital) C*-algebra and
o
(rasp. E' ) the complexification of the underlying involutive real algebra of E (rasp. involutive vector space of E'). We denote by E the complex (unital) C*- algebra obtained from E by reversing the multiplication, i.e. by endowing E with the multiplication
E•
>E,
(x,y),
>yx.
10
4. C*-Algebras
a) /~
becomes a complex (unital) C*-algebra with respect to the norm
]~
~ IR+,
(x, y),
> sup{
Ix +
iYll, IIx -iyll}.
In particular if we restrict the scalars of E to IR then E is a real (unital) C*-algebra. A real C*-algebra which is not of this form (i.e. which is not obtained from a complex C*-algebra by the restrictions of the scalars to IR ) will be called purely real C*-algebra.
b)
The map
[~
~ E • ~,
(~, y) ,
~ ((~ + iy), ( x - iy)*)
is an isometry of complex (unital) C*-algebras (a) and Example ~. 1.1.6). o
E'
is an involutive complex Banach space with respect to the norm
~ ~§
E'
(~',Y')'
~ll~'+iy'll+ II~'-iy'll-
o
d) C i v ~
(~', y') e E' , d ~
(~,, r
[~ --~ r
(~, y) , ~ (~ + iy, ~' + iy') + (x - iy, ~ ' - i r o
Then for each (x', y') e E' , (x', y') e (E)' and
(x', y') e (E)'+ r
~' + iy', ~' - iy' e E §
and the map o o
E' ~
(E)' ,
(~', y') ,
~ (x', y')
is an isometry of involutive Banach spaces. e)
If u : E -+ E is an isomorphism of real C*-algebras which is conjugate linear, then the map
is an isomorphism of complex C*-algebras.
4.1 The General Theory
11
f) If T is a topological space then the map C(T)
>C(T) x C(T),
(x,y),
>(x+iy,5+iy)
is an isomorphism of complex C*-algebras. I f T is locally compact, then we may replace C by Co in the above map.
a) ,b) ,c) ,d) By Example 2.3.2.35 and 2.3.4.22, we need only prove that /~ is a complex C*-algebra. Take (x, y) E E . We have
li(x, y)*(x, y)ll = ii(x*,-y*)(x, y)ii = ii(~*~ + y'y, ~ * y - y*~)li = = sup{[[(x*x + y'y) + i(x*y - y*x)[[, [[(x*x + y*y - i(x*y - y*x)[[} = = sup{ [I(x + iy)* (x + iy)If, [[ (x - iy)* (x - iy)tO } = = sup{ll~
+ iyll ~ , IIx - iyJl ~} = II(~, y)ll ~.
e) Put v" E
>E ,
xz
>u - l x *.
Since U - 1 is an isomorphism of real C*-algebra~ which is conjugate linear, v is an isomorphism of complex C*-algebras (Proposition 2.3.1.50). Then v -1 is an isomorphism of complex C*-algebras and v(uy)* = u - l u y = y,
v - l y = (~y)* for every y E E . Hence for x,y E E , v-~(x
- iy)* = ( u ( ( x - i y ) * ) *
-
u(x
-
iy)
-
ux
+
iuy.
By b), then map >ExE,
(x,y),
> (x + iy, ux + iuy)
is an isomorphism of complex (unital) algebras. f) follows from e). Remark. with
m
A unital real C*-algebra E is not purely real iff there is an x C E ~
x*---x,
x2--1.
12
4. C*-Algebras
E x a m p l e 4.1.1.9
Let E be a complex C*-algebra, A an algebraic basis of
the underlying complex vector space of E , and F the real vector subspace of E generated by A . If F is an involutive real subalgebra of E , then the map
~"
> E,
(x,y),
~ ~+iy
is a isomorphism of involutive complex algebras. If, in addition, F is closed, then F is a real C*-algebra and E may be identified with its complexification. m
E x a m p l e 4.1.1.10
Let E be a commutative complex C*-algebra and endow
Re E with the induced structure of an involutive real algebra. Then the map o
u:ReE
(x,y) ,
is an isomorphism o.f involutive complex algebras. In particular, Re E is a real C*-algebra and E may be identified with its complexification,
Example 4.1.1.11
m
Let E be a complex unital C*-algebra. Take x C E such
that x 2=-1,
x*=-x,
and that x and 1 are linearly independent. Put
Then F is an involutive commutative real subalgebra of E and a division algebra and the map
~"
~ E(~),
(y,z) ,
> y + iz
is an isomorphism of involutive complex algebras. In particular, F is a commutative real C*-algebra, which is not a Gelfand algebra, and E(x) may be identified with the complexification of F . The map
is an isometry of C*-algebras (Example 4.1.1.8).
{ 1, x} is an algebraic basis of the underlying complex vector space of E ( x ) . By Example 4.1.1.9, F is a real C*-algebra and E(x) may be identified with the complexification of F . Since
4.1 The General Theory
13
~ e o~((x, 0)), F is not a Gelfand algebra.
Remark.
x "=
[01] -1
P r o p o s i t i o n 4.1.1.12
0
I
E s162
possesses the above properties.
( 0 ) I f E is a C*-algebra, then
yEE#
for every x E E .
We may assume that IIxll = 1. Then IIx*ll- 1 , s o that sup II~yll ~ Ilxx*ll = II~ll~ = 1.
yEE#
The reverse inequality is trivial. T h e o r e m 4.1.1.13
I
( 0 ) (B. Yood)
Let E be a C*-algebra and IK x E
the associated involutive unital algebra of E . We identify E with the involutive ideal { 0 } x E
of lK x E . Then IK x E endowed with the norm
xE
>~+,
x
> supllxyll-supllyxll yEE#
yEE#
if E has no unit and with the norm
]K x E
> IR+,
(a,m),
> sup{Ic~l, IIx+c~lEll}
if E has a unit 1E is a unital C*-algebra inducing on E its given norm. I f E has no unit, then the above IK x E is called the unital C*-algebra associated to E . I f E has a unit, then the unital C*-algebra associated to E will be E itself.
Case1
E has no unit
The map xE
>~+,
x,
>supllxyll yEE#
is obviously a seminorm. By Proposition 4.1.1.12, it induces on E the original norm of E . Assume that
14
4. C*-Algebras
sup II(~, x)yll = o yEE#
for some element (a, x) e IK x E , (c~,x) r (0, 0). Then c~ ~ 0 and
sup lily + x y l l - o, yEE#
Hence
ay + xy = 0, 1
--xy=
y
Ol
for every y E E . Hence - 1O~- x is a unit of E (Proposition 2.3.1.13 c)) contradicting our hypothesis. Thus the map
lKxg
>IR+,
x,
>supllxyll yEE#
is a norm. Since E is complete, this norm is also complete (Corollary 1.2.5.6). Let x, y E I K •
Then y z E E
and z E E .
and
]]yzll < Ilyll Ilzll. Hence
II(~y)zl, = IIz(~z)ll < I1~11 lyzll < II~ll I1~11 I1~11 and
II~yll _< I1~11 Ilyll. Hence IK • E endowed with the above norm is a unital Banach al gebra. Take x E IK • E . We want to prove that
IIz I: __ II~*xl 9 We may assume that l i x l l - 1. Take c~ < 1. There is a y C E # with
Ilxyli ~ > ~. Since xy, x*xy E E , it follows that
< II~yll ~ -]l(~y)*(~y)ll
= Ily*~*~yll < Ily* I II~*~yll < I1~*~11.
4.1 The General Theory
15
Since a is arbitrary, we deduce that
II~ll~ = 1 <_ II~*~ll. By Proposition 4.1.1.3, IK • E is a C*-algebra with respect to the above norm. Take x E IK • E . Then Iixil = IIx*ll =
s u p IIx*yi] = s u p ]l(y*x)*ll = s u p Ily*xl] = s u p Ilyxi]. yEE# yEE# yEE# yEE#
Case 2
E has a unit
Let F be the C*-algebra IK • E defined in Example 4.1.1.6. P u t
u']KxE
>F, ( a , x ) ,
>(a, a l + x ) .
Then u is linear and
~((~, x)(9, y)) = ~ ( ~ 9 , ~y + Z~ + zy) = (~Z, ~ Z l + ~
+ Z~ + xy) =
= (a13, ( a l + x)(131 + y)) = (a, a l + x)(~, 131 + y) - u(a, x)u(13, y ) ,
u(~, z)* = ( ~ , ~ i + x)* = ( ~ , ~ 1 + x*) =
u(~,x*) =
(~(~, x)*)
for every (a, x), (/3, y) E IK • E . Hence u is a bijective involutive algebra homomorphism. The map defined in the enunciation of the theorem is the transported norm of F , and IK • E normed with it is a C*-algebra inducing on E its initial norm.
Remark.
I
Let E be a complex C*-algebra and let F be the underlying real
C*-algebra of E (Theorem 4.1.1.8 a)). If E is not unital and if E , F denote the unital C*-algebras associated to E and F , respectively, then F is purely real and so it is different from the underlying real C*-algebra of E . Convention
The above result allows us to use the unit for C*-algebras even if
the C*-algebra is not unital. In this case, it is understood that the unit belongs to the associated unital C*-algebra. C o r o l l a r y 4.1.1.14
Let T be a locally compact non- compact space and T*
its Alexandroff compactification. Extend each function in Co(T) continuously to T*. Then C(T*) is the unital C*-algebra associated to Co(T). I
16
4. C*-Algebras
Remark.
Put R'=[-1,+1],
T'=[-1,0[U]0,+I].
Then R is homeomorphic to the Alexandroff compactifications of S and T, respectively. By the above corollary, C(R) is isometric to the unital C*-algebras associated to C0(S) and Co(T), respectively. Since Co(S) and Co(T) are not isometric, it follows that non-isometric, non-unital C*-algebras may have isometric associated unital C*-algebras. P r o p o s i t i o n 4.1.1.15
( 0 )
If E is a complex C*-algebra then c
for every x C R e E .
We may assume E to be unital (Theorem 4.1.1.13). Step 1
i ~ a(x)
Assume that i E a ( x ) . Take a E IR+ and put P "= a -
it Er
Then + 1 = ~ - i 2 = P(i) e P ( a ( x ) ) = a ( P ( x ) ) - a(c~l - ix)
(Theorem 2.1.3.4 b)) so that
+ 1 ~ r(~l-
ix) ~
(Proposition 2.2.4.1). We deduce that
2~ + 1 <__ I1~= I, which is a contradiction. Step 2
a(x) c IR
3.1 The General Theory
17
Take a, 3 C lR with a + i3 C a ( x ) . Assume that ~ # 0. Put P ' - - ~ 1 (t - ~) e r
Then
i=-~1 (~ + i~
- ~) = P ( ~ + i~) e P ( a ( x ) ) = a ( P ( x ) ) = a(-~1 (x - oel))
(Theorem 2.1.3.4 b)) which contradicts Step 1. Hence Z = 0 and a(x) C JR. m
T h e o r e m 4.1.1.16
( 0 ) Let x be a self-normal element of a C*-algebra.
e b) r ( x ) = Ilxll. c)
(x quasinilpotent) r
( a ( x ) = {0}) v:~ (x = 0).
a) follows from Proposition 4.1.1.2 b). b) If the C*-algebra E is real then
o~((x, 0)) c (Proposition 4.1.1.15). By a) and Theorem 2.2.5.4, r(x) = lim Ilxnll ! = fix I" n---+ (x)
c) By b) and Proposition 2.2.4.21 d),
(x = 0 ) ~
(x q u a s i n i l p o t e n t ) ==~ (a(x)= {0}) (~(x) = O) ~
(llxll = 0) ~
(x =
0).
Remark. The condition that x be self-normal cannot be dropped. Indeed, /:(IK 2) is a C*-algebra (Example 4.1.1.7) and X "--
[01]
C
~(]K2)
.
0 0
We have x 2 = 0 so that a(x) = {0} (Proposition 2.1.3.16 c)). Corollary 4.1.1.17 self-normal, then
Let E be a C*-algebra and take x , y C E . If xy is
Ilxyll < Ilyxll.
18
4. C*-Algebras
We may assume E to be unital (Theorem 4.1.1.13). Then = r(xy)
=
r(yx) <
(Theorem 4.1.1.16 b), Proposition 2.1.3.10 c), Proposition 2.2.4.1). Corollary 4.1.1.18
II
Every C*-algebra is semi-simple.
Let E be a C*-algebra with radical R. Take x E R. Then x*x E R and by Corollary 2.1.3.24, o(~*~)
= {0)
By Theorem 4.1.1.16 c), I1~11~ = I1~*~11 = 0 ,
so that x = 0
Coro;lary 4.1.1.19 orE,
i
and hence R = { 0 } .
Let E be a unital C*-algebra, x a self-normal element
a e IK, and r > O. I f
o(~) uy(~), then
We have that o(x - ~1) = o(~) - ~ c u~(0),
and so fi x - c~lil = r ( x - c r l ) < r
by Theorem 4.1.1.16 b). Corollary 4.1.1.20 algebra, F
II
( 0 ) (Rickart, 1946) Let E be an involutive Banach
a C*-algebra, and u : E -+ F
phism. Then u is continuous and Ilull <_ 1.
an involutive algebra homomor-
4.1 The General Theory
19
Given x c Re E ,
Iluxll = rF(ux) ~ rE(X) ~ Ilxll (Proposition 2.3.1.7, Theorem 4.1.1.16 b), Corollary 2.1.3.12, and Propositions 2.2.4.1, 2.2.1.3). Take x E E . Then x*x is selfadjoint, and so
Iluxll ~-
II(ux)*(ux)ll = Ilu(x*x)ll ~ IIx*x[I ~ IIx*ll Ilxll = Ilxll ~
by the above remark. Hence
I1~11 < II~il.
m
On the basis of this corollary, we find it convenient to call homo-
Remark.
morphism of C*-algebras every involutive algebra homomorphism of two C*algebras. /
C o r o l l a r y 4.1.1.21
( 0 ) Let E , F
be C*-algebras and u" E -+ F a bijec-
tive involutive algebra homomorphism. Then u is an isometry. In particular, the isomorphisms and the isometries of C*-algebras coincide and there is at most one norm on an involutive algebra making it to a C*-algebra.
II
Let E be a C*-algebra and E the C*-algebra obtained from E by
Remark.
reversing the multiplication. By the above corollary and by Proposition 2.3.1.50, ++
E and E are isomorphic iff there is an isomorphism of real C*-algebras E --+ ++
E which is conjugate linear. In particular if ]K = lR then E and E are isomorphic. P r o p o s i t i o n 4.1.1.22
Let E be an involutive normed complex algebra and
P the set of seminorms on E , for which p(xy) < p ( x ) p ( y ) ,
p(x) ~ < p(x* x) ,
p(~) ~ Ilxll for every x, y C E . Define q:E
~ IR+,
x~
~ supp(x) . pEP
20
4. C*-Algebras
a) q C P .
b) ~1(o) c)
~ an involut~v~ ~d~at of E .
We have that q(x) = q(y) for every A 9 E/ql(0) and x, y 9 A . Define p(A) := q(x) .
d)
The map E/-ql(O)
~ IR+,
A,
>p(A)
defines a norm on E/-q'(O). e) E/-ql(O) is an involutive normed complex algebra. The quotient map E --+ -1
E~ q (0) is continuous with norm not exceeding 1 and the completion of E/-ql(O) is a complex C*-algebra, called the C*-hult of E . f)
Every involutive algebra homomorphism of E into a complex C*-algebra may be factored through the C*-hull of E . This factorization is an involutive algebra homomorphism and it is unique in the class of involutive algebra homomorphism. a) is easy to prove. b) follows from a) and Proposition 4.1.1.2 a). c) and d) are easy to see. e) Given A , B 9 E/-ql(O), p(AB) < p(A)p(B), p(A) 2 < p(A*A).
By b), Proposition 2.3.1.21 c) and Proposition 4.1.1.2 a), E/q1(0) is an involutive normed algebra. The above inequalities hold also for the completion of .
t
E/-ql(O). By Proposition 4.1.1.3, this completion is a C*-algebra. f) Let u : E ~ F be an involutive algebra homomorphism of E into a C*-algebra F . Define
4.1 The General Theory
pE
>~+,
x,
21
>lluxll.
By Corollary 4.1.1.20, p E P . Hence -1
q (0) C Ker u.
Thus u may be factored through E/-ql(O). This factorization is obviously an involutive algebra homomorphism. Since it is continuous (Corollary 4.1.1.20), -1
it may be extended continuously to the completion of E~ q (0). By continuity, this extention is an involutive algebra homomorphism. The uniqueness follows from Corollary 4.1.1.20. E x a m p l e 4.1.1.23
I
Let E
be a finite-dimensional symmetric, involutive,
semi-simple, complex Gelfand algebra and u the Gelfand transform on E . Then the C*-hull of E is E endowed with the norm
E
~+,
x,
>ll~xll
By Proposition 2.4.2.3 a :=~ c, u is involutive and by Corollary 2.4.1.14, u is injective. Since E is finite-dimensional, u is bijective and the assertion now follows from Corollary 4.1.1.21. I P r o p o s i t i o n 4.1.1.24
( 0 ) Let E be a unital C*-algebra, U the topologi-
cal group of unitary elements of E (Proposition 2.3.1.8, Theorem 2.2.4.14), and the topological group of unital isometries E -+ E of C*-algebras (Example 2.2.2.4). Given x E E , define 5"E
>E ,
y~
) xyx*.
Then ~ = 5" E ~ for every x E U, the map U
~,
x:
;5
is a continuous group homomorphism, and EC= {x E E ] ~ = identity map} = {x E U I ~ = identity map}.
Take x E U. Then 5 is linear, bijective, 51 = xlx* = 1,
~y* = xy*x* = ( ~ j ) *
= (~y)*,
22
,~. C * - A l g e b r a s
for every y E E , and
~(yz) = ~ z ~ * = ~y~*xz~* = (~y)(~z), for all y , z C E .
Hence ~ is an involutive unital algebra homomorphism and
= 5*. By Corollary 4.1.1.21, 5 is an isometry of C*-algebras. Take x, y E U . Then x~"~z = x y z y * x *
= ~z
for every z E E . Hence xy = xy
and the map U
>s
z,
~5
is a group homomorphism. Given z E E ,
I1(~- y ~ ( z ) l l - Ilxzx* - yzy*ll ~ liszt* - yzx*ll + Ilyz~* - yzy*ll
I1~ - yll Ilzll IIx*ll § Ilyll Ilzll IIx* - y*ll-- 21Ix - yll Ilzll (Proposition 4.1.1.4), so t h a t
I1~- ~11 ~ 21Ix- yll and the map U
>~-,
xt
>~
is continuous. We now prove the last assertion. If x C E c , then ~y -
xyx*
-
yxx*
= y
for every y C E . Hence ~ is the identity map. Take x E E with ~ the identity map. Then for any y C E y = xy = xyx
In particular, x x * = 1 and
.
4.1 The General Theory
23
x*x = x(x*x)x* = (xx*)(xx*) = 1, i.e. x E U. It follows yx = xy for every y E E , i.e. x E E c. P r o p o s i t i o n 4.1.1.25
( 0 )
I Let E be a C*-algebra, (F~)~ei a family of
C*-subalgebras of E such that p~F~ = {0}
for all distinct t, A E I , F the C*-subalgebra of E generated by U F~, and tel
y~ F~ the C*-direct sum of the family (F~)~E,. Then (x~)~E, is summable for every x E ~ F~ and the map tE I
,F,
x,
tEI
tel
is an isometry of C*-algebras. If I is finite, then the assertion is obvious since the map
tel
tel
is a bijective involutive algebra homomorphism and therefore an isometry of C*-algebras (Corollary 4.1.1.21). In particular
for every (x~)tei E 11 Ft. tEI
Now suppose I is infinite. By the above considerations, (x~)~ex is summable for every x E ~ F~ and tEI
tEI
The map
tel
tel
is an involutive norm-preserving algebra homomorphism, so that Im u is a C*subalgebra of F . Since it contains U Ft it is, in fact, F and u is an isometry tEI
of C*-algebras.
I
24
4. C*-Algebras
Proposition 4.1.1.26 then E = { O} . Step1
( 0 ) If E is a C*-algebra such that R e E -
{0},
xCE~x*=-x
We have x* + x c R e E = {0}, so t h a t X*
Step 2
---- - - X .
x E E :=v x 2 = 0
B y the first step, ~
= _(z~) 9 = _(z,) ~ = _z~
so t h a t x2_-0.
Step 3
x c E =~ x = 0
By the first two steps, Ilxll ~ = II~*xll = I I - x~ll = 0 , so t h a t x=0.
Proposition 4.1.1.27 and consider
( 0 ) Let E be a real C*-algebra. Take x'C (E)',
~ E
>~,
~) (~, y) 9 E ~ I1(~, Y)ll < I1~11+ IlYll b)
~' e Re o
(~)' r ~'~'
i
9 Re E ' I
c) ~' 9 (E)% r ~'~' e E+.
x~ ~ ( ~ , 0 )
4.1 The General Theory
d)
25
If ~ is an approximate unit of E then ~(~) is an approximate unit of o
E and o
o
x' 9 T(E) r
~'x' e T ( E ) .
a) We have
I1(~, y)ll = I1:~ + ivyll _< I1~11 + II~yll = I1~11+ Ilyll. b) ~ is involutive and so ~' is also involutive (Proposition 2.3.2.22 d)) and we get that x' E Re (/~)' ==~ ~'x' 9 Re E ' . Assume ~'x' E Re E' and take (x, y) 9 Re/~. Then x*=x,
y*=-y,
so that ( ( x , y ) , x ' ) = (~x + i~y,x') = (~x,x') § i ( ~ y , x ' ) - (x,~'x') + i(y,~'x') e IR o
and x' e Re (E)' (Corollary 2.3.1.23 b =~ a). c) Take x' e (/~)'+ . By b), ~'x' E Re E' . For every x C E
(x*~, :'~')= (~(:x),x')
= ((~x)*(:~), ~') e ~ §
and so ~'x' C E~_. o
Assume 99'x' e E ~ . By b), x' C Re (E)'. For every (x, y) C/~, ((x,y)*(~,y),~')
- ((x*x + y * y , ~ * y -
= (~(~*~ + : y ) , x') + i ( : ( x * y = (x*x + y * y , ~ ' x ' ) +
i(~*y-
y*~),~') -
: ~ ) , x'> -
y*x,:'~') e
~§
so that x' e (/~)'+ d) Take (x,y) e E and z e E . By a), II(x, y) - (~, y):zJl - ll(x, y ) -
(~z, yz)lJ = II(x - ~z, y - yz)ll < Jlx - ~zll + I f y - y~ll.
26
4. C*-Algebras
It follows that lim II(x, y) - (x, y)~ozll - o , z,~
i.e.
lim(x, y)~oz = (x, y) . z,~
Similarly, lim(~oz)(x, y) = (x, y). Z,6
Hence ~o(~) is an approximate unit of 1~. o
Take x' E (g)~_. By Proposition 2.3.4.10 a),
lim((~x)*(~x),x') = lim(~(x*x) x'} = lim(x*x, ~p'x'} = x,~ x,i~ ' x,i~ By c),
o
x' e ~ ( E ) ~ = * E x a m p l e 4.1.1.28
Put
-1
1
'
~'x' e ~ ( E )
I
v=[111]1
Then
sup{llull, Ilvll} < Ilu*u + v*vll~ < II(u, v) l-Ilull + Ilvll (Examples ~. 1.1.7). We have (Example 1.2.2.8) = ~
1 + 1 + 1 + 1 + V/(1 + 1 + 1 + 1) 2 - 411 + 112 = 2,
1(
I1~11~ - ~ 1 + 1 + 1 + 1 + v/(1 + 1 + 1 + 1) 2 - 411 +
1] 2
)
= 2,
I1(~, v)ll ~- = --1(2+2+2+2+4(2+2+2+2)22
- 41(1+ i)2 + (1 -
i)212)
=
8,
Ilu*u + v ' v i i - 4. Remark. sharp.
This example shows that the inequality of Proposition 4.1.1.27 a) is
4.1 The General Theory
Proposition
4.1.1.29
27
Let (E,)~ei be a family of Gelfand C*-algebras and
E its C*-direct sum. Given t E I and x~' E a(Et ) , define z,''E
z,
Let ~ a(E~) denote the the topological sum of the family of topological spaces tel
(a(E~))~e, . a)
E is a Gelfand C*-algebra.
b)
x~' C a ( E ) for every t E I and x~ E a(E~); we set
!
~
. o(E~)
!
~ o(E),
!
~,
~ x~
for every t C I . c)
The map ~o(E) eCI
defined by = ~
for every t E I is a homeomorphism. a) follows from Proposition 4.1.1.15. b) is easy to see. c) It is clear t h a t ~ is injective and continuous. For t C I and x~ c E~ let x-~ be the element of E defined by
~eI
0
iS t r A.
Take x' E a ( E ) . Given ~ E I , define
x~''E~ !
>IK,
x~, >x'(~). !
There is a t E I such that x~ # 0. Then x~ E a(E~) and there is an x~ E E~ with
~:(~) r o. Let A E I \ { 0 } . Then for every x~ C E~, we have
28
4. C*-Algebras
xLx:~ - O,
so that 0 = ~'(~)
= ~'(~)~'(~)
9 '(~)
= ~:(x~)x'(~),
= o.
It follows that Xt
--
r XL
and the map ~ is surjective. The above construction shows that ~-1 is also continuous, m E x a m p l e 4.1.1.30
Assume ]K = ~ . Let T be the Cantor set and 1
TI"-TN[0,5],
,1].
T2"-Ta[
Given x c C ( T ) , define xl"T1
x2"T2
>r
>r
t,
t:
>x(3t),
) x(3t - 2).
Given (x, y) e C(T) , define N
(x, y) " T
>q; ,
) f xl (t) + iyl (t) x2(t) - iy2(t)
i.f t C T1 if t C 7'2.
Then the map o
C(T)
>C(T) ,
(x, y) ,
> (~, y)
is an isomorphism of complex C*-algebras (Theorem 4.1.1.8 a)).
m
E x a m p l e 4.1.1.31 ( 0 ) The involutive real algebra ]I-I of quaternions endowed with the Euclidean norm IH
>IR+,
a + ~i +'yj + Sk , ~ (a 2 + ~2 +'Y2 + 52)
is a simple real C*-algebra.
4.1 The General Theory
29
The assertion follows from Example 2.3.1.46 (and Example 4.1.1.7, Corollary 2.1.4.17). m E x a m p l e 4.1.1.32 Let E be a C*-algebra. If we replace the multiplication in E by the multiplication x 9 y := - x y for all x, y E E , then we get a C*-algebra.
The assertion follows from Example 2.4.3.8 a),g).
m
P r o p o s i t i o n 4.1.1.33 ( 0 ) Let (x~)~e, be a family in a C*-algebra E . Then the following are equivalent: a)
(x~x~)~e, is absolutely summable.
b)
(x~x;)~c, is absolutely summable.
If E is finite-dimensional, then the above assertions are equivalent to the following ones:
c)
(x~x~)~e, is summable.
d)
(x~x[)~ei is summable. a
r
b follows from e I ~
ar
c and b r
=
d follows from Proposition 1.1.6.14.
m
If E is infinite-dimensional, then c) and d) may not be equivalent (see Remark of Proposition 5.3.2.13).
Remark.
30
4. C*-Algebras
4.1.2 T h e S y m m e t r y of C * - A l g e b r a T h e o r e m 4.1.2.1
( 0 ) (Kaplanski) Every C*-algebra is strongly symme-
tric. It is sufficient to prove the theorem for IK = ~
(Proposition 2.3.1.40). Let E
be a C*-algebra. We may assume E unital (Theorem 4.1.1.13). By Proposition 4.1.1.15,
for every x E Re E . By Corollary 2.4.2.4 b =v a, E is symmetric. By Theorem 4.1.1.16 b),
~(~) = I1~11 for every x E Re E , so that the assertion follows from Proposition 2.4.2.9 c3 =~ cl. m
( 0 ) A commutative real C'-algebra all of whose elements me selfadjoint is a Gelfand C*-algebra. In particular, a C*-subalgebr~ of a Gelfand C*-algebra is a Gelfand C*-algebra.
C o r o l l a r y 4.1.2.2
Let E be a commutative real C*-algebra such that Re E = E . By Theorem 4.1.2.1, /~ is symmetric. Thus by Proposition 2.3.1.29,
for every x E E . Hence E is a Gelfand C*-subalgebra of E .
Corollary 4.1.2.3
( 0 )
L~t ~ b~ ~ C*-~gg~b~a ~
m
A a ~omm=t~t~v~
subset of E . If IK = qJ (IK - JR), then we assume that A U A* is commutative (that A C Re E ). Then the C*-subalgebra of E generated by A is a Gelfand C*-algebra. Moreover, A is contained in a maximal Gelfand C*-subalgebra of E. The complex case follows from Corollary 2.3.2.14 a),b), so we may assume that IK - IR. It is clear that the subalgebra of E generated by A is commutative and that all of its elements are selfadjoint. Its closure, F , has the same properties. By Corollary 4.1.2.2, F is thus a Gelfand C*-algebra. The last assertion follows from Zorn's Lemma (and Corollary 2.2.1.10 and Proposition 2.3.2.11).
9
4.1 The General Theory
31
C o r o l l a r y 4.1.2.4 ( 0 ) Let E be a (unital) C*-algebra and take x a self-normal element of E with E = E ( x ) (E = E(x, 1)). Then the map
~0(E) (o(E))
~o(x),
~'~ ~, ~'(x)
is a homeomorphism.
By Theorem 4.1.2.1 and Corollary 4.1.2.3, E(x, 1) is a symmetric involutive (unital) Gelfand algebra, so the assertion follows from Proposition 2.4.2.10. m C o r o l l a r y 4.1.2.5 ( 0 ) (Gelfand, Naimark, 1943) The Gelfand transform of any (unital) Gelfand C*-algebra is an isometry of (unital) C*-algebras. In particular, a Gelfand C*-algebra E is unital iff a ( E ) is compact. Let E be a Gelfand C*-algebra. By Theorem 4.1.2.1, E is symmetric and by Theorem 4.1.1.16 b),
r(x) = I1~11 for every x C E . By Corollary 2.4.2.7 a ==>b and Theorem 2.4.1.13, the Gelfand transform of E is an isometry of C*-algebras. m Remark. M.H. Stone proved (1940) the above result for Gelfand C*-subalgebras of the C*-algebra of operators on a given Hilbert space.
C o r o l l a r y 4.1.2.6 Let E be a (unital) C*-algebra and F a (unital) Gelfand C*-subalgebra of E . Then the following are equivalent: a) F is separable. b)
There is an x E S n E such that F = E(x) (F = E(x, 1)).
If F is finite-dimensional then
Dim F = Card a ( x ) . In particular, if E = ]Kn,n for some n E IN, then
Dim F < n. a =~ b. We may assume E and F unital. By Theorem 2.4.1.3 d), a ( F ) is metrizable. Hence there is a y e C(a(F)) generating C(a(F)) as C*-algebra (Theorem of Weierstrass-Stone). By Corollary 4.1.2.5, there is an x E F such that F = E ( x , 1). b =v a follows from Corollary 2.3.2.14 c). The final assertion follows from Corollary 4.1.2.4. I
32
3. C*-Algebras
C o r o l l a r y 4.1.2.7 ( 0 ) Let E be a complex (real) C*-algebra. Take x E E and x ~ E E ~ (x~E ReE~). a) b)
If E is unital, then or(x) C ~ ( T ( E ) ) . If E
is unital and x self-normal then ~(7(E))
is the convex hull of
a ( x ) , the extreme points of a(x) are contained in ~(To(E)), there is an x' E To(E) such that
Ix'(~)l = I1~11, and
I1~11 = I1~:11- I1~: I ~o(E)ll. c) If E is unital then x' E E~_ ~ d)
If x ~ E R e E ,
x ' ( 1 ) - I1~'11 9
then there are y',z' E E+ such that
x'= y'- z', e)
E = {0} ** r
IIx'll = Ily'll + IIz'll.
= 0.
a & b & c & d. By Theorem 4.1.2.1, E is strongly symmetric and by Theorem 4.1.1.16 b) r(y)
-
Ilyll
for every self-normal element y of E . With the exception of a) for IK - JR, all other assertions follow from Proposition 2.3.5.15 a =~ b & c & d & e, Corollary 2.3.4.7 and Proposition 2.3.5.10 c) (and Theorem 4.1.1.13). We now prove a) for IK = JR. Take c~ E aE(X). Consider ~'E----~/~,
x,
>(x, 0). o
By Corollary 2.1.5.14, c~ E a~(~x). By the above proof, there is an y' E r(E) such that (~gx, y~) -- ct. By Proposition 4.1.1.27 d), ~'y' E T(E), so
4.1 The General Theory
33
= ( ~ , y ' ) = (~, ~'y') 9 ~(~(E)). e) Assume that E ~ {0}. By Proposition 4.1.1.26, Re E ~= {0}. Take x 9 Re E \ { 0 } . By Proposition 2.3.2.22 l), there is an x' 9 Re E' such that I I x ' l l - 1,
x ' ( x ) = Ilzll.
By d), T(E) ~ {0}. The reverse implication is trivial. Remark.
a) holds even
m
x' 9 E' (Proposition 4.3.6.1.g)).
Co r o llar y 4.1.2.8 Let F be a C*-subalgebra of the unital complex (real) C*-algebra E . Take x' c E' (x' C Re E') such that y' : =
x'lF C F+.
a) If 1 c F then 9' 9 E ~ ~
b) If F is unital then =
x'
a) By Corollary 4.1.2.7 c), y'(1) = x'(1) = IIx'll _> IlY'II-> y'(1), so that
II~'lt = lly'll. b) Denote by lo the unit of F . If lo = 1, then by Corollary 4.1.2.7 c), x'(1) = y ' ( 1 ) = IlY'II = Ilx']l, So assume lo
x' C E+.
1. Then (1 -
10) 2 = 1 -
10-
l o + 10 = 1 -
I11- Ioli = I.
10,
33
~4. C*-Algebras
Take 99 E]0, 2] and x E E such that I1~11 = s i n ~ .
Then
99)21o
x(1
1.
By the hypothesis of b) and Corollary 4.1.2.7 c),
so that 99 Since x is arbitrary, 99 99 99 2sin ~ cos ~11(1 - lo)x'II = sin 991I(1 - lo)x'lI _< 2sin 2 -~ IIx'll
Since 99 is also arbitrary, =
(1 - 10)x' = 0 ,
0 = (1, (1 - lo)x') -- (1 - lo, x') -- x'(1) - x'(lo),
x'(1) = x'(10) = y'(10) = {]Y'[I = []x']l by the hypothesis of b) and Corollary 4.1.2.7 c). By Corollary 4.1.2.7 c) again, !
x' E E + . Corollary
4.1.2.9
I
If T is a locally compact space and # is a bounded Radon
measure on T , then # is positive iff
, ( T ) : Ilall.
4.1 The General Theory
35
If T is compact, then the assertion follows from Corollary 4.1.2.7 c). If T is not compact, let T* be its Alexandroff compactification. Then # may be identified canonically with an element of C(T*)' and the assertion follows from the above remark (and Corollary 4.1.1.14). C o r o l l a r y 4.1.2.10 x 6 E.
I
( 0 ) Let E be a Gelfand unital C*-algebra and take
Then the following are equivalent:
a)
x is not invertible.
b)
x is a topological divisor of O.
a ==v b. By Corollary 4.1.2.5, we may assume that E = 0 ( T ) , where T is a compact space. There is a to C T , with x(to) = 0. For each n C IN, put Xn "-- eT -- e T A InxI ,
where A denotes the infimum. Then x, E C(T), O <_ Xn <_ eT,
I(xx.)(t)l __ for every n
and t
IIX,I I = I ,
xn(to) = l ,
n
~ll
T . Hence
IIx~.ll _< lll~ll n
for every n C IN and lim x x n = O.
n---+oo
Hence x is a topological divisor of 0. b =v a follows from Proposition 2.2.4.25. C o r o l l a r y 4.1.2.11
I
Let F be the C * - h u l l of the involutive c o m m u t a t i v e com-
plex B a n a c h algebra E . Let u" E -+ F be the canonical map and T the closed subspace
{x' e a ( E ) I x' is involutive} of a ( E ) .
Then F is commutative, u ' ( a ( F ) ) : T , and the map a(F)
is a h o m e o m o r p h i s m .
>T ,
y' ,
> y' o u - u' y'
36
4. C*-Algebras
F is commutative by Proposition 2.1.1.13 e). Take y' C a ( F ) . Then y' is involutive (Theorem 4.1.2.1, Proposition 2.4.2.3 a =~ d), so that u'y' E T , i.e.
u'(a(F)) C T . By Proposition 4.1.1.22 f), the map a(F)
~T,
y' ,
~ u'y'
is bijective. This map is obviously continuous and if a(F) is not compact then
u'y ~ converges to the Alexandroff point of T whenever yt converges to the Alexandroff point of a ( F ) . Hence the above map is a homeomorphism. I Theorem 4.1.2.12
( 0 ) (Rickart, 1946)
Let F be a C*-subalgebra of the
C*-algebra E . Then o~(~) = o~(~) for every x E F . We may assume that E and F are unital (Theorem 4.1.1.13). Assume that x is not invertible in F . We want to show that x is not invertible in E . Case 1
F is a Gelfand C*-algebra
By Corollary 4.1.2.10 a :=v b, x is a topological divisor of 0 in F . Then x is afortiori a topological divisor of 0 in E . Hence by Corollary 4.1.2.10 b =~ a, x is not invertible in E . Case 2
x self-normal
By Corollary 4.1.2.3, F(x, 1) is a Gelfand C*-algebra. Since x is not invertible in F(x, 1), it is not invertible in E by the first case. Case 3
The General Case
Assume that x is invertible in E . Then x* is invertible in E as well (Proposition 2.3.1.14). Hence x*x and xx* are invertible in E (Corollary 2.1.2.6). By Case 2, x*x and xx* are invertible in F and so x is invertible in F (Corollary 2.1.2.6), which is a contradiction. Hence x is not invertible in E . By the above result, c
The reverse inclusion follows from Corollary 2.1.3.14 a).
I
( 0 ) Let E be a (unital) C*-algebra. Then the following are equivalent for every x C Sn E :
C o r o l l a r y 4.1.2.13
4.1 The General Theory
a)
37
x is selfadjoint ( x is unitary).
a v:> b. We define
(F "= E ( x ,
F "- E ( x )
1)).
Then F is a Gelfand C*-algebra (Corollary 4.1.2.3) and o~(z) = o ~ ( z ) ,
by Theorem 4.1.2.12. By Corollary 4.1.2.5 (and Corollary 4.1.2.6), we may assume that F = C(T), with T a locally compact (compact) space. But in this case the assertion is trivial. I C o r o l l a r y 4.1.2.14
Let E be a symmetric involutive complex Banach alge-
bra, F a complex C*-algebra, and u : E -+ F an algebra homomorphism. I f
u ( R e E ) C No F (e.g. F is commutative), then u is involutive and continuous and Ilull <_ 1.
Take x E Re E . Then
~(~x)
~ ( ~ ) u {0}
(Corollary 2.1.3.12, Proposition 2.3.1.29), so u x E Re F (Corollary 4.1.2.13 b => a). By Corollary 2.3.1.23 b ~ a, u is involutive. By Corollary 4.1.1.20, u is continuous and IiuII < 1. P r o p o s i t i o n 4.1.2.15
I
( 0 ) Let E , F
be Gelfand unital C*-algebras and
u" E --+ F a unital algebra homomorphism.
a)
There is a unique map
~- o(F)
>~(E),
such that F
E
ux--xop for every x C E .
b)
u is continuous and involutive, I}ull = 1, and
38
4. C*-Algebras
c) ~ is continuous. d)
u is surjective (injective) if] (p is injective (surjective).
e) u is bijective iff ~ is a homeomorphism. f)
If there is an x e E , with E = E(x, 1) and F = F(ux, 1) and if we identify a(E) with aE(X) and a(F) with a f ( u x ) via the homeomorphisms E
F
and ~"~, respectively (Corollary ~.1.2.~), then a(F) C a(E) and ~(~) = ~
for every a E a ( F ) . We denote by v and w the Gelfand transforms of E and F , respectively. Then v and w are isometries of unital C*-algebras (Corollary 4.1.2.5), and so uo . -
~ o ~ o v-'
. C(~(E))
--,
C(o(F))
is a unital algebra homomorphism. a),b),c),d),e) follow from Proposition 2.4.3.6 and Proposition 2.4.1.17 a),b). f) We have that o(e)
- ~(~)
c ~(~)
- o(E)
(Corollary 2.1.3.12). Take c~ C aF(UX). Let y' C a(F) such that F
~(y')
-
~.
By a), E
. = ~(~(~))
so that ~(~)
Corollary 4.1.2.16 C*-algebra E .
-
~.
1
( 0 ) Let F be a unital C*-subalgebra of the unital
4.1 The General Theory
a)
39
There is a unique map ~" a ( E )
~ a(F),
such that E F X=XO~O
for every x c F .
b) x' C a(E) :=> ~(x') = x ' l F . c) ~ is continuous and surjective. d) ~ is bijective iff E = F . e)
Every character of F can be extended to a character of E .
f) If there is an x e F with F = F(x, 1) and if we identify a(F) with a(x) F
E
via the homeomorphism ~ (Corollary 4.1.2.4), then ~ -
~.
a),b),c), and d) fol!ow from Proposition 4.1.2.15 a) ,b) ,c) ,d) ,e) . e) follows from b) and c). f) By E
F
~(~') = ~(~(~')) F
for every x' e or(E). Since we have identified or(x) with a(F) via ~ we see E
~. Corollary 4.1.2.17
m
( 0 ) Let E be a unital Gelfand algebra and ~ an
upward directed set of unital C*-subalgebras of E the union of which is dense in E . Define ~F,C" ~(G)
>~ ( F ) ,
x', ~ ; x'lF
for all unital C*-subalgebras F, G of E with F C G.
a) (o-(F), PF,a)~ is a projective system of compact spaces such that (PF,a is surjective .for all F, G E ~ , F C G ; call T its projective limit.
b) ~F,E = PF,C o ~G,E for all F , G C ~ , F C G ; call ~ " or(E) --+ T the projective limit of (~F,E)F~;~-
40
4. C*-A19ebras
c)
~ is a homeomorphism.
d)
/f a ( F )
is totally disconnected for every F
E ~
(e.g. F
is finite-
dimensional), then a ( E ) is also totally disconnected.
a) follows from Corollary 4.1.2.16 a),b),c) (and Theorem 2.4.1.3 c)). b) is obvious. c) 99 is continuous. By a), it is surjective. Let x', y' E a ( E ) with 99(x') = 99(y'). Then x' = y' on
U F . Since this set is dense in E , we get x' = y' FE~ (Proposition 2.2.4.19). Hence 99 is injective and so it is a homeomorphism. d) follows from c).
D e f i n i t i o n 4.1.2.18
m ( 0 )
The selfadjoint idempotent elements of a C*-
algebra are called orthogonal projections. If E is a C*-algebra, then we denote by Pr E the set of orthogonal projections of E .
If F is a C*-subalgebra of E , then Pr F C Pr E . The reason for the name "orthogonal projection" will appear in Proposition 5.3.2.8. Example
4.1.2.19 Pr s
[ = {0, 1 }
COS2 99 e i0 cos 99 sin 99 ] e -/~ cos 99 sin 99 sin 2 99
If p, q E P r s with s
2) =
99,0
IR ,
(~e2z q ~ : ~ )
~
with pq = O, then p + q = 1. If we identify s
2)
' as in Example 2.3.5.3, then T0(E(IK2)) coincides with
Pr s Take a, 5 E IR and /3 E ]K. Then we have
[
/~
=
~
[
~+191 ~ 9(~+~) ~(~ + ~) ~ + IZl ~
]
In particular,
[
cos 2 99
e i~ cos 99 sin ~
e -w cos 99 sin 99
sin299
]2[ =
cos 2 99
e iO cos 99 sin 99
e -w cos 99 sin 99
sin299
l
4.1 The General Theory
41
for all ~, 0 CIR. If
[
a
fl ] e P r s
1}
then
{
~2+lfll ~_~, 6 ~ + I;~1~- = 6,
fl(~+6)=fl ? ( ~ + 6) = ?
'
so that
a, S e l R + , Hence there are ~ , 0 c I R ( w i t h a=cos 2p,
a+5=1, r9r
Ifll 2 = a 5 .
~ iflK=IR)
6 = s i n 2w,
with
fl=e i~
This proves the first assertion. By the above considerations, there are ~ , r
C IR
0 p (with ~,~
E
if IK = IR), with
P=
q
[
--
e -~~ sin
cos r e -~p sin
]
0sin l
] [cos ~p e i~ sin ~p].
Suppose first that sinqa#O,
sinr
It follows from pq = 0 that cos qa cos ~b + e i(~176sin ~ sin ~p = O, so either
~p
C ~ or
O-p-Tr 2~ C 2Z.
In the former case,
cos(~ - r
=o
cos(~ + r
= o.
and in the latter,
We get p + q = 1 in both cases. The same is true if s i n p = 0 or s i n ~ = 0. The last assertion follows from Example 2.3.5.3.
m
42
4. C*-Algebras
E x a m p l e 4.1.2.20
Let E be a Gelfand C*-subalgebra of E(IK 2) of dimen-
sion n . Then n < 2, E is generated by a family (Pi)ieINn in Prs with PiPj = 0 for all distinct i , j C INn, and every character of E is the restriction to E of a pure state of/~(IK2). /f n = 2, then the unit of/Z(IK 2) belongs to E .
By the Gelfand transform of E , a ( E ) contains exactly n points and there is a family (Pi)~e~n in Pr/~(lK2)\{0} generating E , with PiPj = 0
for all distinct i , j E INn. By Example 4.1.2.19, n <_ 2 and every character of E is the restriction to E of a pure state of L:(IK2). The last assertion follows from Example 4.1.2.19. P r o p o s i t i o n 4.1.2.21
m ( 0
)
Let x be a self-normal element of a C * -
algebra. Then the following are equivalent:
a)
x is an orthogonal projection.
b)
x is idempotent.
c)
a(x) C {0,1}.
In this case
Ilxll
e {o, 1}.
a =v b is trivial. b =~ c follows from Corollary 2.1.3.7 (and Corollary 2.1.3.2). c =v a. By Corollary 4.1.2.13 b =v a , x is selfadjoint. But then x 2 - x is selfadjoint and by Theorem 2.1.3.4 b),
o(x ~
x) c (o},
so that r ( x ~ - z ) = O.
By Theorem 4.1.1.16 b),
X2--x--O. Hence x is idempotent. The last assertion follows from Theorem 4.1.1.16 b).
4.1 The General Theory
[
Corollary 4.1.2.22
(
0
43
) If x is an element of a C*-algebra E , then the
following are equivalent: a)
x*x is idempotent.
b)
xx* is idempotent.
C)
XX* X -- X.
a ==> b. By Proposition 2.1.3.10 b), ~(x~*)\{0}
- ~(x*~)\{o}.
Hence xx* is idempotent (Proposition 4.1.2.21 c r
b).
b =~ c. We have Ilx-
xx*xll ~ = I1(~ - ~ x * ~ ) ( x * - x * x ~ * ) l l =
= Ilxx* - x x * z x * - x ~ * x ~ * + ~ x * ~ x * ~ x * I I -
0,
so that X :
XX*X.
c =~ a. We have
(X*X)2--X*XX*X--X*X.
Proposition 4 . 1 . 2 . 2 3
1
( 0 ) If x , y are elements of a C*-algebra and p is
an orthogonal projection, then Ilpx(1 - p) + (1 - p)ypll = s u p { Ilpx(X - p ) I I ,
I1(1 - p)ypll } .
Put u " - (1 - p)x*px(1 - p ) ,
v "= py*(1 - p)yp.
Then u and v are selfadjoint and UV :
VZt - - O .
Hence the C * - s u b a l g e b r a of E generated by {u, v} is a Gelfand C*-algebra (Corollary 4.1.2.3). By means of the Gelfand transform, we get
Ilu + vii- sup{l[~ll, Ilvll}
44
4. C*-Algebras
(Corollary 4.1.2.5). We deduce I l p x ( 1 - p ) + (1 - p)ypll 2 =
=
= I1(1 -
I1((1 -
p)x*p + py*(1 - p))(px(1 - p) -4- (1 -
p)x*px(1 - p) + p y * ( 1 - p ) y p l l = sup{llpx(1
Proposition
4.1.2.24
p)yp)ll
Ilu + vii = s u p { l l ~ l l ,
- p)ll 2 , I1(1 -
p)ypll~}.
=
Ilvll} -
I
If E is a unital C*-algebra, then x*(Pr E ) x = Pr E
.for every unitary element x of E . Take p C Pr E . Then x*px is selfadjoint and
(x*px) 2 - x*pxx*px - x*p2x - x*px, so that x*px c Pr E , x*(Pr E ) x C Pr E ,
x ( P r E)x* C Pr E ,
Pr E = x*x(Pr E ) x * x c x* (Pr E ) x c Pr E ,
I
x*(Pr E ) x = Pr E . Proposition
4.1.2.25
If E is a unital C*-algebra and A "- ReE N UnE,
then the map PrE
)A,
p~
)2p-1
is bijective and .~PrE,
is its inverse.
x:
,'~l ( x +
1)
4.1 The General Theory
Now 2 p - l 9
45
and (2p- 1)2= 4p2_ 4p+ 1-
1,
so that 2 p - 1 9 A, for every p 9 P r E . We have 5l(x + 1) 9 R e E and ( ~(x +) 1) 2 1= ~(x 2 + 2x + 1) -- 1 ( x + 1 ) , so that 51(x + 1) 9 P r E for every x 9 A
I
Let E be a Gelfand C*-algebra. If a ( x ) \ { 0 } is discrete for every x 9 E , then a(E) is discrete.
P r o p o s i t i o n 4.1.2.26
Take x' c a ( E ) . Assume that {x'} is not open. We construct inductively a strictly decreasing sequence (Kn)nE~ such that for each n C IN, Kn is a compact open set of a(E) containing x'. Take n E IN and assume the sequence has been constructed up to n -
1. Take y' 9 K n _ I \ { x ' } (Ko := a(E)) and
f 9 Co(a(E)) with f (x') = l ,
f (y') = O ,
SuppfcKn_l.
Since the Gelfand transform is surjective, there is an x C E with ~ -- f . By hypothesis, a ( x ) \ { 0 } is discrete. Then --1
Kn "-- f (1) is a compact open set of a(E) containing x' and strictly contained in Kn. This completes the inductive construction. Let g denote the function on a(E) which takes the value n-__A1 on K n - I \ K n n for every n C IN and 1 on
N Kn. Then y C Co(a(E)). Using the surjectivity nEIN
of the Gelfand transform again, there is a y C E such that ~ - g. But then =
n
nE]N
U{1}
and 1 is not an isolated point of a(y). This contradicts our assumption, so {z'} must be open and hence a ( E ) is discrete. P r o p o s i t i o n 4.1.2.27
I
( 0 ) Let E be a complex (real) C*-algebra without
unit, ~ an approximate unit of E , and IK x E the unital C*-algebra associated to E .
36
,4. C*-Algebras
a)
(x' 9 R e E ' ) .
lim x' (x* x) exists for every x' 9 E' x,~
For x' 9 E' (x' 9
Re E') define x 1' "IKxE
>IK,
(a,x):
;x'(x)+c~limx'(y*y)
and for y' 9 (IK x E)' define !
Yo E
~ IK,
x,
~ y'(O,x).
b) (c~,x) 9 IK x E =~ I_~ < II(~,x)ll- lim IIx-4-~Yll I~1 < II(~,x)ll. c)
x' 9 E' ~ x~ e (IK x E)', IIx'll = Ilxill, (xl)* - (:r
d)
For every y' 9 (IK x E)' and (c~,x) 9 IK x E ,
x ' = Xlo.
( ( a , x ) , y ' - y;,) = cr(y'(1 O) - limy'o(y*y))
Ily'- yg~ll- ly'(1 o ) - limy~o(y*y)l y,~ !
!
Yo = Yolo. e)
x ' E E~_ =~ X '1
9
(z
X
E)~
.
v'
'
(~
E)~
g)
y' c a~ (~K • E)' ~ IlY'II = l Y;II + I 1 r
h)
x' 9 Re E', y' e Re (IK x E)', x ' = y~, IIx'll = Ily'll ~ y ' = x~.
i)
For every x' C E' (x' C R e E ' ) , x' E E+ ~
Y;~II.
limx'(x*x) = x,~:
j)
I1~'11.
Themap E'
> (IK•
x',
!
>x 1
is an involutive homomorphism of E-modules (where E is identified with
(o} • E).
4.1 The General Theory
47
By Proposition 2.3.4.13 b), E' is an ordered Banach space and E~v is the set of continuous positive linear forms on E . a) If x' is positive, then by Proposition 2.3.4.10 a) lim x'(x*x) = x,~
IIx'll 9
If x' is selfadjoint, then by Corollary 4.1.2.7 d), it is a difference of two positive linear forms and the above limit exists in this case as well. If ]K = C , then the general case follows from the decomposition
x' = re x' + i im x ~. b) and c)
Step 1
I1(~,~)11 =
limllx +~Yll y,~
Take y, z E E # . Then
II~z + o~yzll < IIx + ~yll Ilzll < I1~ + ~yll <__ IIx - xyll + Ilxy + ~yll--= I1~- ~yll + I1(~, x)(O, y)ll ~ IIx- ~yll + I1(~, x)ll I1(o, y)ll
IIx- ~yll + ll(~,x)ll, so that
I1(~, x)zll - Ilxz + ~zll
-
lim
Ilxz + o~zyll < lim inf fix + ~Yll
and
ll(~,x)ll-
sup II(~,x)zll ~ lim infllx + c~y[I z C E--/#
Y ,~
lim sup IIx + ~Yll ~ lim sup IIx - ~Yll + I1(~, ~)11 ~ I1(~, x)ll y,~
y,~
(Theorem 4.1.1.13). Hence
I1(~, x)ll = lim I x + ~YlI. Y,~
Step 2
I~1 _< il(~,~)ii
Put
y'.~•
>~(,
(Z,y),
~Z.
.~8
4. C*-Algebras
Then y' C ~-(IK • E) so t h a t
I~1 = ly'(~,z)l < i](~,~)l]. Step 3
c)
I(z'(x) + ~z'(y*y)i = I~'(~ + ~y*y)l < II~'ll IIx + ~y*yil for every y E E . Thus Ix~(a,x)l = lim y,~ ]x'(x) + ax'(y*y)] <- lix']l lim y,~ lix + ay*yll = [Ix'l] I](a, x)ll ' by Step 1 and Proposition 2.3.4.9. Hence x~ is continuous and
il~',il < li~'ll. The reverse inequality and the relation ~--- Xl0
are trivial. Given (c~, y) E IK • E , ((a, y), (x'l)*) = ((~, y*), x~) = x'(y*) + alimx'(z*z) = z,~
= x'*(y) + alimx'*(z*z) = ((a,y), (x'*)i), z,~
so that (z',)*
Step 4
=
(z'*),
.
I1~1__]_< ii(a ' x)l]
Take x' C E # with
IlzJl- ~'(~) 9 Then
9 '(~) - ~'1(0, ~) < I~'1(0, ~) + ,'~(~, o)1 + I~'1(~, 0)1 =
= Ix'l(c~, x)] + Is ]limx'(y*y)l < lix'l]l I](a,x)ll + lal _< Y,~
4.1 The General Theory
(Steps 2 and 3). It follows t h a t <
d) We have t h a t
((a,x),y'-
!
Yol) = y'(o~,x) - y'(0, x) - a limy~o(y*y) = Y,$
= c~(y'(1, O)-
limyo(y*y)) y,~
so, by b), '
y,~
-
-
< II(a,x)ll ly'(1 ' O) - l i m y'o(y*y)l. y,~ Hence
IlY'- Yolll -< lY'(1, O) -limy~o(Y*Y)l y,~
=
!
= I((1, O ) , y ' - Y ~ I ) I - < IlY'-Yo~ll,
Ily' - y~,ll = ly'(1, 0) - lim y~(y*y)J. y,~
By the first equality, !
!
Yo - Y01o -- 0.
e)
If ( a , x ) C IK • E , then ~'~ ((~, ~ ) * ( ~ , x)) = x',(l~l~, ~
+ ~x* + ~*~) -
= x' (~x + c~x* + x ' x ) + [c~l2 lim x' (y'y) Y,~
- lim x' (-~y*x + o~x*y + x*x + lal2y *y) = lim x'((o~y + x)* (a.y + x)) E lR+ y,~ y,~ by Proposition 2.3.4.9, so t h a t x~ is positive.
49
50
4. C*-Algebras
f) It is obvious that y~) is positive. By e), Y~I is positive as well. By d) and Corollary 2.3.4.7, if (a, x) C lK x E , then
((~,~)
9
(~,x),y'-
yo~) = ((1~1~, ~x + ~x* + x * x ) , y ' - yo~) =
= Ic~12(y'(1, O ) - limY~o(y*y)) >
> lal2(lly'l[- Ilimy'( 0 y*y)[) > o so that y ' - Y ~ I is positive. g) First suppose that y' is positive. By f), Y~I and y ' - Y{n are positive, so that Ily'll = Ily;,ll + Ily'- Gll = Ily;ll + Ily'- y;,ll
by c) and Corollary 2.3.4.14. Now let yt be arbitrary. By Corollary 4.1.2.7 d), there are u t, v' C ( ~ x E)~ , with y'= ,'-
v',
Iiy'II = II-'II + Iiv'II.
Then I
I
I
YO1 "-" ~01 -- Vo1 ,
and, by the above considerations, Ily'll < Ily'- y;,ll + Ily;,I <
< l l - ' - ~o, ' ll + II~o~ll ' + IIVo~ll' ' l l + l l v ' - Vo,
= II-'ll + IIv'll = Ily'll,
IlY'II ~ - I l Y ' - -
Y;1 ]--~-IlY;lll 9
h) By g), I
Ily'- yo~]l = o,
so that
4.1 The General Theory
!
Y' -- Y01 =
51
! Xl
9
i) By c),e), and Corollary 4.1.2.7 c), (x' 9 E+) ~
(x'1 9 (IK • E)+) ~
Ilx~ll)
(X'l(1 ' 0 ) =
~==> ( lxi~ m x ' ( x * x ) = IIx'll).
j) Take (x,x') 9 E • E ' . If (c~,y) 9 IK • E , then
( ( , , y), (0, x)x'l> - ((~, y)(0, x), ~ ) - ((0, ~
+ yx), ~'1> =
= ((~x + yx, x'> = (yx, x'> + ~ lim(z*zx, x') = z,~
= (y, xx'> + ~rlim(z*z, xx') - ((~ y) (xx')x) (Proposition 2.3.4.9), so that (xx')~
= (o, x)x'l .
Similarly, (Xt X)l --- Xl' (o, x).
Hence the map
E'
>(]K•
x',
~x~!
is a homomorphism of E-modules. By c), this map is involutive.
Remark. Because of c) and j), we may identify E' with an E-submodule of (]K • E ) ' . i) holds even for IK = ]R and x' e E' (Proposition 4.3.6.1 g)). E x a m p l e 4.1.2.28 ( 7 ) Let E be a nite) family in Pr E such that
(unital)
C*-algebra, (p~)~, an (infi-
PtP,x -- 5~pt for all ~, A E I , and F (resp. G ) the (unital) C*-subalgebra of E generated by {p~ ] ~ E I } . We endow I with the discrete topology (and denote by I the Alexandroff compactification of I ). a) F and G are Gelfand C*-algebras.
52
b)
4. C*-Algebras
Given (a~)~e, e co(I), (aLp~)~c, is summable and the map
~o(I) ~
F,
(~<)<~" >Z~
is an isomorphism of C*-algebras. c)
The isomorphism of b) may be extended uniquely to an isomorphism c(I) ~ G of unital C*-algebras.
d) a(F) (resp. a(a) ) is homeomorphic to I (to I ). a) follows from Corollary 4.1.2.3. b) follows from Proposition 4.1.1.25. c) Let ~ be the filter on I of cofinite subsets of I , i.e.
~d "= { I \ A I A E glI(I)}. Given a C c(I), put := lima,. e,~
By b), the map u" c(I)
> C,
a:
~ ~l + E ( a ~ - ~ ) p ~
is well defined and bijective. It is obviously linear, involutive, and extends the map of b). Take a, 13 C c(I). Then
= ~Zl + Z(~(Z<
- Z) + (.< - ~ ) Z + (.< - ~)(z< - Z))p< -
LEI
eel
-- a'--~l + E ( ( a I 3 ) L - a-'~)pL- u(c~13). LEI
Hence u is an algebra homomorphism. By Corollary 4.1.1.21, u is an isomorphism of C*-algebras. d) follows from b),c), Example 2.4.3.2, and Example 2.4.3.5. I
4.1 The General Theory
Proposition
4.1.2.29
(
)
53
unital C*-algebra, F a unital C*-subalgebra of E , and (P~)~I a family in P r E such that Let E
be a
F w {p, I~ 9 I} c {p, I~ e I} ~ and such that
Hp,#o, ~p,#~ t6J
tEJ
for every J 9 q3S(I), J :/: O. Let further G and H be the unital C*subalgebras of E generated by {p~ I ~ 9 I} and F U {p~ [ ~ 9 I } , respectively.
a) G is a Gelfand C*-algebra and the map a(G)
>{0,1}1,
x',
> (x'(p,)),ei
is a homeomorphism.
b)
The unital C*-algebra C(a(G),F) of continuous maps of a(G) into F (Example 4.1.1.6) is isomorphic to H .
c)
The following are equivalent for every (a,)~ei 9 IK''
el)
(OQ)t.6I 9 e l ( I ) .
c2) /3 9 ]K ==> (oz,(/31 + P,)),eI is summable. %) (a,(1 +p,)),c, is summable. c4) (a~p~)~I is summable. If these conditions are fulfilled, fl 9 IR+, and ((~)~I 9 IRI then:
c~) II E ~ (ill + p~)ll = (1 + ~) tel
E ce~.
t6I
Take J 6 q3S(I ) . We denote by Gj and Hj the unital C*-subalgebra of E generated by {p~ I ~ 9 I} and F U {p~ It 9 I}, respectively, and set
for all K C J , where l-IP~ = H ( 1 - p ~ )
= 1.
54
4. C*-Algebras
Then pJ, l( C P r E for all K c J ,
PJ,KPJ,L = 0 for all distinct K, L C g l ( J ) , and EPJ, KCJ
K =
1.
a) Let J e ~ i ( I ) . Then
a~ = { ~ ~pj,~ ( ~ ) ~
e ~(~)}.
Hence G j is a Gelfand C*-algebra and the map a(Gj)
> {0,1} J,
x',
>(x'(p~))~ea
is a homeomorphism. (Gj)jeVi(1) is an upward directed family of unital C*-subalgebras of G the union of which is dense in G. Hence G is a Gelfand C*-algebra and by Corollary 4.1.2.17 c), the map
o(a)
~{0,1}', x', ~(~'(p~))~,
is a homeomorphism. b) Let J E 9i3i(1). The map
F ~(J)
). H j ,
(XK)KCJ'
>E
XKpJ, K
KCJ
is an isometry of unital C*-algebras (Corollary 4.1.1.21). We identify Hj with F v(J) via this map and set
"2" a(G)
>F,
x',
>E
x'(pj, K)XK
KCJ
for every x C H j . (Pj, KF)Kcj is a family of C*-subalgebras of E , H j is the C*-subalgebra of E generated by
I,.J Pj, KF, and KCJ
(pj, KF)(pj, LF) = {0} whenever K, L C g3(J) are distinct. By Proposition 4.1.1.25, I[~l]- sup [IXK]] = Ilxll KCJ
4.1 The General Theory
for every
55
x E Hj. C (a(G), F) the unital C*-algebra of continuous maps of a(G)
Denote by
into F . By the above considerations, the map
U
Hj
;C(a(G),F),
x,
~
J E~.f (I)
is an involutive unital algebra homomorphism preserving the norms. Since U
Ha is dense in H , the above map may be extended to a homomorphism
JE~f(I)
of C*-algebras u: H
> C(a(G), F )
which preserves the norms. We have to show that u is surjective. Take
y C C(a(G), F) and e > 0. Since y is uniformly continuous, there is,
by a), a J C gig(I) such that for all K C J and --1
x', y' E Pj,K(1) we have
Ily(x') - y(y')ll < ~. -1
Given K C
J, take x Ig C Pj, K(1). We set x := ~
y(x~)p,,~.
KCJ
Then x E H j and
I 1 ~ - yll < ~Since e is arbitrary and since
u(H) is closed, we see that y C u(H). Hence u
is surjective. cl =:~ c2, c2 =v c3, and c2 =v c4 are trivial. c3 ==~ cl,c4 =v cl, and c2 :=v c5. By a), there is an
x' c a(G) such t h a t
x'(p~) - 1 for all ~ C I . By Proposition 1.2.1.16, (a~)~e/ is summable. Moreover, if (a~)~e/e IR / and /3 e IR+ then
(l +13) Z a~ = ~,eI x'(a~(131+ P')) = x' ( ~
~EI
LCI
aL(t31+ ) )~P
LEI
I
56
4. C*-Algebras
4.1.3 F u n c t i o n a l calculus in C * - A l g e b r a s L e m m a 4.1.3.0
( 0 ) Let K be a compact set of IK such that K\{0} is
discrete. Given P 9 IK[t], define IK,
P'K
a,
>P ( a ) .
Put ,.,o
~ ' : = { P I P 9 IK[t]}, :=
{PIP
e IK[t], (0 9 g
==~ p ( 0 ) - 0 ) } .
Then yz is dense in C(K) and G is dense in {x 9 C(K) I O 9 K ==~x(O) - 0}.
Take a 9 K\{0}. Put A "= {/3 9 g ll~ I >_ lal,/3 :# a} x'K
>IK,
/~J
>5~,~,
and
11 ( t - ~) P~(t) := for every n 9
Then Pn 9 ~,
11 ( ~ - ~ ) BEn
~
Pn(a) = 1,
Pn - 0
on A for every
n 9 IN, and (P~)~e~ converges uniformly to x. Hence x 9 G. The assertions now follow easily. I T h e o r e m 4.1.3.1
( 0 ) (Gelfand-Naimark, 1943) Let x be a self-normal element of a unital C*-algebra E . Put F := E(x, 1),
j:a(x)
>IK,
~,
>~,
and let v be the Gelfand transform of F .
a)
There is a unique involutive unital algebra homomorphism
~:c(o(~)) with
~ E,
4.1 The General Theory
b)
57
u(C(a(x))) = F and the map
is an isometry of unital C*-algebras. In particular, f (x) is self-normal
fo~ ~w~y : e c(~(~)). c) f 9 C(a(x)) ==~f ( x ) = v -1
(
f o
, f~('x) = f o ~.
d)
E ( x ) = { f ( x ) ] f 9 C ( a ( x ) ) , (0 9 a(x) :=v f(O) = 0 ) ) .
e)
If a(x)\{0} is discrete, then
{f(x) I f 9 C(a(x)) , (0 9 a(x) ==~ f(0) = 0)},
9 C(a(x))}) is the closed (unital) subalgebra of E generated by x .
f)
Let F be an E - m o d u l e and a 9 F ax = ax* = O
such that (xa = x*a = O).
Then a f (x) - 0
( f (x)a -- O)
for every f 9 C(a(x)) with
0 e a(x) ==~ f(0) = 0. a) F is a Gelfand C*-algebra (Corollary 4.1.2.3), v is an isometry of unital C*-algebras (Corollary 4.1.2.5), and F
for every y E F (Theorem 4.1.2.12, Corollary 2.4.1.7 a)). We define u
:
Then u is a continuous involutive unital algebra homomorphism with
58
4. C*-Algebras
?.tj - -
X.
Let 9c(o(x))
~E
be an involutive unital algebra homomorphism such that wj:x.
Given P c lK[s, t], put
P- o(.)
IK,
a,
~ P(a,~)
and Y" := { P I P
9 IK[s, tl}.
Then .7" is an involutive unital subalgebra of C(a(x)) which separates the points of a ( x ) . By the Weierstrass-Stone Theorem, .7" is a dense set of C(a(x)). We have that u P = P ( u ( j ) , u(j)) - P ( u ( j ) , u(j)*) - P ( x , x * ) -
= P ( w ( j ) , w(j)*) - P ( w ( j ) , w(~)) = w P for every P E IK[s, t]. Thus u and w coincide on .T. Since they are continuous (Corollary 4.1.1.20) and .7" is dense in C ( a ( x ) ) , they coincide. b) and c). Since the map F
o(r) ---+ o(~),
x',~ ;~(~')
is a homeomorphism (Corollary 4.1.2.4), the map F
C(a(x)) ~
C(a(F)),
f,
~f o
is an isometry of unital C*-algebras. Hence the map F
C(a(x)) ~
F,
f,
> v - l ( f o~) - f ( x )
is also an isometry of unital C*-algebra. Given f e C( a ( x ) ) , normal since F is a Gelfand C*-algebra. d) We put
f ( x ) is self-
4.1 The General Theory
~0 := {f 9 ~ ' ] 0 9 a(x) ~ where F
59
f(0) -- 0},
is the set defined in the proof of a). By Corollaries 1.3.5.15 and
1.3.5.16, ~'0 is a dense set of
{ f 9 C(a(x)) I0 9 a(x) ====vf(0) = 0}. Given f 9 ~o, f (x) 9 E(x) and so {f(x) I f e C(a(x)), (0 9 a(x) ~
f(0) = 0)} C E ( x ) ,
by a). On the other hand, { f ( x ) i f e Y0} is a dense set of E ( x ) , so that
{f(x) l f e C(~(x)), (0 C ~(x) ==> f(0) - 0 ) } - E ( x ) , by b). e) follows from b) and Lemma 4.1.3.0. f) If 9v0 is the set defined in the proof of d) then
af(x) = 0
(f(x)a = O)
for every f C $'0. It follows
a f (x) = O (f (x)a = O) for every f e C(a(x)) with 0 e o-(x) ::::=> f(O) -- O.
I
Remark. Let F, G be unital C*-subalgebras of the unital C*-algebra E . Let x be a self-normal element of E belonging to F N G and take f C C(a(x)). Then
E(x, 1) = F(x, 1) = G(x, 1) and so the element f ( x ) defined in the above theorem by means of E, F , and G coincide. In particular, if E is a real C*-algebra, x c Re E , f C C(a(x)), and /~ denotes the complexification of E , then f ( x ) defined with respect to E on E coincide. ( 0 ) Let E be a C*-algebra without unit. Let F be the unital C*-algebra associated to E . Take a self-normal element x of E and take f e C(a(x)). Let f ( x ) denote the element of F defined in Theorem ~.1.3.1. If f(O) = O, then f(x) C E (Theorem ~.1.3.1 d)).
D e f i n i t i o n 4.1.3.2
60
4. C*-Algebras
C o r o l l a r y 4.1.3.3
( 0 ) Let E be a C*-algebra. Take x E R e E and put .T" := {f E C ( o ( x ) ) l f ( a ( x ) ) C IR},
G := {f E .7"10 E a(x) ~
f(0) = 0}.
a)
{ f (x) l f E ~} is the closed real subalgebra of E generated by x and it is contained in Re E .
b)
If E is unital, then { f (x) l f e Y:} is the closed unital real subalgebra of E generated by x and it is contained in Re E . By Corollary 4.1.2.13 a ~ b , a(x) C l R . a) (rasp. b)). Let F be the closed (unital) real subalgebra of E generated
by x. Since {f(x) [ f E G (f E ~)} is a closed (unital) real subalgebra of E containing x, it contains F . Take f E G (f E ~'). By Corollaries 1.3.5.15 and 1.3.5.16, there is a sequence ( P n ) n ~ in JR[t] converging uniformly to f on a(x). If 0 r a(x) or if 0 E a(x) and f(0) = 0, we may additionally assume that Pn(0) = 0 for each n E IN. Thus, by Theorem 4.1.3.1 b),
y(x) = lim Pn(x) E F fq Re E . n--~.oo
Hence
F= {f(x) I lEG
(fE~)}CReE.
I
( 0 ) Let E , F be C*-algebras and u " E--+ F an involutive algebra homomorphism. Take x E Sn E and f E C(a(x)) such that
C o r o l l a r y 4.1.3.4
0 E or(x) ~
f ( O ) = O.
Then u f (x) : f (ux) . If E, F , and u are unital, then we may drop the hypothesis that o e o(z) ~
f(O) = O.
If u is injective, then
II~yll = Ilyll for every y E E .
4.1 The General Theory
61
Since
u(E(x))
F(ux),
(u(E(x, 1))C F(ux, 1))
(Proposition 2.3.1.20, Corollary 2.3.2.13, Corollary 4.1.1.20), we may assume that E=E(x),
( E = E ( x , 1),
F=F(ux),
F=F(ux,
1))
(Theorem 4.1.3.1 b)). Then E , F are Gelfand C*-algebras (Corollary 4.1.2.3) and the assertion follows from Proposition 4.1.2.15 a),f) and Theorem 4.1.3.1
c),d). We now prove the last assertion. First assume that y is self-normal and take f e C(aE(y)) with f(0) = 0 if 0 E aE(y). Then
o~(~y)
~.(y)u
{0}
(Corollary 2.1.3.12). From
uf(y) =/(~y) and the injectivity of u we get
f (uy) = 0 ==v f (y) = O . It follows that
and that
II~yli = ilyl] (Proposition 2.3.1.16, Theorem 4.1.1.16 b)). Now let y be arbitrary. Then
il~yli : = il(~y)*(~y)ii = II~(y*y)ii = liy*yit = ilyil ~, il~yll = liylf,
m
62
~. C*-Algebras
C o r o l l a r y 4.1.3.5 ( 0 ) Let F be a Gelfand unital C*-subalgebra of the unital C*-algebra E . Take x 9 F and f 9 C(cr(x)). Then A
F
F
a ( f ( x ) ) = f(a(X)),
f(x) = f o~,
f(x) 9
f(x) 9 UnE ~
real,
f ( a ( x ) ) C {a 9
m: I I~1 = 1}.
Given y 9 E(x, 1), define ~" a(E(x, 1))
>IK,
y' ,
~ y'(y).
There is a continuous map qa" cr(F)
~ a(E(x, 1)),
such that F
yA = y o ~ for every y 9 E(x, 1) (Corollary 4.1.2.16 a),c)). By Theorem 4.1.3.1 a),b),c) AF
F
f (x) = f (x) o ~ = ( f o ~) o q~ -- f o'~,
so that a(f(x)) = f(x)(cr(F))= f
~'(cr(F))
= f(a(x))
(Corollary 2.4.1.7 a)). We deduce that (f(x) C Re E) ~
(cr(f(x)) C IR) r
( f ( x ) 9 UnE)r
(cr(f(x))C {a 9
( f ( ~ ( x ) ) C {a C
(Corollary 4.1.2.13).
( f ( a ( x ) ) C IR) ~
( f real),
~11~1 = 1})r
~11~1-
1}) I
4.1 The General Theory
C o r o l l a r y 4.1.3.6
63
Let x be a normal element of a complex C*-algebra and
define f:a(x)
>r
a,
(resp. i m a ) .
rrea
Then f(x) = rex
(resp. f ( x ) = imx)
and a(re x) = {re a
a ( i m x ) = {im a
1
P -= 5(s + t) e r
t]
9 ~(x)},
I~ e
~(x)}.
1 (, resp. P "= 2~
t) er
t])
"
Then f (a) = P ( a , ~) for every a 9 a ( x ) , so that
f ( x ) = P(x,x*) = r e x
(resp. i mx )
(Theorem 4.1.3.1 a)). By Corollary 4.1.3.5, o-(re x) = {re a [ a 9 a(x)}, o-(imx) = {ira a I c~ 9 o ( x) }. C o r o l l a r y 4.1.3.7
m
( 0 ) (Russo-Dye, 1966) Every element of a unital com-
plex C*-algebra may be represented as a linear combination of four unitary elements. It is sufficient to show that every selfadjoint element may be represented as a linear combination of two unitary elements (Proposition 2.3.1.22). So let x be a selfadjoint element of a unital complex C*-algebra. We may assume that Ilxll < 1. Then
~(x) c [-1,1]
64
4. C*-Algebras
(Corollary 4.1.2.13, Theorem 4.1.1.16 b)). Put f " a(x)
>T, ,
a,
> a + i ~ / 1 - a 2.
By Corollary 4.1.2.13 and Corollary 4.1.3.5, f ( x ) and f ( x ) are unitary. We have 1
x = -~(f(x) + -f(x))
(Theorem 4.1.3.1 a)). Corollary 4.1.3.8
m ( 0 ) Let E be a unital C*-algebra. Take x E ShE and
P, Q e IK[s, t] such that Q ( x , x * ) is invertible and define f " a(x)
) IU,
~ ,
P ( a , -5) Q(~,~)
"
Then f ( x ) = P ( x , x * ) Q ( x , x ' ) -1
We put g " a(x)
) IK ,
a,
) P ( a , -5) ,
h" a ( x )
) IK,
a,
) Q(a,-5).
Then g(x) = P ( x , x ' ) ,
h(x) = Q ( x , x * )
(Theorem 4.1.3.1 a)). By Corollary 4.1.3.5, 0 r a(h(x)) - h(a(x)).
Hence h does not vanish on a(x) and f is well-defined. We have that g=fh,
so that P ( x , x * ) = g(x) - f ( x ) h ( x ) = f ( x ) Q ( x , x * )
(Theorem 4.1.3.1 a)) and f ( x ) - P ( x , x * ) Q ( x , x * ) -~
m
4-1 The General Theory
Corollary 4.1.3.9
65
( 0 ) Let E be a unital C*-algebra and take x E S n E .
oo
Let ~_, ant n be a power series in IK with radius of convergence strictly greater n--O
than Ilxll and define oo
f ~
' ~,
>Z ~ "
~'
n--O
Then (x:)
f (x) = ~ ~ n--O
(Proposition 2.2.3.1, Proposition 2.2.3.2).
We define akt k 9 IK[t],
P~(t) - ~ k--0
f~.o(x)
>~,
~,
>P~(~)
for every n c IN. Then lim IIfn - fll = o ,
n---+ (x)
and so (x)
f ( x ) = lim f n ( x ) = n ~ - ~ cx)
lira P n ( x ) - E 7t----~ (:X:)
anx~
n--O
(Theorem 4.1.3.1 a),b)), Corollary 4.1.3.10
m
( 0 ) If x is a normal element of a unital complex
C*-algebra, then e ix is unitary iff x is selfadjoint.
We define f " a(x)
>r
a,
> e i~
Then eiX : f (x)
(Corollary 4.1.3.9) and a(f(x))-
f(a(x))
(Corollary 4.1.3.5). Thus, by Corollary 4.1.2.13, (e ix E Un E) ~
( f ( a ( x ) ) C {a E ~ ] ] a ] = 1}) ~=> (a(x) C IR) r
(x selfadjoint).
m
66
4. C*-Algebras
C o r o l l a r y 4.1.3.11
( 0 ) Let E be a unital C*-algebra. Take x c S n E ,
f C C(a(x)), and g E C ( a ( f ( x ) ) ) .
Then
g(f
f (x) is self-normal and =
(Theorem 4.1.3.1 b), Corollary 4.1.3.5), so that g ( f ( x ) ) defined. We have g o f ( x ) = (g o f ) o ~ -
and g o f are well-
g o ( f o ~) = g o f ( x ) - g ( f ( x ) ) ,
where the Gelfand transform is taken with respect to E ( x , 1) (Corollary 4.1.3.5). Thus g
f (x) = g ( f ( z ) )
(Corollary 4.1.2.5). Corollary 4.1.3.12
I Let x
be a unitary e l e m e n t of a unital complex C*-
algebra E . I f
~(~) # {~ 9
1}
then there is a y C Re E such that eiy - x .
There is a Oo E IR, with
~(~) c {~'~
< o < Oo + 2~}
(Corollary 4.1.2.13 a =~ b). We define f : o(x)
>]0o,0o-+27c[,
e i~ ,
~ O,
y := f ( x ) . By Corollary 4.1.3.5, y is selfadjoint. Since for each a E a ( x ) , eiS(~) = a we see that
4.1 The General Theory
67
x = e i/(x) = e i y
(Corollary 4.1.3.9, Corollary 4.1.3.11). Remark.
m
The condition
o(~) # {~
=
1}
is not necessary but it cannot be dropped. Indeed, if E := C(T) and x:T-----~,
>~,
c~,
then x ~ e iu for every y e Re E . On the other hand, if E := C([0, 2rr]) and y:[0,2rc]
>r
c~,
>c~,
then y c Re E , e iy is unitary, and
C o r o l l a r y 4.1.3.13
Take n C IN. If x is a unitary element of f~(~n) (Ex-
ample 4.1.1.7), then there is a y C Re s
n) with e iy = x .
The assertion follows from Corollary 4.1.3.12, since or(x) contains at most n points. 9 P r o p o s i t i o n 4.1.3.14
( 0 )
Let x
be a s e l f - n o r m a l e l e m e n t of a unital
C*-algebra and take P C IK[s, t]. T h e n
P(x,~*) = 0 /ff c~(x) c {c~ ~ IK I P(c~, ~ ) = 0}. We define
f:~(x)
>~,
~,
>P(~,~).
Then f ( x ) is self-normal (Theorem 4.1.3.1 b)) and a(P(x,x*)) = a(f(x)) = f(a(x))
(Theorem 4.1.3.1 a), Corollary 4.1.3.5). By Theorem 4.1.1.16 c), ( P ( x , x * ) = O) r
( ~ ( P ( x , x * ) ) = {0}) r
(~(x) C {~ e IK I P ( ~ , ~ ) = 0}). m
68
4. C*-Algebras
[
Corollary 4.1.3.15
\
( 0 )
If x is a self-normal element of a unital C * -
algebra and ~ E ]K, then
x=c~l
o(~) = { . } .
Define P := t - c~ C IK[t]. Then x = al ~
a(x) = {at
P(x) =
by Proposition 4.1.3.14.
Corollary 4.1.3.16
m
If x is a self-normal d e m e n t of a unital C*-algebra,
then eX=l
i# a(x) C 27ri~.
Define f:cT(x)
>IK,
ot,
) e ~.
Then eX= f ( x ) (Corollary 4.1.3.9), a(e x) = a ( f ( x ) ) = f ( a ( x ) )
(Corollary 4.1.3.5), and f ( x ) is self-normal (Theorem 4.1.3.1 b)). Hence ~ = 1~
by Corollary 4.1.3.15.
o ( ~ ) = {1} r
~(~) c 2 ~ i ~
m
4.1 The General Theory
Proposition
4.1.3.17
69
1
( 6 ) Let x be a normal element of a unital complex
C*-algebra, a an isolated singularity of the resolvent of x , and f : a(x) .. >r
~,
>5 ~ ,
iI~r
g. '
(
0
Then oo
1 f(x) t -- c~
E(t
-- o~)ng(x)
n-t-1
n--O
is the Laurent series of the resolvent of x at a ,
( a l - x ) f ( x ) = 0, and g ( x ) ( a l - x) = f (x) - 1. In particular, a is a pole of order 1 and the residue is an orthogonal projection.
For some r > O,
u := ~ ( ~ ) \ { ~ } c r Take t E U. Define f':a(x)
>r
g':o(z)
/3,
>r
Z,
>
>
f(/3)
t-/~'
1-
f (/3)
t-~
Then f' (/3)
oo
g'(fl) =
1 - f(~) - _~(t_.)~ ( t - ~) - (~ - ~)
-
-
f(~) t-a oo
1 - f(/3) = - ~ ( t - ~ ) n g ( ~ ) (~ - ~)~+~
n--O
f' (9) + g' (~) =
n--O
t -13
.+~ ,
70
4. C*-Algebras
for every /3 E a(x), so that oo
(tl - x ) - ' = i f ( x ) + g'(x) =
1 f(x) - E(tt -- oz
a ) n g ( x ) ~+'
n--O
by Theorem 4.1.3.1 a), and Corollary 4.1.3.9. Hence a is a pole of order 1 and the residue is an orthogonal projection (Theorem 4.1.3.1 a),b), Proposition 4.1.2.21 b =~ a). The equalities (al-x)f(x)=O,
g ( x ) ( a l - x) = f ( x ) - I
follow by the functional calculus. C o r o l l a r y 4.1.3.18
Let x
be a n o r m a l element of a unital complex C * -
algebra and a, 13 distinct isolated singularities of the resolvent of x . Let y, z be the residues of the resolvent of x at a and ~ , respectively. Then yz-O.
Define
g:a(z)----~r
7,
>5~.
By Proposition 4.1.3.17, y - f (x) ,
z - g(x) ,
so that y z : f (x)g(x) : ( f g)(x) : 0
(Theorem 4.1.3.1 a)). E x a m p l e 4.1.3.19
1 Let
be a s e l f - n o r m a l element of /:(HA(2) (Example 4.1.1.7) and take f E C ( a ( x ) ) . Put p : : v / ( ~ - ~)~ + 4 ~ ,
4.1 The General Theory a+5+p
a :=
2
b :=
'
71
a+5-p 2
and assume p ~= O. Then a(x) = {a,b}, /(x)=---Z
v
f (a) - f (b) [ 7
a-
.
--7
7
'--"
=
x + af(b) - bf(a)l '
L~ fl ] + f (a) +2 f ( b ) l = f ( a ) - f ( b ) p
and if IK = ~ , then the residue of the resolvent of x at a is
5-a+p
7
2
It is obvious that a(x) = {a, b}. Moreover, given c C {a, b},
f (a) - f (b) P
c+
a f (b) - bf (a)
= f(c),
P
so that
f (a) - f (b)
f(x) =
a f ( b ) -[b f l( a p) 7
S,a[o-b P
7
5-b
2
P
-7-
-7
-7
--7
+
7
0 0 ]= 1
5
ti-a+p2
= f (a) - f (b) [ 7
a-5
2
= _
a-5+P2
--62/3~] + f (a) + f (b) l
The last assertion follows immediately from the previous one and Proposition 4.1.3.17.
1
P r o p o s i t i o n 4.1.3.20
( 0 ) Let E be a unital C*-algebra. Take and let f" A --+ IK be a continuous map. Put B := {x ~ S n E l l ( x )
c A}.
Then the map B
is continuous.
>E,
x,
>f(x)
A C IK
72
4. C*-Algebras
(Xn)nEIN be
Take x E B and let
a sequence in B which converges to x.
Put
By Corollary 2.2.5.3, K is compact. Let e > 0. By the Weierstrass-Stone Theorem, there is a P E IK[s, t], with sup I f ( c d - P(a,~)l < E. aEK
Then
II/(x)-/(x.)ll ~ II/(x)- P(x,x*)ll+ + l l P ( x , x * ) - P(Xn, X*=)ll + I I P ( x . , x L ) - / ( x . ) l l <
< 2c + IIP(x,x*) - P ( x . , x L ) l l for every n E IN (Theorem 4.1.3.1 a),b)), so that lim sup Ilf(x) - f ( x ~ ) l
2c.
~
rt---+ OQ
I
Since e is arbitrary, (f(x,~))ne~ converges to f ( x ) . Corollary 4.1.3.21
Let E be a unital C*-algebra. Take x E S n E and let
U an open set of IK with u n o ( z ) # O. Then there is an c > 0 such that u n a(y) r 0 for every y E S n E
with I I x - y i l < e -
Take f E C(1K) with Supp f C U,
II/Io(x)ll
Then
f (x) :/= O
r
o.
4.1 The General Theory
73
(Theorem 4.1.3.1 b)). By Proposition 4.1.3.20, there is an ~ > 0, such that
f(y)
o
for every y e Sn E with I1~- yll < c. Thus
~(y)
Supp
r 0
a(y)
0
and so
U
for every y 9 Sn E with I I x - yll < ~. P r o p o s i t i o n 4.1.3.22
m
Let E be the C*-direct sum of the family
( 0 )
(E~)~eI of C*-algebras (Example ~.1.1.6). Take x 9 S n E and f 9 d(cr(x)). Then
and f (x), -- f (x~) for every ~ 9 I . First observe that z~ 9 Sn E~ for every ~ 9 I . Take
Then there is an c > 0 such that
Thus
for every ~ 9 I
(Theorem 4.1.3.1 b)). Hence
7~
4. C*-Algebras
The reverse inclusion is trivial. The relation t e I ==> P(x,x*)~ = P(x~,x:)
holds for every P c IK[s, t]. By continuity, f (x), = f (x~)
for every t E I . P r o p o s i t i o n 4.1.3.23
I ( 0 ) Let E be a real C'-algebra. Take x C R e E
and f e C(a(x)). If (x, 0) denotes the element of the complexification E of E , then f ( ( x , 0)) -- ( f ( x ) , 0).
We first remark that by Proposition 4.1.1.15 (and Corollary 2.1.5.14),
%((x, 0))=o~(x), so that f((x, 0)) is well-defined. The assertion is trivial if f is a polynomial (Corollary 4.1.3.8). Since the set of polynomials is dense in C(a(x)), the assertion also holds for arbitrary f . I
4.1 The General Theory
4.1.4
75
The Theorem of F u g l e d e - P u t n a m
Theorem 4.1.4.1 E-module
( 0 ) (Fuglede, 1950, Putnam, 1951) Take x, y C Sn E
o v e r the unital C * - a l g e b r a E .
Let F be a unital
and a C F
with
xa - ay. Then f(x)a=af(y) f o r every f C C(cr(x) U or(y)). I f F - E
Step 1
IK - ~ ==> x * a -
t h e n x*a = ay* .
ay*
By complete induction, xna
~
ay n
for every n C IN. Hence cr
i n o~n
~
n=O
i n o~n
.
n=O
a -- e - m Z e m X a
--
e - m X a e my
for every c~ C ~ . Define f 9 (~
~ F,
o~l
) e - i c ~ x * a e lay*
Then f is analytic (Proposition 1.3.10.3, Corollary 1.3.10,10). Since ~ x + cex* and ~y + c~y* are selfadjoint, e -i(~x+~') and e i ( ~ + ~ ' ) are unitary (Corollary 4.1.3.10), so that
for every c~ C 9 (Proposition 4.1.1.4). Hence
llf(~)ll - lle-i~**ae i~" II -
tl e - i ~ * e - i ~ a e i ~ e i ~ y *
tl -
for every ce E 9 (Proposition 2.2.3.7 a)). Thus f is bounded and therefore, by Liouville's Theorem, it is constant. Thus
76
4. C*-Algebras
a = f(O) = f(c~) = a + a ( - i x * a + lay*) + . . .
for every c~ E r (Proposition 1.2.9.6), so that - i x * a + iay* = 0
(Corollary 1.3.3.10) and x*a = ay*.
Step 2
f (x)a = a f (y)
If IK = IR, then P(x)a = aP(y)
for all P C IR[t]. If IK =ff~, then, by Step 1, it follows that P ( x , x * ) a = a P ( y , y*)
for all P E C[s, t]. By the Weierstrass-Stone Theorem, f (x)a = a f (y) .
Step 3
The Last Assertion
The assertion follows from Step 1 by complexification. Remark.
a) The above ingenious proof of the first step is due to M. Rosenblum
(1958).
b) Take g C C(a(x) U ~(y)) with f = g on a(x) N a(y). Then, it is easy to see that g(x)a = f (x)a = a f (y) = ag(y) .
C o r o l l a r y 4.1.4.2
( 0 ) Let A be a subset of a C*-algebra E such that A c A* u S n E ,
and G the C*-subalgebra of E generated by A . Let F be an E-module.
4.1 The General Theory
a)
Take a E F with x a = ax f o r every x E A .
77
Then xa = ax f o r every
xcG.
b)
A ~ and A ~ are C*-subalgebras of E and A C = G ~.
c)
I f A is commutative (and contained in Re E if IK = JR), then A ~
is
commutative, G is a Gelfand C*-algebra, and A is contained in a maximal Gelfand C*-subalgebra of E .
d)
Every s e l f - n o r m a l element of E
is contained in a m a x i m a l Gelfand C * -
subalgebra of E .
a) Define H := {x e E l x a = a x } . Then H is obviously a closed subalgebra of E . Take x C A. If x r A * , then x* r A, and so x* c H . If x r S n E , then x* a = ax* ,
by the Fuglede-Putnam Theorem, so that x* C H . Hence A U A* C H and H contains the subalgebra of E generated by A U A*, i.e. the involutive subalgebra of E generated by A (Proposition 2.3.1.19). Finally, H contains G (Corollary 2.3.2.14 a)). b) By a) and Corollary 2.2.1.7, A ~ and A ~ are C*-subalgebras of E . Thus AC
GcA
cc ,
and A c = A ccc C G c C A c,
AC
__
G c
by Proposition 2.1.1.17 b),d). c) By Proposition 2.1.1.17 e), Acc is commutative, and so, by b), G is a Gelfand C*-algebra (Corollary 4.1.2.3). The last assertion follows from the last assertion of Corollary 4.1.2.3. d) follows from c).
I
78
4. C*-Algebras
Corollary 4.1.4.3
Let P (resp. Q ) be a polynomial in two (resp. five) va-
riables. Let A be a subset of IK with
oz, ~ e A ==v P(c~, fl) 6 A, and f " A -+ IK a continuous function with ~, ~ 6 A ===~Q(c~, fl, f(c~), f ( ~ ) , f(P(c~, fl))) = O. Further let x, y be self-normal elements of a unital C*-algebra such that xy = y x ,
a(x) U a ( y ) C A .
Then Q(x, y, f (x), f (y), f (P(x, y))) - O .
Let F be the C*-subalgebra generated by {x,y}. F is a Gelfand C*algebra (Corollary 4.1.4.2 c)). For z 6 F define
~.o-(F)
~,
~',
~,~'(z).
Then a(P(x,y))-
P(x,y)(a(F)) - (P(~,~))(a(F)) C A
(Corollary 2.4.1.7 a), Proposition 2.4.1.2). Hence f ( P ( x , y ) ) Furthermore
~(z,y,f(z),S(y),Z(P(z,y)))-
is well-defined.
O ~,g,S(z),S(y),f(P(z,y)
=
= Q ( ~ , ~ , f o ~ , f o~',f o P(~, ~')) = 0
(Proposition 2.4.1.2, Corollary 4.1.3.5), so that Q(x, y, f(x), f(y), f (P(x, y))) - 0
(Corollary 4.1.2.5 ).
Corollary 4.1.4.4
I ( 0 )
If x is a self-normal element of a unital C*-
algebra, then (x} ~ C { f ( x ) } c for every f 6 C(a(x)).
4.1 The General Theory
79
Let E be the unital C*-subalgebra generated by x. Then f(x) 9 E (Theorem 4.1.3.1 b)), and so {x) c = E c C {f(x)} ~ (Corollary 4.1.4.2 b)). P r o p o s i t i o n 4.1.4.5
I ( 0 ) Let E be a unital C*-algebra. Let A , B
subsets
of IK and f : A ~ B a homeomorphism. Put
A0:-{x 9
CA},
B0:={x 9
)CB}.
Then { f ( x ) l x e Ao} = B0, the map Ao
>Bo,
x,
>f ( x )
is a homeomorphism, and f-l(f(x))
: x,
{x}C= { f ( x ) } c,
E ( x , 1) = E ( f ( x ) , 1)
for every x E Ao. If in addition
0CA~
f ( O ) = O,
then E(x) = E(f(x)).
Take x e Ao. Then f ( x ) is self-normal (Theorem 4.1.3.1 b)), a(f(x)) - f(a(x)) C B
(Corollary 4.1.3.5), and f-1 (f (x)) - x (Corollary 4.1.3.11). It follows from the last equality that ( x } c - { f ( x ) } ~,
E ( x , 1) - E ( f ( x ) , 1)
(Corollary 4.1.4.4, Theorem 4.1.3.1 d)). By Proposition 4.1.3.20, the maps Ao ----> Bo ,
B0
>A0,
x J > f (x) ,
x,
>f-l(x)
are continuous and so they are homeomorphisms.
80
4. C*-Algebras
[
T h e o r e m 4.1.4.6
(
0 )
Let E be a unital C*-algebra. Take x E E and
let A be a subset of ]K which only has isolated points. Then the following are equivalent:
a) x is self-normal, a(x) C A U {0}, and 0 E A if it is an isolated point of a ( x ) .
b)
There is a family ( x ~ ) ~ A in P r E such that:
1) x = E ~ x o , aEA
2) a,/3 E A =r x~x~ = 5~Zx~, 3) a(x) finite => y]. xo = 1. aEA
If these assertions hold, then:
c)
Given a E A , define
fo:a(z)
~,
>6~,:~.
f.(z) = In particular, the family (x~)~eA from b) is unique and
a(x)\{0} = {a E A\{0} I x~ -7/=0}. E A} generate the same closed subalgebra of E and d) x and {x~ E(x, 1) is the unital C*-subalgebra of E generated by {xa I a E A } . It is finite dimensional iff a(x) is finite. e)
Given any f E C(a(x)) such that f(O) = 0 if a(x) is infinite, we have that
E a :=:>b. For a E A, let f~ be the function defined in c) and put
x~ := f.(x)
4.1 The General Theory
81
(f~ is continuous since a is an isolated point of A). By Corollary 4.1.3.5
~(x~)- ~(f~(~))= f~(~(~)) c {0, ~}, so that x~ 9 P r E f~ - 0 ,
for every a 9 A
(Proposition 4.1.2.21 c ==> a). We have
so that x~ = 0 for every a 9 A \ a ( x ) and xo~z = fo(x)f,(~)
= (fof,)(z)
- 5ozf.(x)
= 5.,x.
for all a, fl 9 A (Theorem 4.1.3.1 a)). Take E > 0
and
Since a(x) is compact (Corollary 2.2.4.5) and A consists of isolated points only, B is a finite subset of A. Let C be a finite subset of A containing B and put
f "o'(x)
> IK,
a,
>a.
Then x_
:
lxl-
:
so that f - ~-~f~
aEC
(Theorem 4.1.3.1 a),b)). Since c is arbitrary, we deduce that E O~Xc~= X. aEA
If a(x) is finite, then ~--~f~ - 1, aEA
so that
xo: aEA
1 aGA
b =~ a &: c & d & e. Let F be the unital C*-subalgebra of E generated by {x~ I a c A}. F is a Gelfand C*-algebra and x E F (1), 2), and Corollary 4.1.4.2 c)). Hence x is self-normal. For y C F put
82
4. C*-Algebras
~-o(r)
~1K, y', ~y'(y)
and for a E A put
Am "- {t E a(F) [ ~ ( t ) = 1}. Then
x,~x~ = x,~xz = 5 ~ for all a, fl E A (2) and Proposition 2.4.1.2). Thus ~
= 0 on a(F)\A,~ (1),
Proposition 4.1.2.21 a =~ c, and Corollary 2.4.1.7 a)) and
A~AA~ =0 for all distinct a,/3 E A. From these relations and 1) we deduce that ~ = a on Am for every a E A and ~ = 0 on a ( F ) \ U A,~. Hence c~EA
a(x) = ~(a(F)) C A U {0} (Corollary 2.4.1.7 a)). If 0 is an isolated point of a ( x ) , then a(x) is finite and
aEA
aEA
by 3) (and Proposition 2.4.1.2). In this case,
U Am = o(F) otEA
and
0 E a(x)= ~(a(F))C A (Corollary 2.4.1.7 a)), which completes the proof of a). We next prove c & d. If 0 E A, then a(x) is finite and
E•
I
eo(F)
aEA
by 3), i.e. U Am - a ( F ) . c~EA
We deduce that
4.1 The General Theory
83
I"
f,~(x) - f,~ o ' 2 =
I
1
on
A~
0
on
a(E)\A~
whether or not 0 C A (Theorem 4.1.3.1 c)), so that
A(X) = X~,
f , ( x ) = X~
for every a C A (Corollary 4.1.2.5). It follows t h a t x and {xa I a E A} generate the same closed subalgebra of E (Theorem 4.1.3.1 e)) and F = E(x,
We now prove e). Take a E A . Then
f f~ = f ( a ) f , , and so
f ( a ) x ~ : f ( a ) f a ( x ) : ( f ( a ) f ~ ) ( x ) : ( f f~)(x). We deduce t h a t
~/~/o~z ~/o/xo = ~/~)- o~.~/"~
(~-o~o~'~)/x~
for any finite subset B of a ( x ) . If a(x) is finite, then
f-
~-" f f ~ = O , a~a(z)
so that
Now suppose that a(x) is infinite. Take c > 0. Then f(0) = 0 and there is a finite subset B of a(x), such that
sup If(~)l <~.
~(x)\S Thus
=11( for any finite subsets C of a(x) containing B (Theorem 4.1.3.1). Since c is arbitrary,
f(x)-
~ a~a(x)
f(~)xo,
m
84
4. C*-Al9ebras
C o r o l l a r y 4.1.4.7 and put
Let E be a unital C*-algebra. Take x C E , P C IK[t],
A := {a e IK I P ( a ) = 0}. Then the following are equivalent: a) x is sdS-normal and P ( x ) = O. b)
There is a family (Xs)sC A in P r E such that 1) x = Y'~aXs, seA
2) a , / 3 c A = > x s x z = 6 s ~ x s ,
3) E a . = I . sea
If these conditions are fulfilled, then: c) a(x) = {c~ C A l x s ~ 0}. d)
If o~ c A and
f . : ~(z)
~ ~K,
~,
then
f.(x) =~o. In particular the family (X~)~cA in b) is unique. e) E(x) is the C*-subalgebra generated by {x~ ] a e A } . It is unital and finite-dimensional. f)
f E C(a(x)) => f ( x ) =
y]
f(a)xo.
g) If A C IR (resp. A c { a e I K i l a ] = 1}), then x is selfadjoint (resp. unitary). a=>b&c&d&e&f&g.
Since
o(~)cA (Theorem 2.1.3.4 a)), the assertions follow from Theorem 4.1.4.6 a =v b & c & d & e and Corollary 4.1.2.13 b =v a. b => a. By Theorem 4.1.4.6 b ::v a & e, x is self-normal and P(x)-
E
P(a)x~-O.
I
4.1 The General Theory
Corollary 4.1.4.8
85
Let E be a unital complex C*-algebra. Take x E E and
n E IN. Then the following are equivalent:
a) x is normal and x ~ = 1.
b)
There is a family ( x j ) j e ~ ,
Pr E such that
in
n
1) ~ - E ~ ~ j , j=l
2) j, k E lNn =:> XjXa = (~jkxj, n j--1
If these equivalent conditions hold, then:
c)
IJ
xj
o}
d) I.f j E INn and
(
1
if
a--e.
0
if
a#e~
then
xj = f j ( z ) In particular, the family (xj)je~n in b) is unique. e)
E ( x ) is the C*-subalgebra of E generated by {xj ] j e INn} ; its dimension is at most n .
f) f e C ( a ( x ) ) : : a
E f(a)x~= aCa(x)
E
~=1
f
e n
xj.
xj•o
g) x is unitary,
m
Corollary 4.1.4.9
Let E be a unital complex C*-algebra. Take x C E . Then
the following are equivalent:
a) x is normal and e x - 1. b) There is a family (Xn)nE2E in Pr E such that 1) z -
~
2~inz~ ,
n E 2E
2) m, n c 2Z ~ XmXn = 5mnXm ,
86
4. C*-Algebras
3)
Ez~=I. nE~
If these equivalent conditions apply, then:
c) a(x) d)
=
{27tin I~ 9 2Z, ~ ~ 0}.
If n e 23 and
then h(x)
= ~.
In particular, the family ( x n ) n ~ in b) is unique. e) E(x) is the C*-algebra generated by {xn I n C 23}. It is unital and finitedimensional. f)
f e C(cr(x))=V f ( x ) =
~
f(27rin)xn.
nE~ ::n r
a ::v b & c & d & e &: f. By Corollary 4.1.3.16, a(x) C 27ri23 and the assertion follows from Theorem 4.1.4.6 a =v b ~: c & d & e and Corollary 2.2.4.5. b ::~ a. By Theorem 4.1.4.6 b => a, x is normal and a(x) C 2~i23, so that
eX=l by Corollary 4.1.3.16.
m
Let E be a finite-dimensional unital complex (real) C*algebra. Then the vector subspace of E generated by Pr E is E (is R e E ) . In particular, if A is a linearly independent subset of E (of Re E ), then there is an algebraic basis B of the underlying vector space of E (of Re E ) with Corollary 4.1.4.10
A C B,
B\A c PrE.
4.1 The General Theory
87
Let x E R e E . By Corollary 2.1.3.5, a ( x ) is finite, so by Theorem 4.1.4.6 a =~ b, x belongs to the vector subspace of E generated by Pr E . Since x is arbitrary, the vector subspace of E generated by P r E is E (is Re E ) . The last assertion follows from the first one. Remark.
I
Every finite-dimensional C*-algebra possesses a unit (Corollary
4.2.8.8). E x a m p l e 4.1.4.11
Take n E IN and let u be a s e l f - n o r m a l element of the
involutive algebra IKn,n defined in E x a m p l e 2.o~
Given x, y E IK n , define
n
(xly) - ~ ~ , i=1
[~ly] := [ ~ , ~ ] , , j ~ . Then there are a finite subset A of ]Z n and a f u n c t i o n f " A --+ IK such that
(xly) = 5~,~ f o r all x , y E A and
: E s( )Ex,xl,
a ( u ) - f (A) .
xEA I f u is selfadjoint and
u" ]Kn, n
) ]K,
[~i,j]i,jEINn '
) ~ O~ijUji, i,j--1
then
We first discuss the complex case. Then as involutive algebra, ~n,n may be identified with the C*-algebra /:(r
(Example 4.1.1.7). By Theorem 4.1.4.6
a :=v b & c, there is a family (u~)~e~(u) in Pr s 1 6 2 ?_t - -
~
a~r(u)
OLU(~,
such that
88
4. C*-Algebras
By Example 2.3.1.36, given c~ E a ( u ) , there is a nonempty finite subset A~ of ([~n such that
(zly) = 5~,~ whenever x, y E Am and that
xEAa
Define A :-
U
f:=
~
Aa,
aeAo 9
a~a(u) Take distinct a,~ C a(u) and x C A~, y c A~. Then
i=1
i--1
j=l
k=l
n
Hence
(xiy) = 5x,y for all x , y C A. Then
,:
E aEa(u)
EIx,,I-Es(,)I,I,J, xEAa
xEA
f ( A ) = {a I a E a(u)} = a(u). We now turn to the real case. The matrix u may be viewed as an element of ~ , n . Since in this case u is selfadjoint by assumption, we have that a(u) C ]R and the assertion follows from the considerations above. We now prove the final assertion. Take v E Re IK#,n. By the considerations above, there is a finite subset B of IKn and a function g : B --~ [-1, 1] such that for every x, y E B ,
4.1 The General Theory
89
(xly> - 5x,y
and
v= ~
g(y)[yly] 9
yEB
Then
i~(v)l =
Since v is arbitrary, it follows by Proposition 2.3.2.22 j),
II~ll _< Z
If(x)l.
xEA
Defining g
x ~--.~ ~
h'A
1 ,
L -1,
if f ( x ) > 0 if f (x) < O,
yEA
the above considerations show that
xEA
1. Thus
yCA
xCA
Hence I xEA
P r o p o s i t i o n 4.1.4.12 Let p be an orthogonal projection of the C*-algebra E . If p is not a unit of E , then there is an x C R e E for which x~O,
px = x p = O.
90
4. C*-Algebras
Since p is not a unit of E , there is some y E E \ p E p . yl:=(1-p)Y*Y(1-P),
Put
y2:=(1-p)yy*(1-P)-
T h e n yl, y2 C Re E and PYl = YlP = PY2 = Y2P = O.
If Yl = Y2 -- 0, t h e n Ily(1 - p)ll 2 = I1(1 - p)y*y(1 - p)ll = Ily~ll = o,
I](1 - p)y]l 2 = I](1 - p)yy*(1 - P)]I = ]ly211 = o, so t h a t y = yp = py ,
y = pyp E pEp,
which is a contradiction. Hence either yl ~ 0 or y2 ~ 0. We put x := yl in the first case and x : - Y2 in the second case. Corollary E.
If a(F)
4.1.4.13
Let F
I
be a Gelfand C*-subalgebra of the C*-algebra
is compact and F
does not contain a unit of E ,
a Gelfand C*-subalgebra G of E
containing F
then there is
and an x' C a ( G )
which is
identically 0 on F .
Using the Gelfand t r a n s f o r m on F , we find an orthogonal projection p of F with px
=
xp
=
X
for every x c F . Since p is not a unit of E , there is an x C R e E , with x~:0,
px = xp = O
( P r o p o s i t i o n 4.1.4.12). By Corollary 4.1.2.3, the C * - s u b a l g e b r a G of E generated by F U {x} is a Gelfand C * - a l g e b r a . Using the Gelfand transform on G , we see t h a t there is an x' e a ( G ) such t h a t x ' ( x ) ~ O. It follows x'(p) - O, i.e. x' is identically 0 on F .
I
4.1 The General Theory
91
Let E be a C*-algebra and x E S n E such that E(x) does not contain a unit of E . Then there is a Gelfand C*-subalgebra F of E containing x such that
Corollary 4.1.4.14
inf
z'Ea(F)
Ix'(x)I = o . F
If a(E(x)) is not compact, then we may take F := E(x) since ~ E go(a(F)) (Theorem 2.4.1.3 e)). If a(E(x)) is compact, then by Corollary 4.1.4.13, there is a Gelfand C*-subalgebra F of E containing E(x) and an x ' E a(G) which is identically 0 on E(x). In particular, x E F and x ' ( x ) = O.
I
92
4.2
~,. C*-Algebras
The
Order
Relation
Every C*-algebra is equipped with a canonical ordering with good properties, albeit lacking a lattice structure. Together with the functional calculus, this ordering plays a central role in the theory of C*-algebras. For example every positive element has a uniqueness square root, and so we can define the absolute value of any arbitrary element. Another example is the existence of a canonical approximate unit, which is the upper section filter of an upward directed set. Finally the ordering is also used in the Segal-Schatten Theorem to show that any quotient algebra of a C*-algebra is again a C*-algebra. A consequence of this highlights are the strong properties of C*-algebras: If u : E --+ F is a homomorphism of involutive algebras between the C*-algebras E and F , then u ( E ) is a C*-subalgebra of F and the associated map E / K e r u --+ u(E) is an isometry of C*-algebras. In this section, E is always a C*-algebra.
4.2.1 D efi n i ti on and General P r o p e r t i e s T h e o r e m 4.2.1.1
( 0 ) (Fukamiya 1952, Kelley-Vaught 1953) There is a
unique order relation on E which renders it an ordered Banach space with
E+ = {x e R e E I a(x) C IR+} (= {x 9 N o E I a(x) C IR+} if IK =q:) By Theorem 4.1.1.13, we may assume that E is unital. Since E is symmetric (Theorem 4.1.2.1) and since Ilxil =
r(x)
for every x C Re E (Theorem 4.1.1.16 b)), the assertion follows from Proposition 2.3.2.34 b) (and Corollary 4.1.2.13 b ::v a). Definition 4.2.1.2
( 0 )
II
The order relation on E introduced in Theorem
~.2.1.1 is called the c a n o n i c a l o r d e r r e l a t i o n o f E . In the sequel, all C*-algebras are taken to be endowed with their canonical orders.
The order relation of E remains unchanged if we reverse the multiplication. The example E := Co shows that it may happen that E is order complete but its associated unital C*-algebra is not.
4.2 The Order Relation
Corollary 4.2.1.3
93
If E is unital and x 9 E+, then the following are equi-
valent:
a)
x is not invertible.
b)
There is an x' 9 ~-(E) with x ' ( x ) = O.
c)
There is an x' 9 7o(E) with x ' ( x ) -
O.
c =v b is trivial. b =v a =~ c follows from Corollary 4.1.2.7 b).
Corollary 4.2.1.4
( 0 )
Let E , F
I
be C*-algebras and u "
E -+ F
a
homomorphism of involutive algebras. Then
~(E+) c F+. In particular, x,y G E, x < y ~
ux ~ u y .
Given x E E , ~ ( ~ x ) c o ~ ( ~ ) u {0} (Corollary 2.1.3.12) and the statement follows from the fact that u ( R e E ) C Re F (Proposition 2.3.1.7). The last assertion follows from the first one. I P r o p o s i t i o n 4.2.1.5
If E is not unital and IK
E is its associated unital
C*-algebra, then
~ + x E+ c ( ~ • E)+ c ~ + x E ,
{0} • E+ - ({0} • E ) n ( ~ x E ) +
In particular, if A is a subset of E , x its infimum in E , and ( a , y ) the infimum of { 0 } x A
in IK x E , then a - O
and x - y .
If (a, x) C JR+ • E+, then
(~, ~) - (~, 0 ) + (0, ~) e ( ~ x E)+, so that
Now take (c~,x) E (IK • E)+. Since 0 e a ( ( O , x ) ) ,
94
4. C*-Algebras
e.
+ o((0, ~)) = o ( ( . , ~)) c Ia+.
Hence (IK x E)+ C IR+ x E . The relation
{0} x E+ = ({0} x E ) n ( ~ • E)+ follows from
e E ~
O~x.((0, ~ ) ) = o . ( x ) .
We now prove the last assertion. Take z E A. Then
(.. y) < (o. z). (-a,z-y)
e (lKxE)+ a
Since (0, x) is a lower bound for {0} x A,
(o,~) < (~, y), (a,y-x)
E(IKxE)+CIR+
xE,
a>_O,
ol--O~ x=y.
I
Remark. Take E "- Co and A "- {-en I n C IN}. Then ( - 1 , 0) is the infimum of {0} x A in IK x E and A has no infimum in E . P r o p o s i t i o n 4.2.1.6
( 0 )
x C E+ => [[x[[ E a(x) C [0, []x[]].
This statement follows immediately from
II~ll = r(~) (Theorem 4.1.1.16 b)).
I
4.2 The Order Relation
Corollary 4.2.1.7
95
( 0 ) If f is a continuous increasing positive real func-
tion on IR+ , then
IIf(x) l l - f(llxll) for every x E E+.
We have IIf(x) ll - Ilfl~(x)ll - f(I xll)
(Theorem 4.1.3.1 b), Proposition 4.2.1.6). Proposition 4.2.1.8
I
( 0 ) If F is a C*-subalgebra of E then F+ = E + A F .
In particular, x
in E c = = ~ x ~ y
in F
for all x , y E F .
The first claim follows from x e F ~
o~(x) = ~ ( ~ )
(Theorem 4.1.2.12). The second assertion follows from the first one. Proposition 4.2.1.9
I
( 0 ) If F is a Gelfand C*-subalgebra of E , then F
F
x<_y~===v~<_~
for all x , y E F .
We have F
F
(~(r)) c ~(x) c ~(o(F))u {0} (Corollary 2.4.1.7 a)), so that F
x E F+ v=v~>_ 0 for all x E F . The proposition now follows.
I
96
4. C*-Algebras
Corollary 4.2.1.10
If {xl,x2, yl, y2} is a commutative set of E+ with
Xl _~ Yl,
X2 _~ Y2,
then XlX2 ~_ YlY2.
Let F be the C*-subalgebra of E generated by {xl , x2, yl , y2}. It is a Gelfand C*-algebra (Corollary 4.1.2.3). By Proposition 4.2.1.9, F
F
F
0<_~<~1,
F
0_<~2<~2,
so that F
F
F F
0<~1~<s F
F
XlX~2 ~
y'~,
xlx2 < yly2,
by Proposition 4.1.2.9. Corollary 4.2.1.11
m ( 0 ) Let A be a subset of IR, f an increasing conti-
nuous real function on A . Take x, y C Re E such that a(x) c A ,
a(y) C A ,
x <_ y ,
xy - y x .
Then f (x) <_ f (y) .
Let F be the C ....subalgebra generated by {x,y}. It is a Gelfand C*algebra (Corollary 4.1.2.3). By Proposition 4.2.1.9, F
F
~_<~ so that F
F
F
F
f (x) - f o ~ ~ f o ~ - f (y)
(Corollary 4.1.3.5) and f (x) _< f (y) (Proposition 4.2.1.9).
m
4.2 The Order Relation
C o r o l l a r y 4.2.1.12
97
( 0 ) If x C S n E and f , g E C ( a ( x ) ) , then f ~_ g r
f ( x ) ~_ g ( x ) .
We may assume that E is unital. Put F := E ( x , 1). Then F is a Gelfand C*-algebra (Corollary 4.1.2.3) and the map F
>a(x),
a(F)
x':
>~(x')
is a homeomorphism (Corollary 4.1.2.4). Hence F
f <_ g r
F
F
F
f(x) = f o ~ < g o ~-
g(y) r
f ( x ) < g(x)
by Corollary 4.1.3.5 and Proposition 4.2.1.9. C o r o l l a r y 4.2.1.13
I
( 0 ) Let E be unital and take x c E with z >_ 1.
Then x is invertible and
O < x -~ < 1. We have a(x) = a ( x - 1) + 1 C [1, cx)[.
Hence x is invertible. By Corollary 4.2.1.12, O < x -~ < 1. Remark.
I
This result will be generalized in Proposition 4.2.4.6.
P r o p o s i t i o n 4.2.1.14
Take x e E+ with [[x[I = 1. If
ycE+,y<_x=~
3acIR+,y=ax,
then x is idempotent.
We have I e ~(~) c [0,1] (Proposition 4.2.1.6). Define f:a(x)
>IK,
a,
>a,
98
4. C*-Algebras
g" a(x) ----+ IK,
~,
> ~2.
Then 0 < g <_ f , so that 0 < g(x) ~_ f ( x )
(Corollary 4.2.1.12). Since g(x) = x 2,
f (x) = x ,
(Theorem 4.1.3.1 a)), there is an a E JR+, with X 2
--
OiX.
We deduce that
(Theorem 4.1.3.1 b), Corollary 4.2.1.7), i.e. x is idempotent. P r o p o s i t i o n 4.2.1.15
( 0 ) Given x E R e E , sup a = inf{a E I R l x _< a l } , sEa(x) inf a = sup{a E l R l a l < x}. sEa(x)
Take /3 E IR. Then a(/31-x)=/3-a(x),
a ( x - /31) = a ( x ) - /3,
so that
x<_/31r
CIR+ 4==> sup a_
31<_xc::::vx-31>_Ov::::va(x)-3clR+r
C o r o l l a r y 4.2.1.16 are equivalent:
b)
-al_<x_
inf a : > / 3 . sEa(z)
I
( 0 ) Take x E R e E and a E IR+. Then the following
4.2 The Order Relation
99
a , , b. F := E(x, 1) is a Gelfand C*-algebra (Corollary 4.1.2.3). Thus F
-c~l _< x ~ c~l r
F
--aea(F) _< x _< ctea(F) ~
]]xll --< ct ~
lixll
<
(Proposition 4.2.1.9, Corollary 4.1.2.5). b r c follows from Proposition 4.2.1.15. C o r o l l a r y 4.2.1.17
b)
x
c)
1-xEE+ ~ .
I
Given x E E+, then following are equivalent:
( 0 )
a r b follows from Corollary 4.2.1.16 a r b =~ c. Since
b.
0<1-x<1, c) follows from b => a. c => b is trivial. C o r o l l a r y 4.2.1.18
I
( 0 )
Take x, y E R e E .
If
-y<x<_y, then
IIxll <_ Ilyll. By Corollary 4.2.1.16 a ,~ b,
-Ilvlll <_ - y < x _< y _< Ilvlll,
Let (x,)~e,, (Y~)~e, be families in E such that
C o r o l l a r y 4.2.1.19 ( 0 ) (Y~),eI is summable and
-y~<x~
~EI
~EI
~EI
~EI
~EI
100
4. C*-Algebras
The assertion follows immediately from Corollary 4.2.1.18 and Corollary 1.7.1.6 b). I C o r o l l a r y 4.2.1.20 ( 0 ) If E is a finite-dimensional C*-algebra then there is an a E IR+ such that
for every family (x~)~e, in E for which (x2x~)~, is summable.
The assertion follows from the Corollaries 4.2.1.18 and 1.7.1.7. C o r o l l a r y 4.2.1.21 A of R e E , then
I
If E is unital and x is the supremum in E of a subset
sup a ( x ) = sup sup a(y), yEA
Ilxll <_ sup IlYlI. yEA
Define a := sup sup a ( y ) . yEA
By Proposition 4.2.1.15, a l is an upper bound for A. Hence x < ,~1. Again by Proposition 4.2.1.15, sup a(y) <_ sup a(x) <_ a , for every y E A, so that a _< sup a ( x ) , a = sup a ( x ) . We set fi/"-inf a ( x ) . By Proposition 4.2.1.15, inf a(y) _~/3 for every y E A. Hence Ilxll - r(x) = sup{lal, 1/31} < sup sup{ l sup ~7(y)], l inf a(y)l} = sup Ilyll. yEA
P r o p o s i t i o n 4.2.1.22
I
yEA
( 0 ) P r E C E+.
The assertion follows from Proposition 4.1.2.21 a ~ c.
I
4.2 The Order Relation
101
4.2.2 M o r e a b o u t t h e O r d e r R e l a t i o n T h e o r e m 4.2.2.1
( 0 )
Given x E E , the following are equivalent:
a) xEE+. b)
There is a y E E+ with x = y2.
c)
There is a y E E with x = y*y.
d)
xt(x) E IR+ whenever x' is a continuous positive linear form on E . a =~b. Put f:a(x)---~lR+,
>x/a,
y :-- f ( y ) . Then y E E+ and y2 = f2(x ) = x (Theorem 4.1.3.1 a),b), Corollary 4.2.1.12). b ~ c =~ d is trivial. d ::v a. Let E be the unital C*-algebra associated to E ,-..-i 4.1.1.13). Then x'(x) E IR+ for every x' E E+. Thus
(Theorem
~(x) = ~ ( ~ ) c ~(~(~))c ~+ by Corollary 4.1.2.7 a).
I
Remark. In the complex case, the above theorem can be deduced also from Proposition 2.4.2.9 c). C o r o l l a r y 4.2.2.2
( 0 ) Let (x~)~c, be a f a m i l y i n inf IIx~]I = 0 tel
then
A x~:O.
LEI
In particular,
A(;l x )
nEIN
for every x E E+.
--0
E+. If
102
4. C*-Algebras
Let y be a lower bound for (x~)~ei. Let x' be a continuous positive linear form on E . T h e n
x'(y) ~ x'(x~) ~ I1~'11IIx~ll for every c C I (Theorem 4.2.2.1 a :=~ d) and so
z'(y) ___o,
9 '(-y) e ~ + . By T h e o r e m 4.2.2.1 d =v a,
- y E E+,
y<0. Hence
A xt - - 0 .
tel
Corollary 4.2.2.3 particular, ~, y, z e E , z
( 0 ) If x C E+, then y*xy C E+ .for every y E E . In
z*~z < ~*yz, -II~l ~*z < z*xz < IIxllz*z.
< y~
By Theorem 4.2.2.1 a ~
b, there is a z E E+ with z 2 = x. Thus, by
Theorem 4.2.2.1 c =v a,
y * z y - y*z~y = (zy)*(~y) e E +. The final relation follows from the preceding one and Corollary 4.2.1.16 a =v b.
I Corollary 4.2.2.4
( 0 )
Taking x , y C E and z C E+, x*y + y*x ~ x*x + y ' y ,
x*zy + y*zx ~ x* zx + y*zy.
4.2 The Order Relation
103
We have o <_ (~ - y ) * ( ~ -
o < (~-
y)*z(x-
y) = z * x - x*y - y*~ + y ' y ,
y) = ~* z x - ~*zy - y*zx + y*z~
(Corollary 4.2.2.3). Hence x*y + y*x < x*x + y ' y , x*zy + y*zx <_ x*zx + y*zy. C o r o l l a r y 4.2.2.5
m
( 0 ) If x C E#+ then for every y E E # , ((y*xy)n)ncIN
is a decreasing sequence in E#+ . By Corollary 4.2.2.3, y*xy E E+ and from ]ly*xyll <-IlY*II lixi] liY]l-< 1 it follows that y*xy E E#+. Since r ( y * z y ) = ]IY*zY]I < 1
(Theorem 4.1.1.16 b)), the claim follows from Corollary 4.2.1.12. C o r o l l a r y 4.2.2.6
m
( 0 ) If x' is a continuous positive linear from on E ,
then
i~'(~y)l ~ < Ilzll iyll:il~'llx'(x) for every x E E+ and y C E . By Theorem 4.2.2.1 a ==~ b, there is a z E E+ with x - z 2 . Thus I~'(xy)l ~ < x ' ( z z y ) ~ < x ' ( ~ z * ) ~ ' ( ( z y ) * ( z y ) ) =
= ~ ' ( x ) ~ ' ( y * ~ y ) < x'(x)ll~llx'(~*y) <_
< ~'(~)llxll IIx'll Ily*yll = ]lx]l Ilyll~llz'il~'(x) (Proposition 2.3.4.6 c), Corollary 2.3.4.8).
i
104
4. C*-Algebras
Corollary 4.2.2.7
( 0 ) If x, y C E and O C ]If, then ]](cos O)x + (sin 0)yi] 2 < I]x*x + y*y]l.
By Theorem 4.2.2.1 c =~ a, 0 < ((cos O)x + (sin O)y)*((cos O)x + (sin O)y) <_ <_ ((cos O)x + (sin O)y)*((cos O)x + (sin O)y)+ +((sin O ) x - (cos O)y)*((sin O ) x - (cos O)y) = (cos 20)x* x + (sin 20)y*y + cos 0 sin O(x*y + y ' x ) + +(sin 20)x*x + (cos 20)y*y - cos 0sin O((x*y + y ' x ) = x*x + y*y. By Corollary 4.2.1.18, II(cos O)x + (sin 0)yll 2 -II((cos O)x + (sin 0)y)*((cos O)x + (sin 0)y)l I _< <_ I1~*~ + v'vii 9
Proposition 4.2.2.8
m
Let x' be a positive linear form on E and A a non-
empty downward directed set of E+.
a) {x e E[ inf x'(x*yx) = 0} is a vector subspace of E . yEA
b) / f l K = r
andif
inf x' (x *yx) - 0
yEA
for all unitary elements x of E , then
inf x' (x*yx) = 0
yEA
for all x C E .
a) Let x , y C E such that inf x'(x*zx) - inf x'(y*zy) - O.
zEA
zEA
We have (~ + y)*z(x + y) = x* z~ + y* zy + x*zy + y*z~ < 2(~* z~ + y* zy)
for all z E E+ (Corollary 4.2.2.4). Hence inf x'((x + y)*z(x + y)) = O.
zcA
b) follows from a) and Corollary 4.1.3.7.
I
4.2 The Order Relation
Theorem 4.2.2.9 a)
( 0 )
105
Take x E R e E .
There are uniquely d e t e r m i n e d y, z C E+ such that x=y-z,
yz=zy=O.
Define X§ :-- y,
and call x + and x -
X-
:=
Z
the p o s i t i v e and t h e n e g a t i v e p a r t o f x , respec-
tively.
b) x § - f(x),
Iixll- sup{tlx§
x- - g(x),
II~-II},
where
>IK, a ,
f:cr(x) g:a(x) c)
x and (x +, x - )
>IK,
a,
(-x) + = x - ,
e)
Let F
>sup{-c~,O).
generate the s a m e closed real subalgebra of E
is the C*-subalgebra of E generated by
d)
>sup{a, 0},
and E ( x )
{x +, x - } .
(-x)-=x +
be an E - m o d u l e and a C F
such that a x = 0 (resp. x a = 0 ).
then ax + - a x - - 0
(resp. x + a -- x - a - 0 ) .
a) and b). Step 1 b) and the "existence" part in a) Since or(x)C IR (Corollary 4.1.2.13 a ==:>b), f and g are well-defined. Put y :- f(x),
z :- g(x).
Then y, z E E+ (Corollary 4.2.1.12) and y-
z = f(x) - g(x) = (f - g)(x) = x,
106
4. C*-Algebras
yz - f ( x ) g ( x ) = ( f g)(x) = O, zy=O,
sup{llyll,
Ilzll)
sup(llfll, Ilgll)
=
=
r(x)
=
Ilxll
(Theorems 4.1.3.1 and 4.1.1.16 b)). Step 2
the "uniqueness" part in a)
Take y, z E E+ with
x=y-z,
yz-zy=O
and let F be the C*-subalgebra of E generated by {y,z}. F is a Gelfand C*-algebra (Corollary 4.1.2.3) and F
F
F
FF
~=~-~,
~'2=o.
Thus
f (x) -- f o ' ~ - ~,
y=f(x),
g(x) = g o ' ~ -
z-g(x).
c) follows from b) and Corollary 4.1.3.3 a), d) follows from a). e) follows from b) and Theorem 4.1.3.1 f). C o r o l l a r y 4.2.2.10
IR+ (and x' C R e E '
( 0 )
I
linear form x' on E is positive iff x'(E+) C
- JR).
If x' is positive, then x'(E+) C JR+ by Theorem 4.2.2.1 a :=~ d. For the converse suppose that x'(E+) C IR+ (and x' E Re E' if IK = JR). Then
x'(z*z) ~ for every x C E by Theorem 4.2.2.1 c ~ a. By Theorem 4.2.2.9 a),
.'(~) = ~'(.+) - x'(~-) e ~ , for every x C Re E . It follows that
x'(x*) - x'(rex - i i m x ) = x'rex) - i x ' ( i m x ) = x'(rex) + iz'(imy) = x'(x) for every x C E . Hence x' is involutive and therefore positive.
I
3.2 The Order Relation
Corollary 4.2.2.11
107
( 0 ) I f u" E -+ F is a h o m o m o r p h i s m of C*-algebras
then u'(F_~) C E ~ . Take y' C F_~ and x C E + . By Corollary 4.2.1.4, ux+ C F+, so that by Corollary 4.2.2.10, (x, u'y') = (ux, y') E IR+ , u' y' C E+ .
Corollary 4.2.2.12
I
The maps
ReE
)E+,
x,
) x +,
ReE
>E+,
x,
>x -
are continuous.
The corollary follows immediately from Theorem 4.2.2.9 b) and Proposition 4.1.3.20. I
Corollary 4.2.2.13
( 0 ) Every positive linear f o r m x' on E is continuous
and IIx'll-
sup xE E#+
x'(x).
By Theorem 4.2.2.9 b), the absolute convex hull of E+# is a 0-neighbourhood in Re E . Thus by Corollary 1.7.1.8 b), x'lRe E is continuous. If ]K = ~ , then by Proposition 2.3.2.7, x' is continuous. Assume IK = IR and let x E E #. Then 89 + x* ) E Re E# and 1 x'(~(x-
x*
))=~'*(~(~
1
_ x*)) = . '
1
(~(~*
-
~1)=
-~'
1
(~ (x
-
Hence
9 '(~1 (x 9 '(~)
x* 1 ) = 0 ,
- x ' ( ~1( ~
+ X*)),
Thus x' is bounded on E # and therefore it is continuous. Take x C Re E # . Then x + , x - C E#+ (Theorem 4.2.2.9 b)) and
~*11.
108
4. C*-Algebras
]x'(x)[ = Ix'(x +) - x'(x-)] < sup{x'(x+),x'(x-)} <_ sup x'(x). xE E#+
By Proposition 2.3.2.22 j), IIx'll ~
sup x'(x). xE E#+
I
The reverse inequality is trivial.
Proposition 4.2.2.14
( 0 )
Let IK = r
(IK = IR). Then the following are
equivalent for every Gelfand C*-subalgebra F of E : a)
F is a maximal Gelfand C*-subalgebra of E .
b)
F=F
~
(F=Re(FC)).
a =~ b. Since F
is commutative, F C F ~. Take x E R e ( F r
Then
F U {x} is commutative (and contained in Re E ) , so that the C*-subalgebra of E generated by FU{x} is a Gelfand C * - s u b a l g e b r a o f E (Corollary 4.1.2.3). Since F is maximal, we get x E F . Hence Re ( F ~) C F and F = F~
(F = ae (r~)).
b =~ a. Let G be a maximal Gelfand C*-subalgebra of E containing F (Corollary 4.1.2.3). By a =v b, G = Gc
(G = Re (G~)).
By b), F c G = G~ c F c= F
(FCG=Re(G
c) C R e ( F c ) = F ) ,
and so F = G and F is a maximal Gelfand C*-subalgebra of E .
Proposition 4.2.2.15
( 0 ) Let E be areal C*-algebraand E its corn-
plexification.
a)
F o r all
I
(Xl, Yl), (X2,Y2) E E, (x~, y~) < (~, y~) ~
~ < z~, (~, y~) < (~, ~ ) .
4.2 The Order Relation
109
For all x C E ,
b)
x _> o ~
(z, o) > o,
(0, x) _> 0 ~ o
x = 0. o
c)
c~ e [ - 1 , 1], ( x , y ) e
d)
Let A be a subset of E such that A x {O} has an infimum ( x , y ) in E . Then y = O
E + =~ (x, a y ) e E + .
and x is the infimum of A in E .
e) g ;~ i~ o~d~r compl~t~ (o~d~r ~-compl~t~)
t h ~ E i~ at~o o~d~r
(order a-complete). a) We have (X2 -- Xl, Y2 -- Yl) _~ 0. By T h e o r e m 4.2.2.1 a ==> c, there is an (x, y) C / ~ such t h a t (X2 -- X l , Y2 -- Yl) - - (X, y ) * (X, y ) .
Thus x2 - xl = x*x + y ' y ,
Y2 - Yl : x*y - y ' x ,
(X2, Yl) -- ( X l , Y2) - - (X2 -- X l , Yl -- Y2) - -
= (z*~ + y'y, - x * y + y*~) = (x, -y)* (x, - y ) . By T h e o r e m 4.2.2.1 c ::v a,
x~ < x~,
(~1, y~)_< (x~, yl).
b) A s s u m e t h a t x _> 0. T h e r e is a y C E with y*y - x . T h e n
(x, o) - (y'y, o) - (y, o)*(y, o) > o. T h e implications (x, 0) _> 0 ~
x _> 0,
(0, x) >
x = 0
co,~pl~t~
i10
4. C*-Algebras
follow from a).
(0, y)_< (~, 0), 0 _< ( x , - y ) . Let /3, 3, E [0, 1] such that f l + , y = 1,
f l - q,-- a .
Then
0 <_ Z(~, ~) + 7(~,-y) = ((Z + 7)~, (Z - 7)y) - (~, ~ ) . d) By a), x is a lower bound for A in E . Let z be a lower bound for A in E . B y b ) ,
(z, 0) is a lower bound for A • {0} in /~.Hence
(~, 0) <_ (~, y). B y a ) , z <_ x , s o t h a t x is the infimum of A in E . (0, y) is the infimum of ( A - x) x {0} in E . By b), 0 is a lower bound for ( A - x) • {0) in /~. Hence
(0, u) > 0 and therefore y = 0 by b). e) follows from d). P r o p o s i t i o n 4.2.2.16
m
( 0 ) Let E be unital. Take B c A c E . Let x be
the infimum of B in A and y an invertible element of E such that y*Ay - A . Then y*xy is the infimum of y ' B y
in A .
We have y *xy ~ y*zy
for every z C B (Corollary 4.2.2.3), so y*xy is a lower bound for y ' B y in A. Let x0 be a lower bound for y ' B y in A. Then Xo ~_ y* zy , (y-1)*xoy-1 ~_ z
for every z C B (Corollary 4.2.2.3), so that (y-1),x0y-1 ~_ X xo <_ y* xy .
Hence y*xy is the infimum of y ' B y in A.
m
4.2 The Order Relation
111
P r o p o s i t i o n 4.2.2.17 ( 0 ) Let E be unital, B a downward directed set of E , and x the infimum of B in E . If one of the assumptions from below is fulfilled then B ~ C {x}~: a) I K = e ; b)
1 K - IR and E is order complete;
c) I K - IR, E is order a-complete, and B is countable. a) Let y be a unitary element of B ~. Then y*zy = y*yz = z
for all z E B, so that y ' B y -
B . By Proposition 4.2.2.16, y*xy = x .
Hence xy = y x ,
y e {x} ~
Since every element of B c is the linear combination of four unitary elements of B c (Corollary 4.1.3.7, Corollary 4.1.4.2 b)) we deduce B ~ c {x} ~
b) or c). By Proposition 4.2.2.15 d), (x, 0) is the infimum of B x {0} in E . By a), (B • {o})~ c {(x, 0)} ~
It follows that B~
{~}~
,,
Definition 4.2.2.18 ( 0 ) A subset A of E is called order faithful if for every downward (upward) directed subset of A , which has an infimum (supremum) in E , the infimum (supremum) belongs to A. A is called order a-faithful if the countable subsets of A have the last property. If E is order complete (order a-complete), then every order faithful (order a-faithful) set of E is order complete (order a-complete).
112
4. C*-Algebras
C o r o l l a r y 4.2.2.19 a)
If IK =(~
( 0 ) Let A be a subset of E .
or if IK -
IR and E is order complete then A c is order
faithful.
b)
If IK = IR and E is order a-complete then A ~ is order a-faithful.
a) (resp. b)) Let B be a downward directed (rasp. countable) subset of A ~ and let x be its infimum in E . By Proposition 4.2.2.17 and Proposition b),c), A C A cc C B ~ C {x} c. Hence x C A ~ and A ~ is order faithful (resp. order a-faithful), C o r o l l a r y 4.2.2.20 E. a)
( 0 )
m
Let F be a maximal Gelfand C*-subalgebra of
If IK = C or if IK = IR and E is order complete then F is order faithful. If in addition E is order complete (order a-complete) then F is also order complete (order a-complete) and a(F) is a Stone space (a a - S t o n e space).
b)
If IK = IR and E is order a--complete then F is order a-faithful, F is order a-complete, and a ( F ) is a a-Stone space.
a) (rasp. b)). By Proposition 4.2.2.14 a =v b, F = F ~ if IK = C and F = R e ( F r if IK = IR. By Corollary 4.2.2.19, F is order faithful (rasp. order a-faithful). If /~ is order complete (rasp. order a-complete), then by Proposition 4.2.2.15 e), E is order complete (rasp. order a-complete). It follows that F is order complete (order a-complete). By Proposition 1.7.2.13 e ==> a, a(F) is a Stone space (a-Stone space).
II
Let F be a maximal Gelfand C*-subalgebra and A a subset of F The above proof shows that under the above hypotheses if A has an infimum Remark.
x in E then x E F , even if A is not downward directed. But it may happen that A has an infimum in F and no infimum in E , as the following example shows. E := s
{[o0] fl
a, f l c I K
}
,
4.2 The Order Relation
A {[10] Eoo 0 0
P r o p o s i t i o n 4.2.2.21
( 0 )
'
0
113
.
1
Let E be unital and A a downward directed
set of E (of E+ ). Let x be the infimum of A in E (in E+ ). Then for every y E E , y*xy is the infimum of y*Ay in E (in E+ ) (Corollary ~.2.2.3). In particular, p E p is an order faithful C*-subalgebra of E for every p C Pr E . Let F be the set of all y C E for which y*xy is the infimum of y*Ay in E (in E+ ). Take y, z e F and let b be a lower bound for (y + z)*A(y + z) in E (in E + ) . Take al,a2 C A . There is an a C A with
a<_al,
a<_a2.
Then b < (y + z)*a(y + z),
b - (y + z)*x(~ + z) < (y + z)* (a - ~)(y + ~) -
-- y * ( a - x)y + z*(a - x)z + y*(a - x)z + z*(a - x)y <_ <_ y * ( a - x)y + ~*(a - x)z + y*(a - ~)y + z*(a - x)z -= 2(y*(a - x)y + z*(a - x)z) (Corollary 4.2.2.4), l(b-
(y + z)*x(y + z)) < y*(a - ~)y + z*(a - x)z <
2
--
--
<_ y*(al - x)y + z*(a2 - x)z - y*aly - y*xy + z * a 2 z - z*xz (Corollary 4.2.2.3). Since al and a2 are arbitrary and y, z E F ,
l ( b - (y + z)*x(y + z)) < 0 b < (y + z)*x(y + z ) . Hence ( y + z ) * x ( y + z )
is the infimum of ( y + z ) * A ( y + z )
in E (in E + ) ,
y + z C F , and F is a vector subspace of E . By Proposition 4.2.2.16, F contains all invertible elements of E . Since every element of E is the difference of two invertible elements of E , it follows that F = E . E
114
4. C*-Algebras
C o r o l l a r y 4.2.2.22
( 7 ) Let E be unital, A a downward directed set of
E + , and x the infimum of A in E + . Then x is the infimum of A in E .
Let y be a lower bound for A in E . Then y E R e E , y - x is a lower bound for A - x in E and 0 is the infimum for A - x in E+. By Proposition 4.2.2.21, 0 is the infimum of ( y - x ) + ( A - x ) ( y - x) + in E+. It follows from ((X -- y ) + ) 3 _. ( y __ X ) + ( y
__ x ) ( y
-- X) +
that ( ( y - X)+) 3 is a lower bound for ( y - x ) + ( A - x ) ( y - x) + in E+. Hence ((y -- X)+)3 -- O,
(y-x)
y-
+ =0,
z - -(y-
x)- <0,
y<x,
m
i.e. x is the infimum of A in E . P r o p o s i t i o n 4.2.2.23
( 0 )
x*x < y*y,
Take x , y , z E E with
y*yz - O
(xx* < yy*, zyy* = O).
Then zz = O
(zz = O).
We have that 0 < z*x*xz < z * y * y z - O
(0 < zxx*z* < zyy*z* - 0 )
(Corollary 4.2.2.3), z*z*zz=O
Ilxzll ~ -
Iz'x'~zll
= o
xz - O
(zzz*z* - O ) ,
(llzxl ~ -
(zz = O) .
Iiz*~*z'll
-
0),
m
4.2 The Order Relation
P r o p o s i t i o n 4.2.2.24
115
Let E be a complex C*-algebra and E the complexio
fication of the underlying involutive real algebra of E . The element (x, y) C E is positive (Theorem ~.1.1.8) iff x + iy and x - iy are positive.
Assume that (x, y) is positive. Then there is an (a, b) 9
such that
(x, y) = (a, b)* (a, b) (Theorem 4.2.2.1 a ::v c). Thus x=a*a+b*b,
y-a'b-b'a,
(a + ib)*(a + ib) - (a* - ib*)(a + ib) = = a*a + b*b + i(a*b - b'a) = x + iy, ( a - i b ) * ( a - ib) = (a* + ib*)(a - ib) = = a*a + b*b - i ( a * b - b'a) = x - iy.
Hence x + iy and x - iy are positive (Theorem 4.2.2.1 c =~ a). For the converse, assume that x + iy and x - iy are positive. Then there are c,d C E with x + iy -- c*c,
x-
iy = d*d
(Theorem 4.2.2.1 a ::v c). We have
(
l(c+d), l(c_d)
=~ =1
)'(
,7(c*-
l(c+d ) l(c_d)
=
c+d, 7
( (c* + d*)(c + d) + (c* - d * ) ( c - d) , 4
)
) _
!((c* + d*)(c - d) + (c* - d*)(c + d)) i
_-1-4 (2c*c + 2d*d ' -1 (2c*cz - 2d'd)) = (x, y).
Hence (x, y) is positive (Theorem 4.2.2.1 c =~ a).
I
116
4. C*-Algebras
4.2.3 E x a m p l e s E x a m p l e 4.2.3.1
Let T be a set. An element x of the C*-algebra g ~
(Example ~. 1.1.5) is positive iff
for every t E T . We have a(x) - x(T) (Example 2.1.4.3). E x a m p l e 4.2.3.2
I
Take n C IN and let s
n) be the C*-algebra defined in
Example ~. 1.1.7. Define n
~" s
n)
>IK,
a,
~E
aOxji
i,j=l
for every x E s where we have identified the elements of s their associated n x n-matrices. Then ~ E s ' for every x C s
n) with the
map L ( ~ ~)
~~(~)',
~,
~~
is bijective, linear, and involutive and for every x C s
n) the following are
equivalent:
a) z_>O. b)
~_>0. The first two assertions were proved in Example 2.3.1.33. a =, b. By Theorem 4.2.2.1 a =, c (and Example 4.1.1.7), there is a
y C IKn,n with
x-yy. Thus
i,j=l
k=l
e=l
4.2 The Order Relation
117
for every a C IK,,n. Hence ~ is positive. b =~ a. By Example 2.3.1.33, x C ReIK,,n. By Example 2.3.4.3, a(x) C
~+. Proposition 4.2.3.3
Take n C IN and [aij] E ]Rn,n. Then the following are
equivalent: a)
The linear form
n i,j=l is positive (Example 2.3.1.32). b)
The linear form
(~n,n
>(~ ,
[O~ij] I
> ~_~ OLijaji 1,j-1
is positive (Example 2.3.1.32). a ~ b. [aij] is selfadjoint and by the last assertion of Example 2.3.4.3, ~([a~j]) c ~ + .
b) follow~ now from Ex~mpl~ 4.2.3.2 ~ ~ b. b ~ a is trivial,
E x a m p l e 4.2.3.4
Take n C IN and let E be the C*-algebra f~(iK n) (Exam-
ple 4.1.1.7). Given x, y E lK ~ and a E E , define
i--1
and
) ~ ~ i,j--1 Then, given a E E "
m
118
4. C*-Algebras
a) ~ E T(E) iff there exist a finite subset A of IK n and a function f " A --+ IR+ such that
= 5~,~ for all x , y E A and E
f ( x ) = 1.
xEA
xEA
b) ~ E T 0 ( E ) iff there is an x E IK n such that a = [~lx],
II~ll = 1 .
a) By Proposition 4.2.3.3. a ==v b, it suffices to prove the statement for the complex case. By Example 4.2.3.2, a is a positive element of s positive. Hence identifying s162
with s162
iff ~ is
via the map
it follows "r(L(Cn)) = {a E s
II
~11- 1}.
Take ~ E T(E.((F,n)). By Example 4.1.4.11, there are a finite subset A of C n and a function f 9A --+ IR+ such that
for all x , y E A and
xEA
f(x) =
- 1.
xEA
The converse is trivial. b) Take x E ~ Take a, b E E
(resp. x E I R ~) with I i x i i = l . Then [xx] E T ( E ) b y a ) .
and c~, /3 E ]0,1[ such that ~,bET(E),
a+fl-1,
[xlx ] : a a + ~ b .
By a), there are finite subsets A, B of IKn and functions f " A --+ ]0, co[, g 9B ~ ]0, co[, such that
4.2 The Order Relation
119
(ylz> - 5~,~
for all y, z E A and all y, z E B a= ~
and f (y) -- 1,
f(Y)[YlY],
yEA
yEA
b- ~
g(z)[zIz] ,
~
zEB
g(z) - l .
zEB
Thus f (y)[yiy] + ~ ~
[xix] = ~ ~ yEA
g(z)[zlz] ,
zEA
and 1 - ([xlx], [xlx])-
yEA
zEJ
= ~ ~f(y)l<xly>l
~ + ~ ~-]9(z)l<xlz>l ~ <
yEA
zEB
f (y) §
yEA
~
g(z) - 1
zEB
by Schwarz's Inequality. Hence I(xly>l - x
for all y E A U B. This implies that A and B contain exactly one element and = b = [xl~].
Thus [xlx] E T0(E). The converse follows from a). Example 4.2.3.5
I
The matrix
[ ~E L /~] (5] K 2 ) ' y is positive (Example 4.1.1.7) iff o~,~E~+,
~--~,
I~1~ < o ~ .
120
4. C*-Algebras
This follows from Example 4.2.3.2 and Example 2.3.4.4 b ca c. Remark.
a) Put y := 1 1
'
[2o] 0 0
Z :'--
I
[00] 0 2
Then x, y, z are positive and x
but there are no positive matrices xl, x2 for which x=xl+x2,
xl <_y,
x2<_z.
In particular, Re s 2) is not a vector lattice. b) The assertion of the above example still holds for ]H2,2. E x a m p l e 4.2.3.6
[
Put
c~ /~ ] e R.e/~(]K 2)
y:=
21
31
E R e E ( ] K 2)
(Example 4.1.1.7).
b)
(x e t:(IK2)+,
< 1) ca
c a ( c ~ , 5 9 [0,1[, c) y c ~(]K2)+,
Ilyl] < 1,
and there is no ? C IR+ with
y<[7 -
o]. 0
0
a) By Corollary 4.2.1.16 a =:v b,
licit + ~, II~ll + ~ ~ ~+,
lgl ~ <_ (ll~ll + ~)(llxll + ~).
b) follows from a), Example 4.2.3.5, and Corollary 4.2.1.16 b =a a. c) follows from b) and Example 4.2.3.5.
m
4.2 The Order Relation
E x a m p l e 4.2.3.7
121
Let
be a selfadjoint element of s
2) (Example ~.1.1.7) which is neither positive
nor negative. Define p . _ v/(c~ _ ~)2 + 4 l # l 2 c
a'--
o~ + 6 + p 2 '
b'-
lit+,
o~ + 6 - p 2
Then p > O, b
6
:
p
[ ] [
~
,~-o+p2
P
~
~
] [ ] b
2
e_~
~ ,1
bl
a and b are the zeros of the polynomial
#
oz-t
#
6-t
/ e ~[t]
j
so that the hypotheses imply b
p>0.
The second claim follows immediately from Example 4.1.3.19 and Theorem 4.2.2.9 b). 1 Remark.
Let E be a C*-algebra. Take x, y E E+ and put z--x-y.
If xy = yx then
122
4. C*-Algebras
z+
z- < z ,
but these inequalities do not hold in general without the above commutativity condition as the example x :=
[11] 1 1
,
i1 ]
y :=
-i
1
shows. 4 . 2 . 3 . 8 If (T, ~, #) is a measure space, then the order relation on the C*-algebra L ~ (#) (Example ~. 1.1.5) coincides with the usual order relation
Example
on L~162 Take x C L ~ ( # ) . By Example 2.2.2.2,
a(x) C IR+ e==* x > O p-a.e. Proposition
4.2.3.9
aj
X 2
b)
I~1 e P r E .
I
( 0 ) If x C Re E then the following are equvalent:
C PrE.
c) z~= Izl. If these conditions are fulfilled then x+ = ~(1~1 1 + x) e Pr E ,
X+X
--
XX
+
--
X +
~
1 x - = ~ ( I x l - x) 9 P r E ,
X -
X
--
XX
_
--
--X
_
.
E(x) is a Gelfand C*-subalgebra of E (Corollary 4.1.2.3) and the assertions follow by Gelfand transform.
I
4.2 The Order Relation
123
4.2.4 Powers of P o s i t i v e E l e m e n t s
Definition 4.2.4.1
( 0 ) Take p > 0 and f " IR+
> IR+,
o~
> aP.
Given x E E + , define x p "-- f ( x ) .
By Theorem 4.1.3.1 a), the above definition coincides with Definition 2.1.1.1 whenever p E IN. P r o p o s i t i o n 4.2.4.2
a)
( 0 ) Take a,/~ E ]R+\{0}.
The map E+
>E + ,
x,
>x ~
is a h o m e o m o r p h i s m .
b) /f x E E + , then x y E E + , x~+Z = x~x z,
I1~~
II~ll~,
x~Z = (x~) z,
{~}~: {~}~,
E(~): E ( ~ )
c) I f x, y E E+ and x y = y x , then x y E E+ , ( x y ) a - x~ya ,
x < y c==~ x ~ < y ~ .
d) I f x E E+ , then x ~ belongs to the closed real subalgebra of E generated by x .
e) x E E + ~ , 0 < ~ < / ~ z Z < z " Take x E E#+ and a s s u m e either IK = ~
or IK = IR and E
a - c o m p l e t e . I f (x 1)nEW has a s u p r e m u m p in E , then
g)
9 s E~_, ~' s E'. ~ ~ '
- ~'~ E E ' . .
p E Pr E .
order
124
4. C*-Algebras
h)
Let F
be an E - m o d u l e ,
x E E+,
a n d a C F such that ax = 0
(resp.
xa = 0). Then ax ~ = 0
(resp. x ~a = 0 ) .
a) follows from Proposition 4.1.4.5. b) Put p := E ( x ) .
Then
x/o+~) - ~ )
= (x)o(~)~ = xox~
F
A =(~o)~
(Corollary 4.1.3.5), so that x~+Z = x ~ x ~,
xOZ = (x~) ~
(Corollary 4.1.2.5). The relations
I1~11 = Ilxll~, follow c) d) e)
{ x } ~ : {x~} ~,
E(x)= E(x ~)
from Corollary 4.2.1.7 and Proposition 4.1.4.5. follows from the Gelfand transform. follows from Corollary 4.1.3.3 a). Since g(x) c [0, 1]
(Corollary 4.2.1.16 a =~ c), the claim follows from Corollary 4.2.1.12. f) Let F be a maximal Gelfand C*-subalgebra of E containing {x~ In ~ IN} (b) and Corollary 4.1.2.3). By Corollary 4.2.2.20, p E F . Using the Gelfand transform on F , it is easy to see that p is an orthogonal projection in E. g) By Proposition 2.2.7.4 a), XX ! =
XtX,
4.2 The Order Relation
125
so that X X ' ) * ~ X ' X = XXt ~
x x ' 9 Re E ' .
Take y 9 E+. By c), y x 9 E+ and so
(y, xz') = (yx, x') e m , x x ' 9 E+ (Corollary 4.2.2.10).
h) follows from Theorem 4.1.3.1 f). Corollary
4.2.4.3 ( 0 )
Given x 9 E + , the following are equivalent:
a)
there are distinct c~,~ 9 JR+\{0}, with x~ = x ~
b)
x ~ = x f o r every c~ 9 ]R+ \ { O} .
c) x C P r E . a => b. Define f:a(x)
>IR,
7'
>7 ~ - 7 z-
Then
f (x) = o, so that f(cr(x)) = cr(f(x)) = {0}
(Corollary 4.1.3.5), o ( x ) C {0, 1}.
b) now follows. b =, c is trivial. c =~ a. By Proposition 4.1.2.21 a =~ c, a(x) C {0,1} and a) now follows.
I
126
4. C*-Algebras
( 0 ) Let E be unital. Take p 9 IR and
Definition 4.2.4.4
f'lR+\{O}
>IR+\{O},
g']R+\{0}
~IR,
a,
a,
>o~p
>10ga.
Given x E E+, if x is invertible, define x p :-- f ( x ) ,
log x :-- g(x). If p > 0, then the above definition of x p coincides with Definition 4.2.4.1. By Theorem 4.1.3.1 a), the above definition of x p coincides with Definition 2.1.2.5 whenever - p E IN. Since x ~ - 1, the above Definition is compatible with Definition 2.1.1.3. By Corollary 4.1.3.9, for every x e U E(0), (1 + x) ~ and log(1 - x) defined above coincide with [1 + x] ~ and log(1 - x) defined in Proposition 2.2.3.13 a) and 2.2.3.9, respectively.
Proposition 4.2.4.5 put
( 0 ) Let E be unital. Take a , ~ C ] R , a ~ = O , and
A "= {x C E+ I x is invertible}. a)
The maps A
A
>A,
x~
) x ~,
~ReE,
x~
)logx,
are homeomorphisms. b)
If x c A then x,+Z _ xoxZ,
elOgX_x,
{~}~-
xoZ _ (x")Z,
logx o-alogx,
{~o}~ : {log z}~,
logx_<x,
E(~) - E(~ ~ : E(~, 1) - E(log ~, 1).
4.2 The Order Relation
c)
127
Given x, y E A with x y = y x , (xy) a = x a y ~ ,
log(xy) = log x + log y,
x < y ~
log x < log y.
a) follows from Proposition 4.1.4.5 and Corollary 4.1.2.13 b r
a.
b) Define F:=E(x,
1).
log
= alog
Then
logx ~ -
log x ~ =
=alogx,
(Corollary 4.1.3.5), so that x ~+z-x"x
~,
x ~ z = ( x ~ ) ~,
logx "-~logx
(Corollary 4.1.2.5). The relations log x < x,
{x} c -
{x"} c - {log x} c ,
elog x ~
X,
E ( x ) - E ( x ~) - E ( x , 1 ) -
E(log x, 1)
follow from Corollary 4.2.1.12 and Proposition 4.1.4.5. c) follows from the Gelfand transform. P r o p o s i t i o n 4.2.4.6
(0)
m
Let E be unital and take x, y C E+ with x <_ y .
I f x is invertible, then y is also invertible and y-1 ~_ x-1
128
4. C*-Algebras
There is an a > 0 such that a l _< x (Proposition 4.2.1.15). Hence a l _< y and y is invertible (Corollary 4.2.1.13). Now 1 = x - ~1x x
1
8 9_
1
yx-~
(Proposition 4.2.4.5 b), Corollary 4.2.2.3). Thus x~y l x l = (X 89
~)-1 <_1
(Corollary 4.2.1.13) and 1
1
~ = X -1
y-1 ~_ x - ~ l x
(Proposition 4.2.4.5 b), Corollary 4.2.2.3). C o r o l l a r y 4.2.4.7 a > O, then:
m
( 0 ) If E is unital, if x , y E E+ with x <_ y and if
a) x ( a l + x) -1 _< y ( a l + y)-i 1
1
b) 0 <_ x~(al + y)-lx~ < 1. c)
IIx~(al
+ y)-89 < 1.
a l + x and a l + y are invertible and (al + y)-I __ (al + x) -1 by Proposition 4.2.4.6. a) x(al + x ) -1 --~ 1 - a ( a l + x) -1 __ 1 - a ( a l + b) By Corollary 4.2.2.3 and Corollary 4.2.1.12, 1
1
y)-i
I
__
y(al + y)-i
1
0 < x~(al + y)-lx~ < x~(o~l + x)-~x ~ < 1. c) By b) and Corollary 4.2.1.17 b =~ a, I x 89 P r o p o s i t i o n 4.2.4.8 every a C ]0, 1[.
+ y)-89 ( 0 )
= IIx89
+ y)-'x89
_< 1.
Take x , y C E+. If x < y, then x ~ < y" for
Given n E IN, define
/ [0,11yll]
m
>IR,
3'
f 0
/3dt (1 + j3t)t ~ '
4.2 The Order Relation
129
and In -[0, llyl{]
> ]R,
1 ~(2n) ~ j3 > ~- k=l -~1+ ~
/3'
Then f 9 C([0, Ilyll]) and (f~)~e~ is an increasing sequence in C([0, Ilyll]) (since the function
JR+
~ JR,
t~
(1 + 9t)t-
is decreasing) converging pointwise to f . By Dini's Theorem, lim
n----> o o
IIf - A l l -
o.
Since x(1 + tx) -1 ~ y(1 + ty) -1 for every t C IR+ (Corollary 4.2.4.7 a)) we have that --1
A(x)-~:~
_
= A(y)
~:~
for every n E iN. Hence f (x) -- lim fn (x) < lim f~ (y) -- f (y) n - - ~ OO
n - - ~ CK:)
(Theorem 4.2.1.1, Proposition 1.7.1.5 a), Theorem 4.1.3.1 b)). Since f(~) = fo~ every
(1 +/3t)t" =
(1 + s)s"
(1 -~ s)s"
~ ~ [0, llyll], x" _< y".
Corollary 4.2.4.9
Take x , y C P r E .
If, for some a E JR+, x < a y , then
x<_y.
By Corollary 4.2.4.3 c =v b and Proposition 4.2.4.8, x = x z < (c~y)z = aZyZ = aZy for every /3 C ]0, 1]. Hence x< --
lima sy=y. 0<8-+0
130
4. C*-Algebras
C o r o l l a r y 4.2.4.10
Take x, y E Pr E . I f x = y z (x = z y ) f o r s o m e z C E ,
then x < y .
Since =
~
-
yzz*y <__Ilzll~y
(x = ~ = y z * z y ___ Ilzll~y)
(Corollary 4.2.1.16 a =~ b, Corollary 4.2.2.3), x _< y by Corollary 4.2.4.9.
Proposition 4.2.4.11
I f f o r s o m e a E ]1, ec[,
x, y c E + , x < y ~ x then R e E
m
~
is c o m m u t a t i v e .
Step 1
x, y C E + , x < y ~ x
2
By complete induction, x, y c E + , x _ < y : = ~ x
~n <_y~n
for every n E IN [Proposition 4.2.4.2 b)). By Proposition 4.2.4.8 (and Proposition 4.2.4.2 b)), x, y C E + , x < y = = v x
Step 2
2
x, y E E+ :=v re x y C E + , (imxy) 2 < (re xy) 2
Given c > 0, it follows from Step 1 that x~x+cy,
x y + y x + ~y2 ~_ O.
Hence 1
r~ ~y - ~ ( ~ y + y~) > 0.
Since y x y is positive (Corollary 4.2.2.3), 0 < re x ( y x y ) -- re(xy) 2 -- (re x y ) 2 - (im xy) 2 ,
4.2 The Order Relation
131
(im x y ) ~ <__ (re x y ) 2 .
Step 3
E is c o m m u t a t i v e
Put A := {/3 e [1, c~[ I x , y 9 E+ = = , / 3 ( i m x y ) 2 < ( r e x y ) 2 } ,
7 " - s u p ~ _< oo. ~cA By Step 2, 1 E A. Suppose 3' is finite. T h e n 3' E A . Take x, y E E+ and a'--rexy,
b :- im xy.
Then, we see successively t h a t ~,b2 _< a 2 ,
a 2 - 7b 2 , b2 E E+
(Theorem 4.2.2.1 c ::v a),
0 _< 2re b2(a 2 - "yb2) - b2(a 2 - ")'b2) + (a 2 - "yb2)b 2 =
= a 2b2 + b2a 2 _ 2~/b4 '
_ a 2 b 2 _ b2a 2 <_ -2~/b 4
by Step 2. Since y x y , ab2a, a , and r e a ( b a b )
are all positive (Corollary 4.2.2.3
and Step 2) and since x(yxy)
= ( x y ) 2 = (a + ib) 2 = a 2 - b 2 + i(ab + ba) ,
we see successively t h a t ",/(ab + ba) 2 < (a 2 - b2) 2 ,
93b 4 < 3,ba2b < 3,(ab2a + b a 2 b + 2 r e a ( b a b ) ) =
= 9/(ab2a + ba2b + abab + baba) = ~/(ab + ba) 2 < (a 2 - b2) 2 =
132
4. C*-Algebras
= a 4 + b4 -
b2a 2 <_ a 4
a2b 2 -
27b 4 (Corollary 4.2.2.3),
A- b 4 -
(72 + 2 " / - 1)b 4 < a 4 ,
V/72 + 27 - l b 2 < a 2 (Proposition 4.2.4.8),
V/72 + 2 7 -
1 c A,
~/72 + 2 7 - 1 < 7 , 1
1_<7_<~
which is a contradiction. Hence A cannot be bounded, (im x y )
Ilxy-
2 = O,
y~ll = = 12i(imxy)ll 2 -- 4 I ( i m x y ) 2 [ [ - 0,
xy = yx.
By Theorem 4.2.2.9 a), Re E is commutative, P r o p o s i t i o n 4.2.4.12
m
( 0 ) Take x' E E~+. Let A be a n o n e m p t y downward
directed set of E+ with
inf z ' ( x ) - O. xEA
Let ~ be the lower section filter of A and take ~ > O. Then
limx'(x '~) = 0. x,~"
Case 1
c~ >
1
We may assume that E is unital (Proposition 2.3.4.11) and that 1 is an upper bound for A. Then for x E A,
(~'(xO)) ~ -
(z' (~o-~x~
<~'
(Proposition 2.3.4.6 c), Corollary 4.2.1.11), so that
4.2 The Order Relation
133
limx'(x ~) = O. x,~
Case 2
oe < 1
There is some n E I N with a88 >3"1 Given k c i N , d e f i n e k
fk " E+
>E + ,
x~
>x ~
Then, given any k C IN, f k ( A ) is a nonempty downward directed set of E+ and f k ( ~ ) is the lower section filter of f k ( A ) (Proposition 4.2.4.8). By Case 1, lim x'(fl ( x ) ) = 0 . xE;~
Since fk+l--
fl o fk
for every k C IN (Proposition 4.2.4.2 b)), complete induction shows that limx'(fk(X)) --0 x,~
for all k E IN. For k - n, lim x' (x ~ ) - 0 .
I
x,;~
P r o p o s i t i o n 4.2.4.13
( 0 )
Let E be unital. Take x , y C E , z C E + ,
u C E ( z , 1), and c~,~ E IR+ with x*x < z ~ , Then the sequence
(
yy* <_ z ~
,
c~+~>l
1)
x ( n1l + z ) - ~ Y neIN converges and
II ( )1 y < IIz~ll, lim x
-1 + z
(1 )21 _1
limx
n--+oo
-l+z ?'t
z~ = x ,
,,xu,j < jlz~ujr, j,xuyl,< II~z~
134
,4. C * - A l g e b r a s
We have
= IIz~yy*u*z~ll < IIz~uz'u*z~ll = I I za ~ z ~/3l l ~ =
a + ~ i2 II~z~l
(Corollary 4.2.2.3, Corollary 4.2.1.18, Corollary 2.3.2.16). Given m, n c IN, put !
gn " a ( z )
~ IK,
"~ ,
> ")'~
( (~~~) 1-
(1 )~ (1)-~ (1)-,
zn "= x
-1
+ z
n
zmn'--
1
-l
+z
-
Y,
l +z
n
By the above inequalities, Proposition 4.2.4.5 b), and Theorem 4.1.3.1 b), x-x
l+z
z~
=
x
1-
I1(( ~ ) n
+ z
z 2
l+z
z~
<_
-II~(z)ll,
IIzn-zmll IIxzmnyll~IlZmnz~ll II~/Z/ ~/Z~ll for all m, n E IN. By Dini's Theorem, ( f - ) . c ~ and (g.)~c~ converge uniformly to
4.2 The Order Relation
o(z)
~IK,
a+fl-1
~,
135
,
and
a(z)
>lZ,
~/,
>0,
respectively. It follows that lim f n ( z ) =
a§
z
2
n---+ (x)
lim g , ( z ) -- O,
n---+ (X)
(1 )21 _1
x-
lim x
1+ z
n---~c~
z~
rt
(Theorem 4.1.3.1 b)). Hence (z,)nc~ is a Cauchy sequence and II lim z n l l n--4oo
Corollary 4.2.4.14
lim IlZnll ~ lim ]]f~(z)]]-
n---+ cx3
( 0 )
n---} cx3
.
m
Take x c E , z E E + , a n d a C ] 0 , 1 1 2 s u c h that X*X
T h e n there is a y c E
z
~ m Z.
with
-y~,
]lyil < ilz~-~
9
Let E be the unital C*-algebra associated to E . Given n c IN, put
(1)01 1
Yn:=x
-l+z
z ~-~CE
n
for all n E IN. By Proposition 4.2.4.13, (Yn)~e~ converges to some y E E with
llyll _< liz~-~ll 9 Given n E IN,
(1 )21 _1_
ynz ~ - x
-l
+ z
z5
n
so that yz a -
lim YnZ~ = x n---+ o o
(Proposition 4.2.4.13).
i
136
4. C*-Algebras
Corollary
4.2.4.15
( 0 )
(Effros, 1963) Let F
be a closed ideal of E .
Take x C E , y C F+. I f x*x
By Proposition 4.2.4.2 d), y 89
By Corollary 4.2.4.14, there is a z C E with 1
x-zy~
P r o p o s i t i o n 4.2.4.16
I
EF.
(Pedersen, 1969) Take x, y, z E E , such that x*x < y*y + z * z .
T h e n there are u, v E E , such that u*u ~_ y * y ,
v*v ~ z * z ,
xx* - u u *
+ vv*
Let E be the unital C*-algebra associated to E . Given n E IN, define 1
wn'=
w:=y*y+z*z,
-l+w
,
n
'it n
XWny,
Vn "-- XWnZ
lim un,
v:=- lim vn
"--
and set u'-
r t - - + OO
n ----~(:X:)
(Theorem 4.2.2.1 c ==v a, Proposition 4.2.4.13). Then u, v E E . We have that $
UnU
n
-- y*WnX* XWny _~ Y *WnWWny ~ _ y *y
for every n C IN (Corollary 4.2.2.3, Corollary 4.2.1.12), so that u'u-
lim u~un < y * y n----~ OO
(Theorem 4.2.1.1, Propositions 1.7.1.5 a) and 2.2.1.6). It follows V*V ~ Z*Z.
4.2 The Order Relation
137
Now 0 < wnx*xw,~ < wnwwn < 1
(Corollary 4.2.2.3, Theorem 4.2.2.1 c => a, Corollary 4.2.1.12) and 1
-
W n W W
n
-
1
2
- w n
n
so that for every n C IN, - ~*n**: -- v n v :
l -- IIx(1
WnWWnX*..)-
-
w=(yy*
n
+ zz*)wn)x*ll
-
n
n
(Corollary 4.2.1.17 b =~ a). Hence uu* + vv* -
l i m (u,~u~ + v n v ~ ) - - x x * .
I
n---~ c ~
P r o p o s i t i o n 4.2.4.17 F.
Let F be a closed ideal of E and G a closed ideal of
Then G is an ideal of E . 1
1
Take (x,y) C E x G+. Then y89 E G c F , so that x y : i , y ~ x C F and 1
xy = (~y})y~
Assume IK = r
1
9 G,
yx -
y 89
e a.
Since every element of G is a linear combination of four
elements of G+ (Theorem 4.2.2.9 a)), it follows that G is an ideal of E . O
If IK = IR, then F is a closed ideal of /~ and G a closed ideal of (Proposition 2.1.5.11 b)). By the above considerations G is an ideal of E . Hence G is an ideal of E . P r o p o s i t i o n 4.2.4.18
I
( 0 )
Given (E~)~e, a family of C*-algebras, let E
be their C*-direct sum. Suppose that F
is a C*-subalgebra of E
given ~,,~ E I with ~ r ~ and any a~ E E~, there is an x C F and x~ = O. Then R e E C F .
# Take ~ E I and a~E(E~)+. Stepl
~cI\{~}==>
3xCF+ #,x~=a~,x~-O
By the hypothesis, there is a y c F such that !
y~ -- a 2 ,
y~=0.
such that
with x~ - a~
138
4. C*-Algebras
Define f'lR+
>IF[,
a,
>inf{a, 1}
and := S(y*y).
Then x E F+# , XL =
aL,
X~ - - 0
(Corollary 4.2.1.12, Proposition 4.1.3.22).
Step2
{ 3zCF+#,x,-a~, (~CK~x~=0)
KC~S(I\{t}),K#0~
Define n := Card K . By Step 1, for each ~ C K , there is a_ y(~) C F # such that z
y .l = a : ,
yt') = 0
If we endow K with a total order, then
x := H y(~) ~EK
has the required properties. Step3
There is an x C F # , w i t h x L = a ~ , ( ~ C I \ { ~ } = ~ x ~ = 0 )
By the hypothesis, there is a y C F such that !
yL = a~ Take n r IN. There is a finite subset K of I containing ~ such that
1
sup
<
~C I \ K
?2
By Step 2, there is a z E F # such that zL -
---
a~,
$.2 The Order Relation
139
n C K\{~} ==~ z~ = O. Put x (n) :-- y z . Then
x~n) -- y~z~ -- a~ ,
c K\{~}
~
x(2) = y . z ~ = 0 ,
1
,~ e I N K ~
IIx~n)ll = Ily~z~ll ~ l y~ll IIz~ll < -n .
(X(n))nC~ is obviously a Cauchy sequence in F . Then x : - lim x (n) n-+c~
has the desired properties. Step 4
{x C Re E I{~ c I Ix~ ~ o} is finite} c F
The assertion follows immediately from Step 3 (Theorem 4.2.2.9 a)). Step 5
Re E c F
The assertion follows from Step 4. Example 4.2.4.19
m
Take
x . - [ ~~ ~
c (~)+\{0}
and put ab]
1 .--X
b d
(Example 4.2.3.5).
2
'
140
4. C*-Algebras
a)
a - - ~l ( c +
5) __ ~ § v/~a
d=~ l(c+
~) = 5 § v/~5
b=-
b)
ad-
C)
d)
C
IbiS= v"~a- I~1~
x l --"
-~- r
1)
T h e f o l l o w i n g are equivalent:
dl)
a5=1/312,
d2)
x~ = ~ x
1
1
1
d3) x is not p r o p o r t i o n a l to 1 and x~ is p r o p o r t i o n a l to x . e) x e PrlK2,2\{1} iff a5 = I~1 ~ and a + 5 = 1. a) By Example 4.2.3.5, a , 5, a 5 - I~12, a, d, ad - IbJ 2 e IR+
and c r O. We have
[o
5
= [a2b(a + 2d) db,a 1 2+ Ibl 2
so that a
+ Ibl 2 = a
d 2 + Ibl 2 -
5
b(a + d) =
Put t'=a+d.
Then t > 0
and a-
so that
5 = a2 - d2= (a-
d ) ( a + d) = t(a - d)
4.2 The Order Relation
141
a+d=t a-d=
~-~ t
and
'(t+~) d - ~ l(t + -~-~). 7It follows a t e _ a2t2 + ib[2t2 -= ~(t 1 4 -+- 2 ( o -
0 - - t 4 - - 2 ( o z -4-
~)t 2 -t- ( o z - (~)2) 4-I~1 ' ,
~)t 2 -~- ( ~ -- ~)2 +
41N12 :
= t 4 -- 2(o~ + (~)t 2 + (oz + 5)2 _ 4a5 +
41/~12 =
= (t 2 - (a + 6))2 _ 4(a5 - 1 9 1 ~ ) ,
t = - (~ + ~) = 2 v / ~ -I~1 = ,
t - V/~ + ~ + 2 v / ~
a = ~1( c :
+ ~ - ~) =
1 (a+5+2V/aS-I/~[
+ x/~-
1
-I~1 ~ -c.
2+a-5)=
IZl~
1
C
b) By a),
ad_lbl2
1(
~
+ (~ + 5)v/~5
_ i/~1~. + ~
2 ( ~ - I ~ 1 ~ + (~ + ~ ) v / ~ C2
_ I/~1~ - I ~ 1 ~
I~1~
)
-
142
4. C*-Algebras
c~ ~ / ~ 5
c) follows from a). dl => d2 follows from c). d2 => d3. Assume x proportional to 1. Then a x-} = v/-~l
1
1
~#
v~+~
--
5,13 = 0, x -
al,
X.
Hence x is not proportional to 1. d3 => dl. There is a 7 E IK such that !
x2 -- 7x. By c), ( 7 c - 1)x
x/c~5- I/~121
=
so that ~5 -IZl ~ = 0.
e) If x e Pr IK2,2\{1} then by Example 4.1.2.19, a5 = The reverse ~mplication follows from d~ => d2 & d3. Remark.
Inl~ and a + 5 = 1. I
The above assertion still hold for z e (~,~)+\{0}.
E x a m p l e 4.2.4.20
Take
x-
[ a~ ~ ]
E(IK22)+\{1}
and p u t p .-
v/(~
- a)2 +
o~ + 5 + p
a-
~
, 2
b:=
oL + 5 - p
2
T h e n f o r every 7 > O, o~ X 7
--
-
~
b'~
ab'~
_
ba'~
+
P
P
The assertion follows from Example 4.1.3.19.
I
4.2 The Order Relation
143
4.2.5 T h e M o d u l u s Definition 4.2.5.1
( 0 )
Given x E E , define
Ixl := (z**)89 for every x C E (Theorem 4.2.2.1 c ~
a). IxI is called the modulus o f x .
Another candidate for the modulus would be (xx*)89 Now 1
(xx*)~ - (x*z)~
1
iff x is normal. P r o p o s i t i o n 4.2.5.2
The map
E
x,
~lxl
is continuous.
This follows from Proposition 4.2.4.2 a). P r o p o s i t i o n 4.2.5.3
( 0 ) I f x c E , then
I
Ixl
belongs to the closed real
subalgebra of E generated by x ' x , and x and Ixl generate the same closed ideal of E .
The first assertion follows from Proposition 4.2.4.2 d). Let F and G be the closed ideals of E generated by x and IxI, respectively. Then
Ixl2 = x * z c F ,
x*z=lxl 2 c G
Thus Ix[ E F and x C G by Corollary 4.2.4.15. P r o p o s i t i o n 4.2.5.4
( 0 ) Take x C S n E and f:o(x)
Then Ix I -- f (x) and in the complex case
Ire x I + lim xl _> Ixl = ((re x) 2 + (im x) 2) 89_> [re x[ (resp. lim x I ).
144
4. C*-Algebras
Define g'o(x)
9 >IK,
~-+~.
Then f2 = yg, so that f ( x ) 2 = f2(x) - -j(x)g(x) -- x*x . Since y ( x ) is positive (Corollary 4.2.1.12), f(x) = Ixl. Moreover, in the complex case, Irexl + l i m x l -
I~1 =
((rex) 2 + (imx)2) 89>_ Irexl
(resp. limxl)
follows from Corollary 2.3.1.24 c), Theorem 4.2.2.1 c =v a, and Proposition 4.2.1.12. I Remark.
If
then
Ixl=
x=[0001] [00] [1] o
1
'
rex -
0 1 5 0
and so r e x < Izi
does not hold. P r o p o s i t i o n 4.2.5.5
( 0 )
Let E , F
be C ' - a l g e b r a s and u " E --> F
h o m o m o r p h i s m of involutive algebras. Take x C E .
~)
~lxl = I~xl.
a
4.2 The Order Relation
c)
z e E + , c~ e Ia+ ~ ux ~ = (ux) ~
d)
u(E+)
e)
I f in a d d i t i o n u is s u r j e c t i v e t h e n
u(E) .
f o r e v e r y y~ E Re F ~.
a) Since
~lxl e F+
(Corollary 4.2.1.4),
~(1~1) ~ = (~1~1 ~) = ~(~*~) = (~x)* ( ~ ) ,
ulxl = luxl.
b) Now u x + , u x - c F+ (Corollary 4.2.1.4), (~+)(~-)
= ~(z+x-)
= 0,
~ ( z - ) ~ ( x +) = ~ ( ~ - ~ + ) = o ,
ux = u(x + - x-) = ux + - ux- ,
so that (~x) + = ~x + ,
(~x)-
= ~-.
c) follows from Corollary 4.1.3.4 (and Definition 4.2.4.1). d) By Corollary 4.2.1.4, u(E+)
u(E).
Let y E F+ F] u ( E ) and let x E E such that u x = y . By a), y = lyl = I ~ l = ~lxl e ~ ( E + ) , F+ N u ( E ) C u ( E + ) .
e) Assume u ' y ' E E'+. Take y E F+. By d), there is an x E E+ with u x -- y .
It follows (y, y') = (~x, y') = (~, ~'y') c ~ §
145
136
3. C*-Algebras
y'E F~_ (Corollary 4.2.2.10). The reverse implication was proved in Corollary 4.2.2.11.
I
Remark. A similar result to d), namely u(Pr E) = Pr F M u(E) does not hold. While the inclusion from left to right is obvious, the reverse inclusion fails, as shown by the example E'=C([0,1]),
u'E
T h e o r e m 4.2.5.6 ~)
>F,
F:=C({O, 1}), x,
>x[{0,1}.
( 0 ) Take x,y e E.
II Ixl II = I1~11.
b) x E R e E = ~ l x l = x
++x- >x,
d)
xCReE,
IxiEPrE~x
e)
ix + yl ~ < 2(Ixl ~ + lyl~).
f)
Ixl ~< lyl ~ Ilxll ~< Ilyll.
{
~+ = ~(1~1 + ~) 9 -=
~ ( l ~ l - ~).
+,x- EPrE.
II I~111 ~ = II Ixl~ll- IIx*xll = Ilxll ~ (Proposition b) Since ~)
4.2.4.2 b)).
(~+ + x - ) ~ = (~+)~ + ( x - ) ~ = (~+ - x - ) ~ = Z = Ixl ~
(Theorem 4.2.2.9 xl-
z + +z-
_ z +-
z-
= x
(Theorem 4.2.2.9 a), Proposition 4.2.4.8). c) follows immediately from the definition. d) By Proposition 4.2.5.4 (and Corollary 4.1.3.5, Proposition 4.1.2.21
4.2 The Order Relation
147
{1~11 ~ e ~(~)} = ~(1~1) c {0,1}, so that a(x) C { - 1, O, 1}. Thus a ( x +) U a ( x - ) C {0, 1}
(Corollary 4.1.3.5, Theorem 4.2.2.9 b)), and so x + , x - C P r E 4.1.2.21 c :=:> a).
(Proposition
e) By Corollary 4.2.2.4,
Ix + yl ~ - (~* + y*)(z + y) - ~*x + y*y + x*y + y*~ <
2(x*x +
y'y)--
2(Ixl ~ -4-lyl~).
f) By a) and Corollary 4.2.1.18,
Ixl-IIIxll Remark.
m
~< lily II = Ilyll.
a) For no a ~ IR.:., a relation of the form
+ yl < ~(1~1 + lyl) holds as it can be seen from the example
X
"~-
[1~1 0 0
I-
_ [
Y
"1
COS 2 19
[ cos 0 sin 0
with 0 E JR. b) Ifwe set
x=[O1] 0 0
then
r~ ~ ~ ~lxl for all a C IR+.
'
cos 0 sin 0 [ sin 2 0
J
148
4. C*-Algebras
E x a m p l e 4.2.5.7
Let
z:=
~
~
be a selfadjoint element of L:(IK2) which is neither positive nor negative and put
p - - v / ( ~ - a) 2 + 41/~12 9 IR+. Then
- ~
(o~ + 61~
v
a
52
q~
(~+6)~ +
] =
+ ~
The assertion follows from Theorem 4.2.5.6 b) and Example 4.2.3.7. P r o p o s i t i o n 4.2.5.8
If E is a real C*-algebra, then
(0, x) >_ 0 ~
x =0
for every x E E .
We have
(0, ~)*(o, x ) = (0, -x*)(0, x) - (**x, o) - ( ~1=,o)- (t.I, o) = . Since (Ixl,O)is positive,
I(o,~)1 = (1~1, o). Moreover,
(o, z) _> 0 implies that
(o, ~) - I(o,,)1 - (l-t, o). Hence x-0.
m
4.2 The Order Relation
Corollary 4.2.5.9 then (O,x) C ReE
149
Let E be a real C*-algebra and take x E E . I f x* - - x and
(0, x)+ -- ~(Ixl, 1 x),
(0,~)- - ~1( l ~ l , - x ) .
We have 1 1 1 (0, x) • = ~(l(0, x)l + (0, x)l = ~((Ixl,0) + (0, xl) = ~(Ixl, • (Theorem 4.2.5.6 b), Proposition 4.2.5.8).
i
150
4. C*-Algebras
4.2.6 Ideals a n d Q u o t i e n t s of C * - A l g e b r a s P r o p o s i t i o n 4.2.6.1
"( 0 )
and let F be the closed left ideal
Take x E E
of E generated by x . T h e n there is a sequence (xn)nc~
in F M E(ix]) such
that
lim xxn = x n--}co
and O < xn <_ Xn+l <_ 1 f o r every n E IN.
Now Ix[ E F M E+ (Proposition 4.2.5.3). Given n c IN, define fn " a(Ixl)
> IK,
g ~ ~(1~1) ---+ XK,
~inf{na, 1},
a:
~,
~ ~(1- f~(~)),
xn . = f n ( I x l ) e e ( I x l ) n
E§ n F
(Corollary 4.2.1.12, Corollary 4.1.3.3 a)). Then IIx~n - x I~ - I I ( x x n
-
x)*(x~
-
x)ll = II(:~nx* -
x*)(~x~
-
x)ll---
1 = I1( x l ( 1 - x ~ ) ) ~ l l -
II Ix[(1 - xn)ll 2 - I I g ~ ( I x l ) l l ~ - I l g n l l ~ = (4n) 2
for every n E IN, so that lim XXn - x . n - - } (x)
Since O ~ _ f n ~_ fn+l ~_ 1
it follows 0 ~_~ X n ~ Xn+ 1 ~ 1
for every n C IN (Corollary 4.2.1.12).
I
4.2 The Order Relation
C o r o l l a r y 4.2.6.2
151
( 0 ) (Segal, 1949) Every closed ideal of E is involutive.
Let F be a closed ideal of E and take x c F . By Proposition 4.2.6.1, there is a sequence (x=)n~r~ in F+ with x = lim xx,~. n--~ oo
Thus x* -
lim xnx* E F , n - - - ~ (x)
since xnx* E F for every n E IN. Hence F is involutive. P r o p o s i t i o n 4.2.6.3
I
(St0rmer, 1967) If F and G are closed ideals of E ,
then (F + G) N E+ = F N E+ + G n E+ .
Take x E ( F + G) N E + . There is a (y,z) C F • G with x=y+z.
We may assume y, z to be selfadjoint (Corollary 4.2.6.2). Then
_< lyl + Izl (Proposition 4.2.5.6 b)). By Proposition 4.2.4.16 (and Proposition 4.2.4.2 b)), there are u, v E E such that ~*~ <_ lyl,
v*v <_ Izl,
x-
uu* + vv*.
Then lyl c F , Izl c G (Proposition 4.2.5.3), so that u*u C F , v*v C G
(Corollary 4.2.4.15). Hence (uu*)2 _ uu* uu * ~ F ,
~*=
((u~*)~)~ e F
(Proposition 4.2.4.2 d)), vv* E G , x C FNE+ +GNE+
(Theorem 4.2.2.1 c =:> a), ( F + G ) NE+ C F N E + + G N E + .
The reverse inclusion is trivial.
I
152
4. C*-Algebras
P r o p o s i t i o n 4.2.6.4
( 0 ) If F is a closed ideal of E , then [IXI[ = inf { [ [ x - x y l l [ y E F#+}
for every X E E l F
and x E X
Define a := inf { I I x - xyll I y E F+e} 9 Take z E X . Then x - z E F , so, by Proposition 4.2.6.1, there is a sequence (xn)ne~ in F such that O<_xn_
Take n E IN. Then xn, 1 - xn E F+~ (Corollary 4.2.1.17 b =~ a & c). Hence
< IIzll,
___ I I X l .
The reverse inequality is trivial. T h e o r e m 4.2.6.5 E , then E / F
( 0 ) (Segal, Kaplanski, 1949) If E is a closed ideal of
is a L'*-algebra (Corollary ~.2.6.2).
By Proposition 2.3.2.10 (and Theorem 1.2.4.2 e)), E / F is an involutive Banach algebra and the quotient map E --+ E / F is a homomorphism of involutive algebras. Take X E E ,
xEX,and
yEF+ #.WehaVe
x*xEX*X,
x(1-y) EX,
III-YlI_I
(Corollary 4.2.1.17 a :=v c). Thus, by Proposition 4.2.6.4,
4.2 The Order Relation
IIXII = <~ IIx(1 - Y)ll ~ = I1(1 - y)x*x(1
153
- Y)II <~
II1 - yll IIx*x(1 - y)II ~ IIx*x(1 - y)[], IIX]] 2 < inf { I]x*x(1 - y)]] ] y E F+# } - l l X * X I I . Therefore E l F is a C*-algebra in the complex case (Proposition 4.1.1.3). If IK = IR, then the claim follows from the above considerations and Proposition 2.3.1.42. Theorem
1 4.2.6.6
( 0 ) (Gelfand-Naimark, 1948) Let E, F be C*-algebras and u : E -+ F a homomorphism of involutive algebras. Then I m u is a C*subalgebra of F and the associated algebraic isomorphism E/Ker u
> Im u
of u is an isometry of C*-algebras (Theorem ~.2.6.5).
First assume that u is injective. By Corollary 4.1.3.4, ]luxiI = Iixll for every x C E . By Proposition 1.2.1.18 c), I m u is closed. Hence I m u is a C*-subalgebra of F (Proposition 2.3.1.16) and the map E
>Imu,
x,
>ux
is an isometry of C*-algebras. Now let u be arbitrary. Then Ker u is a closed ideal of E (Proposition 2.1.1.10). Let v : E/Ker u
>F
be the factorization of u through E / K e r u. E / K e r u is a C*-algebra (Theorem 4.2.6.5) and v is injective. It is easy to see that v is an involutive algebra homomorphism. By the above proof, I m u = I m v is a C*-subalgebra of F and the map E/Keru
>Imu,
A,
is an isometry of C*-algebras. C o r o l l a r y 4.2.6.7
( -/ )
> vA
I If F is a closed ideal of E and G is a C*-
subalgebra of E then F + G is the C*-subalgebra of E generated by F U G and the C*-algebras (F + G ) / F and G / F M G (Theorem ~.2.6.5) are canonically isometric. If in addition F / 3 G = {0}, then
Ilyll ~ IIx + yll .for all (x,y) C F • G .
154
4. C*-Algebras
It is obvious that F + G is the involutive subalgebra of F generated by F U G (Corollary 4.2.6.2). We show that F + G is closed. Let q : E ~ E / F be the quotient map. By Theorem 4.2.6.6, q(G) is closed and the associated algebraic isomorphism G / ( F n G)
~ q(a) = (F + G ) / F
of qlG is an isometry. Since -1
F + a = q (q(G)), F + G is closed. The last assertion follows from Corollary 4.1.1.20, since the map F+G-----+G,
x+y~
~y
is an involutive algebra homomorphism. C o r o l l a r y 4.2.6.8
I
Let T be a compact space, .T a closed ideal of 12(T) and
s .= n xE.T
Then the factorization 12(T)/.T
~12(S)
of the map t2(T) ~ 1 2 ( S ) ,
x,
) xIS
is an isometry of C*-algebras (Theorem 4.2.6.5). Now = {x e C(T) I x l S = 0} (Corollary 1.3.5.18) and the map C(T) --+ C(S) ,
x:
~ xlS
is a surjective homomorphism of involutive algebras. The assertion therefore follows from Theorem 4.2.6.6. I
4.2 The Order Relation
155
T h e o r e m 4.2.6.9
( 0 ) Let x be an invertible element of E . Then Ix]
is invertible, xlx1-1
is unitary, and there is a uniquely determined (y,z) 9
Un E x E+ such that x-yz.
(y, z) is called the polar representation o f x . Furthermore y-xlxl
-I 9 E ( x ) - E ( x , 1 ) ,
z-
lxl 9 E ( x ) .
x*, x ' x , and Ixl are invertible (Proposition 2.3.1.14, Corollary 2.1.2.6, Proposition 4.2.4.5 a)). Since (xlxl-1)(xlxl-*)
* -
xlxl-=x
* -
x(~*x)-~
* -
xx-~(x*)-~x
*
-- 1,
(x[x[-1)*(x[x[ -1) --[x[-lx*x[x[ - 1 - - [x[-l[x[2[x[-l : 1, xlx1-1 is unitary. This establishes the existence of a polar representation. We
now prove its uniqueness. z 2 = zy*yz = (yz)*(yz) = x*x = Ixl 2
and so
z=lxl, y=~lx1-1 By Theorem 4.2.5.3, Izl G E ( x ) . Thus Ix1-1 G E ( x ) (Proposition 4.2.4.5 b)) and zlz1-1 e E ( x ) = E ( x , 1). 9 Corollary 4.2.6.10
The map
V n E • {z 9 E+ ] z is invertible}
> {x 9 E t x is invertible},
(y, z) ~ > yz is a homeomorphism.
The map is bijective (Theorem 4.2.6.9) and continuous. Since the map {x e E I x isinvertible}
> E+,
x,
~ lxl
is continuous (Proposition 4.2.5.2), the map {x e E I x isinvertible}
~ E,
x,
~ xlx1-1
is also continuous (Theorem 2.2.4.14). Hence the inverse of the above map is continuous (Theorem 4.2.6.9). m
156
4. C*-Algebras
Corollary 4.2.6.11
Take x , y E S n E . Let z be an invertible element of E
and (u, v) the polar representation of z . I f -1
y -= Z X Z
,
then y - uxu
-1
We have, successively, that yz = zx,
y* z = zx*
(Theorem 4.1.4.1), xz* = z * y ,
XV 2 -- XZ* Z ~
z*yz
XV
---- Z * Z X
-- V2X~
-- VX
(Corollary 4.2.4.2 b)), y -- ~tVXU-I?.t -1
P r o p o s i t i o n 4.2.6.12
~_ U X ~ ) U - I ~
-1
~
UXU -1
Take x , y C E with y unitary and x = yl~l
Then x is normal iff yJzl = I ~ l y
If x is normal, then izl~y - ~ , ~ y
= ~*y
= ya~l~*y
and so I~ly - ylxl
(Proposition 4.2.4.2 b)).
= ytzl ~
9
4.2 The Order Relation
157
From y l x l = Ixly
it follows that xx*
=
~ l ~ l ~ y * = I z l ~ y y * = Ixl ~ = ~ * ~ ,
i.e. x is normal,
m
C o r o l l a r y 4.2.6.13
L e t x be an i n v e r t i b l e e l e m e n t o f E
a n d let
(y, z) be its
polar representation.
a) x is n o r m a l i f f y z = z y . b)
x is s e l f a d j o i n t i f f y z = z y
a n d y is s e l f a d j o i n t . I n this case,
x +=y+z=y+x
+,
x- = y - z = y
x
a) follows from Proposition 4.2.6.12. b) Assume x to be selfadjoint. By a), x ~ x* ~ y * z
and so (y*,z) is the polar representation of x. Thus y* = y and the C*subalgebra of E generated by { y , z } is a Gelfand C*-algebra (Corollary 4.1.2.3). By Corollary 4.1.2.5, x +=y+z,
x-=y-z.
Since y+x- = y+y-z
= O,
y-x + = y-y+z
= O,
we see that by Theorem 4.2.5.6 b), y+z = y+(x + + x-) = y+x + ,
y-z = y-(x + + x-) = y x
.
If y z = z y and y is selfadjoint, then x* = zy* -- z y -- y z -- x .
[]
158
4. C*-Algebras
P r o p o s i t i o n 4.2.6.14
Let E , F
be unital C * - algebras, u " E --+ F a ho-
momorphism of involutive unital algebras, and x an invertible element of E .
a)
x e E+,c~ E ]R ==~ u log x --log ( u x ) , ux a -- (ux) ~ .
b)
If (y, z) is the polar representation of x , then (uy, uz) is the polar representation of u x .
a) follows from Corollary 4.1.3.4 (and Definition 4.2.4.4). b) uy is unitary (Proposition 2.3.1.16), uz E F+ (Corollary 4.2.1.4), and (~y)(~z) = ~(~z) = ~,
I
i.e. (uy, uz) is the polar representation of u x . P r o p o s i t i o n 4.2.6.15
Let E, F be C*-algebras and u" E --+ F a surjective
involutive algebra homomorphism. Take x E E+ and y E F . If y'y<_ ux then there is a z E ul (y) such that Z*Z ~ X .
We may assume that E , F , u
are unital (Theorem 4.1.1.13, Proposition
2.1.1.7). Take a E ul(y). Given n E IN, define b'-
( a ' a - x ) +,
c'-b+x,
1
1
cn "=
-1
+ c
,
zn :=
acnx~ .
n
Then a*a < b + x = c
and hence (z,)nc~ converges (Proposition 4.2.4.13). Put z "- lim zn. n----~ ( ~
Since ,
1
1
1
1
ZnZ n - - X - ~ C n a * a C n X ~ ~_ X2CnCCnX2 ~
1
1
x~lx~
= x
4.2 The Order Relation
159
for every n E IN (Corollary 4.2.2.3, Corollary 4.2.1.12), we see t h a t z*z = lim
Z n*Z n
n---~oo
< x
(Proposition 1.7.1.5). For n E IN u x ) + = 0,
ub = (u(a*a - x ) ) + = ( y * y -
(
1
ucn -
m
1 + uc
)-
ll+ux n
_1
uzn = (ua)
ux
(ux)
(Proposition 4.2.5.5 b),c), Proposition 4.2.6.14 a)), so t h a t uz -
lim uzn = u a = y
n--+ ~
(Proposition 4.2.4.5 a)). Corollary 4.2.6.16
Let E , F
be C*-algebras, u " E --4 F
lutive algebra h o m o m o r p h i s m , and ( Y n ) n ~ there is a decreasing sequence (xn),~e~
a s u r j e c t i v e invo-
a decreasing sequence in F + . T h e n
in E+ such that UXn = Yn f o r every
n E IN.
We construct the sequence
(Xn)nE]N
recursively. By Proposition 4.2.5.5 d),
there is an x l E E+ with u x l - Y l . Take n E IN and assume the sequence is -1
constructed up to n . By Proposition 4.2.6.15, there is a z C u
~
Yn+l
such
that z*z <_ x n . Put Xn+l " - z * z . Then xn+l E E + , x,~+l <_ xn, and ?.tXn+l - - ( U Z ) * ( ~ t Z )
Proposition 4.2.6.17
( 0 )
~-
Let F
Yn+l
Yn+l
-- Yn§
be an E - m o d u l e .
.
I f (x,a) E E • F ,
then
IIxall--{I Ixlall,
Ilaxll--Ilalxl II.
We may assume that ]]xlI- 1. Take n c IN. By taking
z--Ixl 2,
~:=
I
n-1 2n
in Corollary 4.2.4.14, we obtain an element Yn C E such that
160
4. C*-Algebras
= y=lxl ~-~,
Ily=ll _< Ilxll ~-
(Proposition 4.2.4.2 b)). Put (
1
)
z. : -
-
1
* + Izl 2n+l OZ n
>~,
~"
>!+ n
Then 1
l) -1
n
I1( )1 x:lLx. x! 11 +
(Corollary 4.2.1.7),
llxoll-IlYn Ixl n-' all-< Ilynll II x~ n-' all-< llxlt: II Ixln-'~ ' -1
n
I1( 1 1 + ix')
II
a
-IIz.xa
I - I l z n l l Ilxall _< Ilxll
Ilxa I.
Since ( f n ) , ~ is increasing and lim fn(Ol) -- C~
n---+oo
for every c~ e rT(ixi), it follows, by Dini's Theorem, that (fn),~e~ converges uniformly to the function ~(Ixl)
>IN,
~,
>~.
Hence --1
aim (ll+,x,)
ix ~ - - n---+ limoo fn(lx[)= Ixl.
n--+oo
We deduce that
Jl~all _< aim Ilxl : II J~l ~-' ~ --
limll(11 )1 + Ix
n-+oo
n
I~1 ~+1 a
II I~ oil-
II
< lim I xll 1 Ixall- II~all,
IIx~ll- II Ixlal -
i
4.2 The Order Relation
Proposition 4.2.6.18
161
Let ~ be a set of closed ideals of E such that
FnC={0} whenever F , G E ~ are distinct and let H be the C*-direct sum of (F)FE~ (Corollary ~.2.6.2). Define Ho := {x E H I { F E ~ l xF # O} is finite},
and ) E,
uo " Ho
x:
~,
y~XF. FE~
Then there is a unique involutive algebra homomorphism u : H -+ E extending no. u preserves the norm. If the vector subspace of E generated by U F is F E~d
dense in E , then u is an isometry of C*-algebras. Since H0 is dense in H , uniqueness follows from Corollary 4.1.1.20. Let F, G be distinct elements of ~ and take (x, y) E F • G . Then
xy E F N G , so t h a t xy - O. Let 6 be a finite subset of ~ and take x E
I-I F such that FEE)
~
-~XF=O.
FEE)
Take G E 6 . By Proposition 4.2.6.1, there is a sequence (Yn)nEr~ in G such that xc--- lim x c y n . n -----~(:X:)
By the above considerations,
xcyn - Z
xFyn = 0
FEe)
for every n E IN. Thus xG - 0. Since G is arbitrary, it follows t h a t x - 0. Hence the restriction ue) of u0 to He):={xEH0
I{FE~IxF:fi0}C6}
is injective. Since HE) is a C*-algebra and since ue) is an involutive algebra homomorphism, it follows that ue) preserves the norm (Theorem 4.2.6.6). Since 6 is arbitrary, u0 preserves the norm. Hence, we may extend u0 to an involutive algebra homomorphism H -+ E which preserves the norm. The last assertion now follows.
I
162
4. C*-Algebras
4.2.7 T h e O r d e r e d Set of O r t h o g o n a l P r o j e c t i o n s Proposition
4.2.7.1
( 0 )
Take x E E
and p E P r E .
T h e n the following
are equivalent:
a) Ixl~p,
Ix*l~p.
b)
x E E #.
px = xp=
c) p x p -
x E E#
I f x is selfadjoint then the above are also equivalent to the following:
d)
-p<_x<_p.
e)
px = x E E #
f)
xp = z E E #
a =~ b. Since
(Theorem 4.2.5.6 a), Corollary 4.2.1.18), x E E # and
~*x
p
(Corollary 4.2.1.12). We deduce successively that 0 _< (1 - p)x*x(1 - p) < (1 - p)p(1 - p) - 0 (Corollary 4.2.2.3),
x(1 - p ) = 0 ,
x -- x p ,
x* = x ' p ,
x - px.
b =v c is trivial. c ==v a. We have, successively, that
4.2 The Order Relation
163
II~*~ll = II~ll~ < 1, 0 < x*x < 1 (Corollary 4.2.1.17 a =a b), (pxp)*(pxp) = px*pxp < p x * x p < p l p = p (Corollary 4.2.2.3),
I x l - Ipxpl - ((pxp)*(pxp)) 89<_ p (Proposition 4.2.4.8). Since px*p = x* E E # ,
it follows from the above relation that Ix* I < p . a =a d follows from Theorem 4.2.5.6 b),c). d =a c. By the implication a =a c above, we have successively that 0 < x + p < 2p, o _ < ~l ( x + p ) <_ p , ) p ( 1(x+p) p=-~l ( x + p ) , x +p,
pxp+p=
pxp = x.
Now ]]xll < Iip]l < 1 (Corollary 4.2.1.18), i.e. x E E # b r e ca f is trivial. C o r o l l a r y 4.2.7.2
( 0 )
I
Take p E P r E
and x E E + .
x
e]0,1] ~
x_< x" _
c~ E [1, cxD[=::~ x ~ < x < p.
If
164
4. C*-Algebras
By Proposition 4.2.7.1 a ::a b, px = xp = x.
Hence the C*-subalgebra F of E generated by {x,p} is a Gelfand C*-algebra (Corollary 4.1.2.3) and the assertions follow by the Gelfand transform on F .
I Corollary 4.2.7.3
( 0 )
Given x E E~+ and p E P r E ,
the following are
equivalent:
a) p_<x. b)
px - p.
c)
xp= p.
d)
p x = xp - p .
e)
pxp - p.
f)
xpx = p.
In particular, every strictly decreasing sequence in Pr E is linearly independent.
By Corollary 4.2.1.17 a =a b & c, x _< 1 and 1 - x E E+# . a~b. 0
1-x-(1-p)(1-x)-
1-x-p+px
(Proposition 4.2.7.1 d =:v e), i.e. p x -- p .
b ==v c ==~ d =v e and d =v f are trivial. e =~ b. We have, successively, that 1
x<x~
< 1
(Corollary 4.2.7.2), p-
p x p <_ px89p <_ p l p - - p
4.2 T h e Order R e l a t i o n
165
(Corollary 4.2.2.3), !
px2p = p,
It
II(
: 1
=p,
px2
1_
px--px2x2
!
!
--px2
--p.
f~a. p = xpx ~ xlx
= x 2 <_ x
(Corollary 4.2.7.2). C o r o l l a r y 4.2.7.4
a)
m
( 0 ) G i v e n p, q C P r E , t h e f o l l o w i n g are e q u i v a l e n t :
pq = qp.
b) p q is t h e i n f i m u m o f p a n d q i n E#+ . c) p + q - p q is t h e s u p r e m u m
o f p a n d q in E#+.
d) pq a n d p + q - pq belong to P r E . e)
pq C
ReE.
a ::~ b & c & d. The C*-subalgebra F of E generated by p and q is a Gelfand C*-algebra (Corollary 4.1.2.3). Identifying F with C o ( o r ( F ) ) by means of the Gelfand transform, we see that pq (resp. p + q - p q ) belongs to Pr E and is a lower (upper) bound for p and q. Let x be a lower (upper) bound for p and q in E+# . Then px - x ,
qx - x
(px - p ,
qx -
q)
by Proposition 4.2.7.1 d =~ e (by Corollary 4.2.7.3 a :=~ b). Hence p q x -- x
( (p + q - p q ) x -- p x + q x - p q x -- p + q - pq)
166
4. C*-Algebras
and x < pq
by Proposition 4.2.7.1 e ~
(p + q - pq <_ x)
d (by Corollary 4.2.7.3 b =v a). Thus pq (resp.
p + q - pq ) is the infimum (supremum) of p and q in E+# .
b :=> e, c =~ e, and d ==v e are trivial. e :=~ a. We have qp = (pq)* = pq.
C o r o l l a r y 4.2.7.5
( 0 )
Take p, q E P r E .
I If
pq=0, then the i n f i m u m of p and q in E+ is O, p + q is an orthogonal projection and it is the s u p r e m u m of p and q in E#+ .
The assertion follows immediately from Corollary 4.2.7.4 a =v b & c & d. I Remark.
p + q is not necessarily the supremum of p and q in E + , as the
following example shows. Put
E'-s
P'-
E10] 0 0
'
q'-
[001 0
1
"
Then p+q--1, but the matrix
v~
2
is larger than p and q and is not larger than 1. p and q have no supremum in E + , since such a supremum would be smaller than 1, i.e. it would belong to E+e and this contradicts the above considerations. C o r o l l a r y 4.2.7.6 a)
p<_q.
( 0 )
Given p, q c P r E ,
the following are equivalent:
4.2 The Order R e l a t i o n
b)
167
p q = qp = p .
c) p q = p . d)
qp=p.
e)
pqp = p.
f)
qPq = p .
g)
q-pC
h)
a p <_ q f o r s o m e
PrE. a > O.
i) p q = a p + 3 q f o r s o m e a e ] K \ { O } In particular,
a n d /3 9 ] K .
e v e r y s t r i c t l y d e c r e a s i n g s e q u e n c e in
Pr E is l i n e a r l y i n d e -
pendent.
By Corollary 4.2.7.3, the assertions a) to f) are equivalent. b =:> g. Since (q _ p)2 = q2 _ pq _ qp § p2 = q _
q-pc
p_
p + p = q_
PrE. g ::~ a =~ h and c ==~ i are trivial. h ~ c. By Proposition 4.2.7.1 d =~ f, o~pq = a p
and so pq=p.
i =~ c. Since p q = p2 q = a p + 3 p q ,
(1 - / 3 ) p q = a p . Since a r O, it follows that 3 --/=1. Thus oz
1
,~p _ pq = pq2 = P
OL
1 - 13pq
__
()2 OL
1 - t3
P,
p,
168
4. C * - A l g e b r a s
p =
Hence either p = 0 or ~
O~
1 _ / ~ p-
- 1 and in both cases pq-p.
Now we prove the last assertion. Take (P~)~e~ a strictly decreasing sequence in P r E
and (C~n)ne~ C ]K (~) such t h a t
E olnPn -- O.
nE IN
By a =~ b , for every m E IN, 0 -- E
OlnPn(Pm -- P r o + l )
-~ E OlnPnPm - E OZnPnPm+l n E IN n E IN
nEIN
OG
-.~ + E n--1
m
oo
-o,.- E-.,~+x + E
n=m+l
n=l
-o~. :
n=m+l
= ( n=lon)
pm+,,
m
E c~n - 0 . n=l
Hence
a,~ =
0 for every n E IN and
(Pn)ne~
is linearly independent.
I C o r o l l a r y 4.2.7.7
( 0 )
Take p,q, rCPrE
with
p<_q<_r. T h e n p + (r - q) is a n o r t h o g o n a l p r o j e c t i o n a n d it is the s u p r e m u m
of p and
r - q in E#+.
By Corollary 4.2.7.6 a :=~ c & g, r - q
C Pr E and
p ( r - q) -- p r - p q -- p -
The assertion now follows from Corollary 4.2.7.5.
p--
O.
I
4.2 The Order Relation
P r o p o s i t i o n 4.2.7.8
( 0 )
169
Take p C P r E . Let F be a C*-subalgebra of
E such that (pF)
(Fp) C F .
Put
A:= {xcFlO<_x<_p} and let q be a maximal element of A .
a) q C P r E . b) x C A => x q -
qx = x .
c) q is the greatest element of A . a) We have 1
q
(Corollary 4.2.7.2), so q89 C A. Since q is ma:dmal, !
q=q2,
q2_q, i.e. q C P r E . b) We have, successively, that O<_x2
(Corollary 4.2.7.2), 0<_ (p - q)x 2 ( p - q )
<_ (p - q)p(p - q) = p -
(Corollary 4.2.2.3, Corollary 4.2.7.6 a =, e & g), q ~_ q + ( P - q)x2(p -- q) ~_ P,
( p - q)x 2 (p - q) = 0
(since q + (p - q)x2(p - q) E A ) ,
q
170
4. C*-Algebras
I[x(P - q)I[ 2 -- I[(P - q) x2 (P - q)II = O,
x ( p - q) = o,
x -- x p -
xq
(Proposition 4.2.7.1 d =~ f), x=qx.
c) follows from a),b), and Proposition 4.2.7.1 b =~ d. P r o p o s i t i o n 4.2.7.9
{'
x~ I x E A
}
I
( 0 ) Let A be a subset of E#+, such that both A and
have the s a m e i n f i m u m y in E + . T h e n y E Pr E and xy = yx = y
.for every x c A .
Since y89 is a lower bound of
x~ i z C A
(Proposition 4.2.4.8),
1
y~ _
y2=y,
y2=y,
yCPrE.
The last assertion follows from the first one and Corollary 4.2.7.3 a =:v d. C o r o l l a r y 4.2.7.10
( 0 ) I f a subset of P r E has an i n f i m u m in E + , then
this i n f i m u m is an orthogonal projection.
C o r o l l a r y 4.2.7.11
I
I
( 0 ) Take x E E#+ . I f (Xn)n~iN has an i n f i m u m y in
E+ , then y is an orthogonal projection and x y -- y x -- y .
4.2 The Order Relation
(Xn)nEIN and
171
(x2n),e~ have the same lower bounds (Corollary 4.2.2.5),
so they have the same infimum in E+ and the assertion now follows from Proposition 4.2.7.9. P r o p o s i t i o n 4.2.7.12
I ( 0 ) Given x C E+ and p , q E Pr E , then following
are equivalent:
a)
x~p,z~q.
b)
n c IN :=> x <_ (pqp)n, x <_ (qpq)r,.
I f x C Pr E , then these assertions are equivalent to the following ones:
c)
n C IN ~ x ~ (pqp)"
d)
n E IN :=~ x ~ (qpq)".
a =~ b. We have x = (pqp)"x(pqp)" < ( p q p ) , p ( p q p ) n = (pqp)2n
for every n C IN (Proposition 4.2.7.1 a ==> b, Corollary 4.2.2.3). Since the sequence ((pqp)n)ne~ is decreasing (Corollary 4.2.2.5), x < (pqp)n
for every n E IN. It follows that x < (qpq)n
for every n E IN. b =~ a. By Corollary 4.2.2.3, x <_ pqp < p ,
x < qpq < q. b ~
c & d is trivial.
c ==~ a. By Corollary 4.2.2.3, x <_ pqp <_ p
so that xp -- p x -- x
172
4. C*-Algebras
(Corollary 4.2.7.6 a =v b). We deduce
xqx = xpqpx = x (Corollary 4.2.7.3 a =~ e) and
x
d =~ a follows from c =:~ a. Corollary 4.2.7.13
( 0 ) Take p,q E Pr E . If ((pqp)n),er~ has an infimum
r in E+, then r is an orthogonal projection and it is the infimum of p and q in E+. By Corollary 4.2.7.11, r is an orthogonal projection. By Proposition 4.2.7.12 c =~ a, r is a lower bound for p , q . Let x be a lower bound for p and q in E + . By Proposition 4.2.7.12 a :=> b, x is a lower bound for ((pqp)n)n~ in E + . Thus x < r . Hence r is the infimum of p and q in E + . P r o p o s i t i o n 4.2.7.14
I ( 0 ) Suppose E is unital. Take A C E#+ and put
B'-{1-xixeA}. Then the following are equivalent: a)
A has a supremum y in E#+.
b)
B has an infimum z in E#+.
If these conditions hold, then y-l-z. In particular, for p, q E Pr E , p V q exists iff (1 - p ) A
( 1 - q) exists and in
this case 1-pVq--(1-p)
A(1-q),
First assume a) holds. Take x C A. Then
x
pVq<_p+q.
4.2 The Order Relation
173
0
#
-~
9
y
Coronary 4.2.7.15
( 0 )
F
m
a C*- balg
a oJ E
that
every x e F+# , the sequence (xn),.e~ has an infimum in E+ which belongs to F . Take p, q E Pr F . Then p and q have an infimum in E+ which belongs to Pr F . If in addition E is unital and F is a unital subalgebra of E , then Pr F is a lattice and for every p,q E P r F supremum in Pr E .
the supremum of p and q in P r F
is its
Let r be the infimum of ((pqp)n)n~N in E + . By Corollary 4.2.7.13, r is the infimum of p and q in E+ and it belongs to P r F .
The last assertion
follows from the first one and Proposition 4.2.7.14. C o r o l l a r y 4.2.7.16
m
If E+ is order complete (order a-complete), then every
subset (countable subset) of Pr E has an infimum in E+ which belongs to Pr E . If in addition E is unital, then Pr E is an order complete (order a-complete) lattice. The assertion follows immediately from Corollaries 4.2.7.15 and 4.2.7.10.
m P r o p o s i t i o n 4.2.7.17
( 0 ) Let F be a C*-subalgebra of E with the pro-
perty that every commutative subset A of F+~ , for which - A
is well-ordered,
has an infimum in E+ which belongs to F . Then every nonempty subset of Pr F has an infimum in E+ which belongs to Pr F . If in addition F is unital then Pr F is an order complete lattice and an order faithful set of E .
1 74
4. C*-Algebras
Let A be a nonempty subset of Pr F . Let ~ denote the set of subsets B of Pr F for which - B
is well-ordered and every lower bound of A in E+ is
a lower bound of B . Given B, C E ~ , define B~C:c=:::~(BcC
and ( x E B , y E C ,
-< is an inductive order on ~ .
x
~ therefore has a maximal element B (Zorn's
Lemma). By Corollary 4.2.7.6 a =~ b, B is commutative. Hence, by assumption, B has an infimum p in E+ which belongs to F . By Corollary 4.2.7.10, p E Pr F . p is larger than any lower bound for A in E+. Take q C A. By Corollary 4.2.7.15, p and q posses an infimum r in E+ which belongs to Pr F . B U {r} belongs to ~ and B _ B U {r}. Since B is maximal, we deduce that r c B and so
p<_r<_q. Hence p is a lower bound for A. It is thus the infimum of A in E+. The last assertion follows immediately from the first one (and Proposition 4.2.7.14).
Remark.
1 The lattice P r F
is not necessarily distributive as shown by the
example E := F "= s
P r o p o s i t i o n 4.2.7.18
Let F be a maximal Gelfand C*-subalgebra of E and F
take p C P r F such that {~=/-0} is singleton. Then p E p C F if ] K - r p(Re E)p C F if IK = IR. F
Let x' be the unique element of {~" :/: 0}. Take x C R e E , y C F , and put 9- x ' ( y ) ,
z:=y-ap.
Then x'(z)
=
~'(y)
-
~x'(p)
=
~ -
~
-
0
and
4.2 The Order Relation
175
so that zp= pz =O.
Thus ypxp = apxp + zpxp = apxp = apxp + pxpz = pxpy .
Hence pxp C F ~ . By Proposition 4.2.2.14 a :=> b, p x p C F . Hence p(Re E ) p C F . It follows p E p C F if IK =(U. P r o p o s i t i o n 4.2.7.19
I
Let (Pk)ke~. be an increasing f a m i l y in E c N P r E f o r
some n E IN such that pl = O and Pn = l .
Then E
is the C*-direct product
of the family
((Pk+l -- pk)E(pk+I -- Pk))kEINn_l . Fix k C INn_l. Then (Pk+l
--
pk)Epk "-- ( P k + l
--
Pk)pkE
--
{0},
p k E ( p k + I - Pk) = E p k ( p k + 1 - p:~) -- {0}
(Corollary 4.2.7.6 a =~ c & d). It follows that
Pk+l Epk+l
--- (
(Pk+l - - Pk) E (Pk+l - - Pk) ) (~ (PkEpk)
(Corollary 4.2.7.6 a =~ g). The statement now follows by complete induction. I P r o p o s i t i o n 4.2.7.20
Assumme
E
is the C*-direct product of the f a m i l y
(E~)~eI of unital simple C*-algebras. For every J C I put pj : E
~ E,
x,
~ ejx.
Then p j E E c N P r E f o r every J C I and the map
q3(I)
;E cnPrE,
is an i s o m o r p h i s m of ordered sets, where ~ ( I )
J,
)pj
is ordered by inclusion.
176
4. C*-Algebras
It is obvious that p j E E c n Pr E for every J C I and that the map . ~ E c n PrE,
~(I)
J ~
~ pg
is injective and increasing (Corollary 4.2.7.6 c =, a). So we have only to prove that the above map is surjective. Take p C E c n P r E
and put
F:={x6EIpz=x},
a:={zeElpz=O}.
F and G are C*-subalgebras of E and for every x E E , pxeF,
(1-p)xeG,
x=px+(1-p)x.
Hence E is the C*-direct product of F and G . Take ~ E I . Since E~ is simple, we have EL C F or E~ C G , where we identified E~ canonically with a C*-subalgebra of E . Put
Then p = pa and therefore the map ~3(I) ~ E
~nprE,
J'
)Pa
is surjective.
I
P r o p o s i t i o n 4.2.7.21 be a family in P r E
( 0 ) A s s u m e E is (U-order a-complete, let (Pj)jes%
with n E IN, and put l ~-~ x := pj. n
a)
(x 88
j=l
i s a n i n c r e a s i n g s e q u e n c e i n E#+. Let p b e i t s s u p r e m u m i n k c IN
b)
pCPrE.
c) p is the supremum of (Pj)je~n in P r E
and in E#+ .
Let y be an upper bound of (Pj)j~INn in E+# . By Corollary 4.2.7.3 a=vb&c, -- PJY - n pj n j=l j=l 1 n 1 yx--YPJ- n pj xy
n j=l
j=l
x x.
E.
4.2 The Order Relation
177
Let F be a maximal Gelfand C*-algebra containing x and y (Corollary 4.1.2.3). Since
n
x
~
1,
j=l
F+# . a) follows e.g. via the Gelfand transform of F . By Corollary 4.2.2.20,
p G F . Using again the Gelfand transform of F , we deduce p C Pr F C Pr E and p < y . For j
INn, 1 -pj < x < p, n
so that by Corollary 4.2.7.6 h =~ a,
pj <_p. P r o p o s i t i o n 4.2.7.22
(" 0 )
A s s u m e E finite-dimensional and let P be
the set of minimal elements of Pr E \ { 0 } . Then for every p E Pr E there is a finite subset A of P such that
q, r E A ==~ qr - 5qrq.
Step 1
3q E P , q < p
Assume the contrary. By recursion, one may construct a strictly decreasing sequence in Pr E \ ( 0 } . By Corollary 4.2.7.6, this sequence is linearly independent and this contradicts our assumption that E is finite-dimensional. Step 2
The assertion
Assume the contrary. By recursion, using Step 1 and Corollary 4.2.7.6 a :=> g, one may construct a sequence (qn)ner~ in P such that n--1
qn < P - ~
qi
i--1
for every n E IN. Then
P-
~ qi
is a strictly decreasing sequence in
178
~. C*-Algebras
4.2.8 A p p r o x i m a t e U n i t Proposition
4.2.8.1
(0) {x 9 E+
I I1~11 < 1}
is upward directed.
Define A := {x 9 E+ f:[O,l[
-~IK,
I Ilxll < 1}, a. ~ 1 --Cg ~
g:lR+
>IK,
c~.~
;
l+a
Then f([O, 1[) C IR+,
g(lR+) C [0, 1[,
9 [0, 1[ = ~ g ( f ( a ) ) = a. Take x, y 9 A. Then a(x)
[0, i[,
a(y) C [0,1[
(Corollary 4.2.1.16 a =~ c), f(x), f ( y ) 9 E+ (Corollary 4.1.3.5), x = g(f(x)) _< g ( f ( x ) + f(y)) e A , y = g ( f (y)) <_ g ( f (x) + f (y)) 9 A
(Corollaries 4.1.3.11, 4.2.4.7 a), 4.1.3.5, 4.2.1.16 c ::v a). Hence A is upward directed. 1 T h e o r e m 4.2.8.2
( 0 ) (Segal, 1947) The upper section filter of {x 9 E+ I Ilxll < 1}
(Proposition 4.2.8.1), i.e. the filter on E generated by the filter base
{ { ~ e E+ I IIxll < 1 , x _ _ _ y } l y e
E+, II~ll < 1}
is an approximate unit of E . it is called the canonical approximate unit of E.
4.2 The Order Relation
179
Define A := {x e E+
< 1}
and let ~ denote the u p p e r section filter of A . Take x C R e E #
and e > 0. P u t F -
E(x), F
K - - {x' e ~ ( f ) I I~(x')l > ~} F
(Corollary 4.1.2.3). Since ~ E C o ( o ( F ) )
0_
( T h e o r e m 2.4.1.3 b), e)) there is, by
such that
Urysohn's T h e o r e m , an f C C o ( o ( F ) )
f_>l-e
on K .
F
Take z C E with ~ = f (Corollary 4.1.2.5). T h e n z E A and F
FF
F
(Corollary 4.1.3.5, T h e o r e m 4.1.3.1 b)). Take y E A with z _~ y. T h e n 0_
(Corollary 4.2.2.3), so t h a t IIx(1 - y)xll < Ix(1 - z)xll (Corollary 4.2.1.18) and ]ix - xyl] 2 - ]ix(1 - y)I] 2 - Iix(1 - y) 89(1 - y) 1 I]2 _<
_< ]ix(1 - y)89
= IIx(1 - y)}(x(1 - y)89
< Iix(1 - z)xll _< IIx -
xzillxil
Hence lim x y -- x . y,iY
= IIx( 1 - y)xII _<
<_ c .
180
4. C*-Algebras
It follows that x
= lim x y -
lim y x .
y,i~
y,~
Now let x be arbitrary. We have, for every y E Re E , iiyx - ~li ~ = II(y~ - ~)(yx - ~)* II = II(yx - z)(~*y - l yxx*y - xx*y-
- ~*)ii -
yxx* + xx*l].
By the above considerations, 0 = Ilxx* - xx* - xx* + xx* I] = lim ]]yxx*y - x x * y Y,~
= l i m ]]yx -
y,~
yxx* + xx*ll =
x]l 2
so that lim
yx
-- x,
x = x** = lim(yx*)* = l i m x y . Remark.
i
The above theorem shows that the state space T(E) and the pure
state space T0(E) arc well-defined for each C*-algebra E . C o r o l l a r y 4.2.8.3
( 0 )
There is a unique order relation on E ' , which
renders E ~ an ordered Banach space and for which E~+ is the set of positive linear forms on E .
We have
li~' + y'li-
IIx'il + iiy'li
for all x', y' E E + . Moreover, E' is order complete and every operator defined on E' is order continuous.
By Theorem 4.2.8.2, E is quasiunital. By Corollary 4.2.2.13 and Proposition 2.3.4.13 b), there is a unique order relation on E ' , with respect to which E' is an ordered Banach space and E+ is the set of positive linear forms on E . By Corollary 2.3.4.14, li~' + y'ii-
ilx' i + ily'll
for all x', y' E E+: The last assertion follows from the final assertion of Proposition 1.7.2.5. i
4.2 The Order Relation
C o r o l l a r y 4.2.8.4
181
If IK = C (IK = JR), then E " = E '~ = E '~ ((E" c E '~ C
E '~ , Re E '~ C E " ) . Moreover, the following are equivalent f o r every nonempty upward directed set A' of El+ 9
a)
A' is bounded above.
b)
A' is bounded in norm.
c)
the upper section filter of A' converges.
d)
the upper section filter of A' has a point of adherence.
e)
A' has a supremum.
I f these conditions are fulfilled, then the upper section filter of A' converges to the supremum x' of A ~ and
9 '(x) = sup y'(z) y~EA r
for every x C E + . In particular, every closed set of E ~ is order complete and an order faithful set of E'
By Theorem 4.2.8.2, E is quasiunital. Hence, except for R e E '~ C E" all the claims follow from Proposition 2.3.4.16. By Corollary 4.1.2.7 d), the absolute convex hull of E~_# is a 0-neighbourhood in Re E t , so that Re E I~ c E" by Proposition 1.7.2.4 b). Remark.
I
In the real case, we have not E '~ C E" in general. Take an arbitrary
linear form on {x' C E ' i x ' * = - x ' } and extend it by 0 on Re E ' . This linear form belongs to E ~ , but it need not be continuous if E ~ is infinite-dimensional. C o r o l l a r y 4.2.8.5 a)
( 0 ) Let E ~ (0} be a C*-algebra and take x C S n E .
The closed convex hull of a(x) U {0} and the closed convex hull of
~(~-(E)) U {0} coincide. b)
The extreme points of or(x) are contained in ~c(T0(E))U {0}.
c)
There is an x' C ~-o(E) with
Ix'(x)l = IIxll
182
4. C*-Algebras
d) IIxll = II~ll = II~lw0(E)ll.
By Corollary 4.1.2.7 e), 7(E) =/= 0 so T0(E) ~= 0 (Theorem of KreinMilman). Let F be the unital C*-algebra associated to E (Theorem 4.1.1.13). Then ~
= ~
a) By Corollary 4.1.2.7 b), the closed convex hull of aE(x) is ~:(T(F)). The assertion now follows from Proposition 2.3.5.5 a). b) By Proposition 4.1.2.7 b), the extreme points of a(x) are contained in ~(7o(F)). The assertion now follows from Proposition 2.3.5.5 b). c) The assertion follows from b) (and Theorem 4.1.1.16 b)). d) follows from c) (and Proposition 2.3.5.8). I C o r o l l a r y 4.2.8.6 Assume IK = ~ (IK = IR). Given x E E (x E Re E ) , a(x) is contained in the convex hull of ~(T(E)) U {0}. Let F be the unital C*-algebra associated to E (Theorem 4.1.1.13). By Corollary 4.1.2.7 a),b),
and by Proposition 2.3.5.5 a), ~:(T(F)) is the convex hull of ~ ( T ( E ) ) U {0}. I C o r o l l a r y 4.2.8.7 lent: a) x C R e E
Assume IK =q3. Given x c E , the following are equiva-
(rasp. x C E+).
b) ~(~(E)) c ~
(~e~p. ~(~(E)) c ~ + )
c) ~(~0(E))c ~
(r~; ~(~0(E))c ~ + )
a =~ b follows from Corollary 4.2.8.5 a) (and Corollary 4.1.2.13 a =~ b). b ~ c is trivial. c =~ a. Since v
imx]z0(E) - 0, it follows from Corollary 4.2.8.5 d) that imx - 0. Hence x C Re E and the claim now follows from Corollary 4.2.8.5 b).
1
4.2 The Order Relation
Corollary 4.2.8.8
183
( 0 ) Every finite-dimensional C*-algebra possesses a
unit. This follows from Proposition 2.2.7.25 b) and Theorem 4.2.812.
m
Corollary 4.2.8.9 Let F be a C*-subalgebra o,f E . Then, given any y' C F~, there is an x' E E~+ with x ' i F = y',
IIx'll = IlY'II-
We may assume E to be unital (Theorem 4.1.1.13). First suppose that F is unital. By Corollary 2.3.2.23, there is an x' C Re E' such that
IIx'll
x ' l F - - y',
= Ily'll.
By Corollary 4.1.2.8 b), x' is positive. Now suppose that F is not unital. Define G := { a l + x I ( a , x ) 9 IK • F } . G is a unital C*-subalgebra of E . By Corollary 2.3.4.12, there is a z' C G~_ such that
z'lF-
y',
IIz'll = IlY'II-
By the above considerations, there is an x' E E~_ with
x'lG = ~',
ii~'ii = liz'll.
Then
9 ' l r = z'lF = y', C o r o l l a r y 4.2.8.10
liz'li = llr
m
Let F be a C*-subalgebra of E . Then for each y' E
~-0(F), there is an x' C ~-o(E) with x'iF = y'. We may assume that E is unital (Theorem 4.1.1.13, Proposition 2.3.5.5 b)). Put
A' : : {x' e ~ ( E ) i x'lF = ~'}. By Corollary 4.2.8.9, A' is nonempty. Take z~,x~ C T(E) and c~,fl E]0, 1[ with
a+ fl-1,
axe+fix'2CA'.
183
3. C*-Algebras
Then 1
-- a
+ fl >_ ~llx'~lFII +/~llx~lFII >__ II(c~x~ + ~x~)lF)ll--
1
(Corollary 4.2.8.3), so that 1,
IIx'~lFII = I I ~ I F I I -
' ' xilF = x2lF E ~(F)
Since
it follows that zl' IF = z2' I F X l,,
X 2, E
=
A'
y'
9
Hence A' is a face of T(E). Since A' is obviously closed in T(E) and since ~-(E) is compact (Proposition 2.3.5.9 a)), by the Theorem of Krein-Milman (Theorem 1.3.1.10 b)), it has a pure state. I C o r o l l a r y 4.2.8.11
If E is a Gelfand C*-algebra, then T o ( E ) = a ( E ) .
E is symmetric (Theorem 4.1.2.1) and quasiunital (Theorem 4.2.8.2), the Gelfand transform of E is bijective (Corollary 4.1.2.5), and Re E ' - E~_ - E~_ (Corollary 4.1.2.7 d)). Hence the assertion follows from Proposition 2.4.2.11 h=~b. I P r o p o s i t i o n 4.2.8.12
Assume ]K = ~
(IK = JR). Take x E E and let
(Xn)nE~ be a bounded sequence in E . a)
If ((xn, x'>),c~ converges (to ( x , x ' ) ) f o r every x' E To(E), then
(<~,~'>)~ ~o~w~g~ (to <~,x'> ) So~ ~w~y z' e E' (~' e R+ E'). b)
/f ((xn, x'>)ne~ converges for every x' E vo(E) and if F is a predual of E , then there is a y E E such that
lim (xn, a> -- (y, a) n--~ oo
for each a E F
(aEReE).
4.2 The Order Relation
a)
Case 1
185
E unital and separable
By Proposition 2.3.5.9, re(E) is metrizable and compact. By Proposition 2.3.5.8, (~ni~'0(E))=e~ is a bounded sequence in C(To(E)) which converges pointwise to a function f on T0(E) (to ~]T0(E)). Take x' C T(E). By Choquet's Theorem (G. Choquet, P.-A. Meyer, Existence et unicitd des reprdsentations int@rales dans les convexes compacts quelconques, Ann. Inst. Fourier, 13 (1963), Corolaire 1~) there is a positive Radon measure # on T0(E) such that
yd# =
#(T(E)\To(E)) = Then, by Lebesgue's Dominated Convergence Theorem,
i fd# = (x, x') =
lim n-----~ o o
i xnd# V =
d# = lim n--~cr
lim (x~,x') n --.+ o o
xnd# = lim (xn, x')
.
n--+cr
By Corollary 4.1.2.7 d), if x' e E' (x' e R e E ' ) , then ((xn, x'))nc~ converges (to (x, x') ). Case 2
E unital
Let F be the unital C*-subalgebra of E generated by {x} U {xnin C IN}. Then F is separable (Corollary 2.3.2.14 c)). Take y' C T0(F). By Corollary 4.2.8.10, there is an x' E T0(E) such that x'iF = y'. It follows that ((xn, y'))ne~ converges (to (x,y')). Given y' e F ' , (y' e R e F ' ) , ((xn, y'))nc~ converges (to (x, y')) by Case 1. In particular, ((x~, x'))~er~ converges (to (x, x') ) whenever x' E E', (x' C Re E'). Case 3
E arbitrary
Let F be the unital C*-algebra associated to E (Theorem 4.1.1.13). Then ((x=,y'))~er~ converges (to (x,y'))whenever y ' E ~-0(F) (Proposition 2.3.5.5 b)). By Case 2, ((xn, y'))~er~ converges (to (x,y')) whenever y' C F ' , (y' C Re F'). Hence ((Xn, X'))ne~ converges (to (x,x')) whenever x ' C E ' , ( x ' C Re E') (Hahn-Banach Theorem and Corollary 2.3.2.23).
186
4. C*-Algebras
b) By a), ((xn, a))~e~ converges whenever a E F (a E R e F , by Proposition 2.3.2.22 e)). By the Alaoglu-Bourbaki Theorem, there is a y E E such that lim (xn, a) = (y, a)
n--+oo
for every a E F (a E R e F ) . T h e o r e m 4.2.8.13
I
( 0 ) (Grothendieck, 1957) For every x ' E R e E ' , there
are uniquely determined x ' + , x ' - E E~+ (Corollary ~.2.8.3), such that 9 '-
~'+
- ~'-,
II~'ll- IIx'+ll + II~'-II,
x ~+ and x ~- are called the p o s i t i v e and the negative part o] x ~, respectively. If u " E -+ F is a surjective involutive algebra h o m o m o r p h i s m of E onto a C*-algebra F then (u'y') + = u'y'+,
(u'y')- = u ' y ' -
for every y~ E Re F ~.
The existence was established in Proposition 4.1.2.7 d). We now prove uniqueness. First assume that E is unital. We may assume that IIx'l]- 1. Take y', z' E E'+ with
Take c > 0. By Proposition 2.3.2.22 j), there is an x E Re E # , such that C2
9 '(x) >
Defining y'-~ l(1-x)
,
1
z'-
~ (l+x) '
we have that y, z E E # and it follows from -1 <x<
1
(Corollary 4.2.1.16 a ~ b) that 0_
0
4.2 The Order Relation
187
C2
~.2
y'(1) + z'(1) = Ily'll + IIz'll = 1 < x ' ( x ) + --~ = y'(x) - z'(x) + -~ (Corollary 2.3.4.7),
1 y'(y) + z'(z)
-
~2
-~ (y'(1) - ~'(~) + z'(1) + z'(~)) < 1-~'
C2
C2
y'(y) < -~--6, z'(z) < 16 Take a C E . We have C2
C2
z'llz'(z) < ~
Iz'(za)l 2 ~
(Corollary 4.2.2.6, 4.2.1.17 b =~ a), so t h a t
lY'(ya)I <_ l all,
Ix'+(Ya)l <_ Ilall,
Iz'(za)l <_ -4 lall,
Ix'-(za)l <_
We deduce
lY'(a) - x'+ (a)l - lY'((Y + z)a) -- x '+ ((y + z)a)l -
= l y ' ( y a ) + y'(za) - x'§
= ly'(ya)-
- x'+(za)l
=
x'+(ya) + z'(za) - x ' - ( z a ) l <_
ly'(ya)l § Ix'+(ya)l + Iz'(za)l + Ix'-(za)l ~ Ellal] 9 Since c is arbitary, it follows t h a t
y'(a) - x '+ (a) and thus
188
4. C * - A l g e b r a s
Now suppose that E has no unit and let ~ be an approximate unit of E (Theorem 4.2.8.2). Take y', z' C E~+ with
x'--- y'-
z',
IIx'll- Ily'll + IIz'll.
Let F be the unital C*-algebra associated to E . Take xl E Re
F'
, Yl, z~ E F~_
as defined in Proposition 4.1.2.27 a),e). Then Yl' E = y' ,
Xl' I E = x' ,
IIx'~ll =
IIx'll,
Z 'l l E = z' ,
Ily'~ll - Ily'll,
IIz'~ll = IIz'll
limz'(x'x)
= limx'(x*x)
(Proposition 4.1.2.27 c)). Thus y'l ( 1 ) - Z'l(1 ) - l i m y ' ( x * x ) -
- X'l(1 )
(Proposition 4.1.2.27 a)), so that Yl
X l =
--
Zl.
Moreover,
]1~'111 = I1~'1 = Ily'll + ]lz'll = Ily'l I + IIz',ll and so I Yl --
,+ Xl
, Zl
,
--
IXl
,
by the first part of the proof. We get
y' - - X l'§
,
~'- X l'-IE
which proves the uniqueness in this case. Now we prove the last assertion. By Corollary 4.2.2.11, !
u ' y ~+ , u ' y ' - C E +
By Theorem 4.2.6.6 and Proposition 1.3.5.2,
II~'y'll = Ily'll = Ily'§
+ Ily'-II = II~'y'§
+ ll~'y'-II
Hence (u'y') + - u'y,+ ,
(u'y') - - u'y'-
.
4.2 The Order Relation
Remark.
189
It is obvious that
(_x,)+ _ ~,-,
( _ ~ , ) - _ ~,+
for every x' E Re E ' . In general, the map ReE'
>E',
x',
>x '+ (resp. x ' - )
is not continuous. C o r o l l a r y 4.2.8.14
Let F be a C*-subalgebra of E
and take y' E R e F ' .
Then there is an x' E Re E' such that x ' l F = y',
IIx'll = Ily'll,
and for any such x ' ,
x '+l F = y ' + ,
z'-I F = y ' -
Tile existence of x' was proved in Corollary 2.3.2.23. Now take x' E Re E' with x'lF - y',
llx'll- lly'll.
Then y' = z' F = z'+l F - z ' - I F ,
lly' = llx'+l F - x'-IFll ~< llx'+IF I + Ilx'-IFll ~< llx'§ + lx'-ll = llx'll- lly'll. It follows that
lly'll- llx'§
+ Ix'-IFll,
so that (by Theorem 4.2.8.13) y,+ _ x ' + l F , Remark.
y'- _ x'-lF"
The above relations
(x'le) + = x ' + l e ,
(x'le)- - x'-le
do not hold in general, as shown by the following example
I
190
4. C*-Algebras
E:=C({O,~}), x''E
~IK,
x,
>x(O)-x(1),
F - = IK1.
Let E be a C*-algebra without unit, F its associated unital C*-algebra, and take x' 9 Re F ' . Then
C o r o l l a r y 4.2.8.15
x'+lE = (x'lE) +,
x ' - I E - (x'lE)-.
By Theorem 4.2.8.2, E possesses an approximate unit. We use the notation of Proposition 4.1.2.27 in this proof. We have X0' :
(X/+)0 - - ( x! - )0,
! __ (Xt+)01 X01
(X/--)01
and by Proposition 4.1.2.27 f), (x'+)o, (x"-)0 9 E~_. By Proposition 4.1.2.27 g), we deduce that
ii~'ii = ii~il + l l x ' - ~0,1i' < < li(x'§
+ ll(z'-)oll + lix '§ - (~'§ = i ~'§
i + li~'- - ( x ' - ) 0 , 1 E -
+ II~'- I = I ~ ' l
Thus
l i z ~ l i - li(~'§
+ ti(z'-)oll,
(x'iE) § = (z0) § - (~'+)0 = z'§ (x'iE)- - (x~)- - (x'-)o - x'-IE. C o r o l l a r y 4.2.8.16
( 0 ) Let (x,x')C R e E x E~+. If XX I --
XlX
,
then x§
'-
x'x § -
(xx') § ,
x-x'=
x'x-
=
(xx')-.
4.2 The Order Relation
191
We may assume E unital (Theorem 4.1.1.13, Proposition 4.1.2.27 e),j)). By the Theorem of Fuglede-Putnam, 1__
(X-~-)2X
i
- - XI(x"~)'~
1
1_
i
(X--)2X
-- XI(X--)2
1
so that
~+x'-(x+)~
1 Xl
(~+)~1 - x ' x
x-x' - (x-)~x'(x-)~
t
§ e E+,
- ~ ' x - c E+
(Proposition 2.3.6.4). Since z x ' = (x § - x - ) ~ ' = ~ + x ' -
z-x'
and
]lx~'ll- I] Ixlx'll- II(x § + x-)x'll = IIx+z'll + ]lx-x'l (Proposition 4.2.6.17, Corollary 4.2.8.3) we deduce x+x ' -
(~x') +,
x-~'=
(~')-
(Theorem 4.2.8.13). E x a m p l e 4.2.8.17
i
Take n C IN. For a C s
n) p u t n
~'s
n)
, IK,
[aii]i,i ~ n ;
"~ E
aijaji.
i,j=l
Then f o r every a C
Re s a + - "d+ ,
a--a
--
By Example 4.1.4.11 (with the corresponding notation), there is a finite subset A of IKn and a real function f on A such that
(zly) - 5~,~ for all x,y E A and a - ~-~ f (x)[xlx] ,
a(a) = f (A) .
xEA
By Theorem 4.1.4.6 e) and Theorem 4.2.2.9 b),
192
4. C* -Algebras
a+ = E
sup{f(x), O}[xlx],
xEA
a- = E s u p { - f ( x ) , 0}[xlx], xEA
By Example 4.1.4.11,
II~ll - ~ If(x)l,
Ila~ll = ~ sup{f(x), 0},
xEA
xEA
l a~-II = ~
s u p { - f ( x ) , 0},
xEA
I1~11- Ila+ll + Ila-II. Since a + , a- are positive (Example 4.2.3.2 a =~ b), it follows ~+ = a + E x a m p l e 4.2.8.18
~- = a -
I
Take q), ~ E IR,
cos cp sin cp c0s2r
y -=
sin2 cp cos r sin r ]
cos r sin r
sin2r
a:=~+r
z'-x-y,
'
3-=p-r
1 [sinasinfl+[sin/~[ u "- ~ sin/3 cos a
sin~cosa ] sin a sin/3 + [ sin fl '
1 [ sinasinfl+lsinfl v := ~ - sin/3 cos a
-sin/3cosa sin a sin/3 + I sin/3 I
Given a E E(IK2), define 2
~c(~)
' ~,
[~J]',J~ ~
Z i,j= l
Then 7, ~ E ~-0(E(IK2)) and
~+=~,
~-=~.
~jaj~
.
4.2 The Order Relation
By E x a m p l e 2.3.5.3 b =a a, ~, ~ E ~-0(s -
[
Z
c~
193
Now 7
99 - cos 2 r
cos 99 sin 9 9 - c o s r sin r ]
[ cos 99 sin 9 9 - cos r
sin 2 99 - sin 2 r
sin r
J
so t h a t Z+ -~-- U~
Z-- - - ' V
by E x a m p l e 4.2.3.7. T h e assertion now follows from E x a m p l e 4.2.8.17.
Remark.
m
If we take r := -99 and set [
J
1
if sin 299_>0
-1
if sin 2 9 9 < 0
Oz
then u = cos 99 sin 99
I~1 11 1 I~1 '
Hence if 27r~ r 1 8 9
v = cos 99 sin 99
[1~1 -1
-1lal ]
then
z#y,
~#x,
v#y.
This is a c o u n t e r - e x a m p l e to the assertion "if E is a C * - a l g e b r a and
9 ', y' 9 ~0(E) are distinct, then
(~'-y')§
Proposition
4.2.8.19
= x',
Let ] I 4 - ~
(z' - y')- = y' "
and n E IN3. Every n-dimensional C * -
algebra is isometric to ~ ( ] N n ) . Let E be an n - d i m e n s i o n a l C*-algebra. By Corollary 4.2.8.8, E possesses a unit. By Corollary 4.1.2.5, it is sufficient to show t h a t E is commutative. This is obvious if n E IN2, so assume n = 3. By Corollary 4.1.4.10, there are p, q C P r E so t h a t { 1, p, q} is an algebraic basis for E . Take a,/3, "7 C 9 with
194
4. C*-Algebras
pq = a l + /3p + Tq. Then
a l + ~p + ,,/q = pq = p2q = ap +/3p 2 + 7Pq = a,),l + (a + ~ + 137)p + 72q,
a l + 13p + ~/q = pq = pq2 = aq + flpq + 7q 2 = a/31 +/32p + (a + ~/+/37)q, so t h a t
l
a=aT=a~
7 = -),2 -- a +-), + fiT. It follows from this system of equations t h a t a,/3,-), C I R . Thus
qp = (pq)* = -~1 + 13p + ~q = pq . Hence E is commutative.
Remark.
m
a) The above result also follows from Corollary 6.3.6.5.
b) ([J is a 2-dimensional real C*-algebra, wich is not isomorphic to g~(IN2), so the above proposition does not hold for IK = IR. Proposition
4.2.8.20
Let E be a finite-dimensional complex C*-algebra. If
E has a commutative C*-subalgebra F of codimension 1, then E is commutative. We may assume t h a t 1 C F . P u t n " - Dim F . Using the Gelfand transform, we find a family an algebraic basis for F such that
pipj = 0 whenever i , j E lNn are distinct and n
E Pii--1
1.
(Pi)ielNn
in Pr F which forms
4.2 The Order Relation
195
By Corollary 4.1.4.10, there is a q 9 Pr E such that {p~[i 9 INn} U {q} is an algebraic basis for E . Take i 9 INn. There is a family (olj)jelN n in ~ and a 9 C such that p~q = ~
ajpj +/3q.
j=l
We get Piq = PiPiq = aipi + ~p~q ,
PiqPi = aiPi + ~qPi.
If 13=1 then a i = O
and qPi = piqPi ,
Piq = (qPi)*
Ifl3#l
=
PiqPi
=
qPi.
then OLi Piq
=
1 - / 3 pi
and we deduce from Corollary 4.2.7.6 i =v b p~q = qPi.
m
Hence E is commutative. P r o p o s i t i o n 4.2.8.21
Let E
be a unital complex C*-algebra and take p C
E\{O, 1 } . / f E ( p , 1) is a m a x i m a l c o m m u t a t i v e proper C*-subalgebra of E , then there is a unital C*-subalgebra of E ~(~)
containing p which is isometric to
.
Take x E Re E \ E ( p , 1). Assume that px(p-
1) = O.
Then px = pxp,
196
4. C*-Algebras
x p -- (px)* -- p x p -- p x .
Since the C*-subalgebra of E generated by (1, p, x) is commutative (Corollary 2.3.2.14 b), this contradicts our hypothesis. Hence y := p x ( p -
1) -~ 0.
Take a, fl, 7, 5 E r with a l + tip + 7Y + 6y* = 0. Then 0 = p(o~l + tip + 7Y + 6y*)(1 - p) = "/y,
0 = (1 - p)(c~l + tip + 7Y + 5y*)p = 5y*. Hence 7 =(f =0,
a=fl=O. Thus (1, p, y, y*) is a basis for a vector subspace F of E . Since py*y = 0 = y * y p ,
the C*-subalgebra of E generated by ( 1 , p , y * y )
is commutative (Corollary
2.3.2.14 b)). By the hypothesis on E(p, 1), it follows that yy* E E(p, 1). Similarly y*y E E(p, 1). Hence F is a unital C*-subalgebra of E containing p. Let u be the linear map E -+/2(C 2) defined by ul "=
~y-Ilyll
E10] [01] 0
1
0 0
,
'
where we have identified q~2,2 with s
up :--
[10] [ j ,
0 0
~y*-Ilyll
o 0 1
0
'
It is easy to check that u is a bijective
h o m o m o r p h i s m of involutive algebras. By Corollary 4.1.1.21, u is an isometry. m
4.2 The Order Relation
Proposition
4.2.8.22
197
Every 4-dimensional complex C*-algebra is isometric
either to g~(INa) or s
Let E be a 4-dimensional C*-algebra. By Corollary 4.2.8.8, E possesses a unit. If E is commutative, then the Gelfand transform shows t h a t E is isometric to g~(lN4). Assume that E is not commutative. Take p E E \ { 0 , 1} (Corollary 4.1.4.10). By Proposition 4.2.8.20, E(p, 1) is a maximal commutative C*-subalgebra of E . Hence by Proposition 4.2.8.21, E is isometric to /:(IK2).
m Remark.
The above result also follows from Wedderburn's Theorem (Corollary
6.3.6.5). Proposition
4.2.8.23
Let E be a C*-algebra without unit and IK x E its
associated unital C*-algebra. Then the map
is a projection of ]K x E onto {0} x E with norm 2 and any such projection has the nerm at least 2.
Let u be a projection of I K x E
onto {0} x E .
We show that Ilull > 2.
Since {0} x E is an involutive set of IK x E , re u is ~lso a projection of ]K x E onto {0} x E (Proposition 2.3.2.22 n)). Since
(Proposition 2.3.2.7), we may assume t h a t u is selfadjoint. Then there is an x C Re E with
~(i, 0) = (0, x). By Corollary 4.1.4.14, there is a commutative C*-subalgebra F of E containing x such that inf Ix'(x) l = 0. x'ea(F) Take x' c e ( F ) . Using the Gelfand transform on F , we find a y C F+# with
~ ' ( v ) - 1. Then the unital C*-subalgebra G of IK x E generated by y is a Gelfand C*-algebra. Using the Gelfand transform on G , we see that
198
4. C*-Algebras
11(1,-2y)11 ~ 1. It follows t h a t
I1~11 ~ Ilu(1,-2y)ll = I1(0, x I~'(x- 2y)l-
2y)ll = I I x- 2yll
Ix'(~) - 21 ~ 2 -
I~'(~)1 9
Since x' is arbitrary, we deduce that
II~ll > 2. By Proposition 4.1.2.27 b) (and Theorem 4.2.8.2),
II(O,x)ll = Ilxll ~ 211(,~,x)ll for every (c~, x) e IK x E . Hence the projection of IK x E onto {0} x E
~•
>~•
(~,~),
~(0,~)
has norm at most 2. By the above considerations its norm is exactly 2. Theorem
4.2.8.24
( 0 )
A real C*-algebraE for which R e E
I
is one-
dimensional is isomorphic to JR,fig, or ]H. Let 9.1 be the set of finite subsets A of E \ { 0 } such that
x E A ~
x+x* =0,
x, y C A , x r y ==~ xy + yx = O. For every A C 92, denote by FA the subalgebra of E generated by A U R e E . Then FA is a finite-dimensional involutive subalgebra of E and therefore a C*-subalgebra of E . By Corollary 4.2.8.8, FA is unital. Moreover, for every x E F A \ { 0 } , x*x is a strictly positive element of FA. Thus there is an a > 0 with
x*x = c~l . By Proposition 2.3.1.47 e), FA is isomorphic to IR, C or 1H. Assume E is not isomorphic to I R , r
or IH. Then FA ~ E for every
A E 9.1. By Proposition 2.3.1.47 c), we can successively find elements x, y, and z of E such that
4.2 The Order Relation
199
{x}, {x, y}, {~, y, z} e ~ ,
The relation z r F{x,y} contradicts Proposition 2.3.1.47 d). Hence E is isomorphic to IR, C or IH.
Corollary 4.2.8.25
I
( 0 )
Let E
be a real C*-algebra. If the dimension
of every Gelfand C*-subalgebra of E is at most 1, then E is isomorphic to
{0}, ~ , ~ ,
or ~ .
Let x C R e E , x ~ 0. Then there is an c~ C IR, c~ ~ 0, such t h a t X2 ~
O/X.
Put 1
y := - x . Ol
Then y2
1 = --~X
---- --Xoz ~- y "
Let x, y be distinct elements of Re E \ { 0 } . We want to show t h a t x and y are linearly dependent. By the above we may assume x 2 : x~
y2 = y.
There are c~,/~ C IR,/~ ~ 0, such t h a t
(x + y)~ = ~(x + y),
( x - y)~ = Z ( z - y).
It follows successively
{
~x + ~y = (~ + y)~ - ~ + ~ + (~y + y~) - x + y + (~y + yx) ~ x - ~ u = (x - y)~ - ~
+ y~ - ( x y + u x ) - x + u - ( x y + y ~ ) ,
(~ + 9 - 2)x + (~-
Z - 2)v = 0,
x and y are linearly dependent. Thus Re E is at most one-dimensional and the assertion follows from Theorem 4.2.8.24.
200
Remark.
4. C*-Algebras
The real Banach space gx endowed with the multiplication g1
> gx,
g~
(x,y):
~0
and with the involution 61
~ 61 ,
x l
) --X
is an infinite-dimensional real involutive Banach algebra such that Re gl _ {0}. This example shows that we cannot replace "C*-algebra" by "involutive Banach algebra" in the above corollary. T h e o r e m 4.2.8.26
( 0 )
Let E be a real C*-algebra such that
xy = O ~ for all positive elements x , y
(x = O or y - - O )
of E .
Then E is isomorphic to {0}, IR,C or
IH.
Let F be a Gelfand C*-subalgebra of E , F r {0}. Suppose or(F) has two distinct points. There are f, g E Co(a(E))+ such that
Ilfll - Ilgll- 1,
f 9 = O.
Then there are positive elements x , y of F with ~ = f , 4.2.1.12). We have ~9
-
fg
-
~-g
(Corollary
0,
so that xy = 0 and therefore x=O
or y = O .
This contradicts the relations
Ixll- Ilfll = 1,
[lyll- [If I [ - 1.
Hence a ( F ) has exactly one point and F is one-dimensional. By the preceding Corollary, E is isomorphic to {0}, IR, C, or IH. I P r o p o s i t i o n 4.2.8.27
( 0 )
real algebra for which R e E (xn)ne~ in E such that:
Let E be an infinite-dimensional involutive
is finite-dimensional. Then there is a sequence
4.2 The Order Relation
201
1) n C iN =~ xn + x n* = 0 , 2) n , p 9 IN, n ~ p ==~xnxp + XpXn
:
O,
3) n, p, q 9 IN , n < p < q =~ xnxpxq = O , IN ==~ ~ xn does not belong to the subalgebra of E
4)
n
c
t
generated by { x m l m 9 INn-l} U Re E .
By Proposition 2.3.1.22, ]K = JR. We denote for every A C E by E ( A ) the subalgebra of E generated by AURe E . The family (xn)n~p in E (with p 9 INU{0, c~} ) is called a Clifford family if for every n 9 INp, $
Xn + X n -- 0 ,
Xn r E({~ml~n e ~ _ ~ } ) . If in addition X m X n - ' t - X n X m ~- 0
for all distinct m, n 9 INp, then (xn)ncsp is called a strong Clifford family.
Step 1
If (Xn)ne~p is a Clifford family in E with p 9 IN U{0}, then there is an Xp+l 9 E such that (xn)ne~+l is a Clifford family in E
By Proposition 2.3.1.49, E ( { x n l n 9 ]Np}) is finite-dimensional. Take x E E \ E ( { x n l n 9 INp}) .
We set Xp+ 1 :-'- X -- X* .
Then Xp+ l ~- Xp+ l - - O .
Since X --~ X* C R e E C E ( { X n I T t e I N p } )
,
202
4. C*-Algebras
it follows
Hence
(Xn)neINp+ 1 is a Clifford family.
Step 2
There is a Clifford sequence in E
The assertion follows from Step 1 by induction. If (xn)ner~ is a Clifford sequence in E then for every p E IN there is a Clifford sequence Step 3
(Yn)ner~ in E such that n E ]Np
~Xn
n G IN\INp
= Yn,
~ XpYn + ynXp = 0
Since XpXn + XnXp C:_.Re E
for every n E IN and since Re E is finite-dimensional, there is a strictly increasing sequence (kj)jeiN in IN such that kl = p and such that the family (XpX n "~ XnXp)nE]Nkj+l \INk 3
is linearly dependent for every j C IN. Hence for every j E iN, there is a family (~jn)nEiNkj+l\iNkj in IR such that kj+l
Z
kj+l 2
+ x.x ) - o,
n--k 3 +1
n--kj +1
We set Yn :-- xn
for all n C INp and kj+ 1 yp+j :--
~ O~jnXn n-kj +1
for all j C IN. Then for every j C IN, * yp+j 4- yp+j --
kj+l ~ n-kj+l
* Otjn(X n -'~ Xn) = O,
r o.
4.2 The Order Relation
203
and kj+l
Xpyp+j + yp+jXp =
~
ajn(XpXn + X~Xp) = O.
n=kj + l
Step 4
There is a strong Clifford sequence in E
The assertion follows from the Steps 2 and 3 by induction. Step 5
~ If (xn)ner~3 is a strong Clifford family t in E , then X l X 2 X 3 E R e E
This follows from XlX2X3)* -- X3X2X 9 9 91 = - - X 3 X 2 X 1 = X l X 2 X 3 .
If (xn)ne~ is a strong Clifford sequence in E , then Step 6
for every p, q C IN, p < q, there is a strong Clifford sequence (Yn)ne~q in E such that n C INq ==:>xn = yn ,
n C IN\INq ==~ XpXqyn = O
By Step 5, (X,XqXq+n)~er~ is a sequence in Re E . Since Re E is finitedimensional, there is a strictly increasing sequence (kj)j~lN in IN such that kl = q and such that the family (XpXqXn )nEINkj+l \INkj
is linearly dependent for every j c IN. Hence for each j C IN, there is a family (O~jn)nEINkj+l\lNkj in IR such that kj+l
kj+l OLjnXpXqX n -- O,
n=kj -4-1
2 ~ OLin # O. n=kj + 1
We set Yn :-- Xn
for all n C INq and kj+l yq+j :--
~_~ OLjnXn n=kj -+-i
204
4. C*-Algebras
for all j E IN. Then for every j E iN, kj+l yq+j h- yq+j
.___., O~jn(Xn W Xn) -~ O, n-kj -t-1
y~+j ~ E ({y~l~ e
]Nq+j_l})
,
and kj+l XpXqyq+j =
E OLjnXpXqXn --~ O ~ n-kj -t-1
kj+l 3gmYq+j + yq+jXm =
for all m E iNkj
E O~jn(XmXn + XnXm) --" 0 n=kj--t-1
(iN\iNkj+,), and therefore ki+l
Yq+iYq+j Jr- Yq+jYq+i --
E
Olin(XnYq+J -1- yq+jXn) : 0
n=ki+l
for all i E iN, i :fi j . Step 7
The assertion of the proposition
Follows from the Steps 4 and 6 by induction. T h e o r e m 4.2.8.28
1
(
0
)
Let E
be a C'-algebra. I f R e E
I is f i n i t e -
dimensional, then E is also f i n i t e - d i m e n s i o n a l .
Assume E is infinite-dimensional. By Proposition 4.2.8.27, there is a sequence (xn)ner~ in E such that" 1) n C iN =::~x,~ + xn = O, 2) n, p C iN, n ~ p ~ xnxp + xpxn = O, 3) n , p , q c iN , n < p < q ==~x~XpXq - O,
4)
iN =~ ~ Xn does not belong to the subalgebra of E n
c
(
generated by {xml m E iNn-l} U Re E .
4.2 The Order Relation
By
205
1), X n2 -~- -x~x,~ C Re E
for every n C IN and by 2), 2
2
2
2
XnX p -- XpX n
for all n,p e IN. Hence the C*-subalgebra F of E generated by {x~ln e IN} is a Gelfand C*-algebra (Corollary 4.1.2.3). Since F C R e E ,
F is finite-
dimensional. Hence a(F) is finite. In particular, F is unital and there are p, q C IN, p < q, such that P
q
Xn : n=l
Xn n=p+l
are strictly negative (Proposition 4.2.1.19). Hence P
E n-1
q 2
Xn :
E
2
Xn
n-p+l
are invertible in F . By 2) and 3), X m=l
X n=p+l
Xq+l = E m=l
E 2 2 2 XmXnXq+l n=p+l
=0
so that by 1), 9 2 Xq+lXq+ 1 -- --Xq_4_l
~--~0
Ilxq+lll 2= IlXq+lXq+l I = 0, Xq+l - 0. This contradicts 4). Remark.
II
a) The real Banach space gl endowed with the multiplication
/71• and with the involution
>/71, (x,y),
~0
206
4. C*-Algebras
is an infinite-dimensional real involutive Banach algebra with Re t~1 = {0}. If E denotes the canonical involutive unital real Banach algebra associated to 61 (Proposition 2.3.2.8), then E is infinite-dimensional and R e E
is one-
dimensional. These examples show that the Theorems 4.2.8.24 and 4.2.8.28 do not hold for arbitrary involutive real Banach algebras. b) In order that E is finite-dimensional it is sufficient to assume that every Gelfand C*-subalgebra of E is finite-dimensional (Proposition 6.3.6.17 e ==~ a).
Proposition 4.2.8.29
Let E be a C*-algebra and
P := {p E E c N Pr E [ E p a)
For every p E P
is one-dimensional}.
there is a unique xp " E ~ IK such that xp = xp(x)p
for every x C E . xp is a non-degenerate positive algebra homomorphism. b)
If E is the C*-direct sum (or product) of a family of unital simple C * algebras, then for every non-degenerate algebra homomorphism x' " E ]K there is a p C P such that X ! --
X pI
.
In this case, there are exactly Card P factors of the form IK in E . a) The existence and uniqueness of xp' is obvious. Take x , y E E and a, ~ C IK. Then
9 '~(~ + Z y ) p - ( ~ + ~ y ) p - ~(~p) + Z(yp) =
= .x',(z)p + Z~'~(y)p- (.x'~(z) + Z~',(y))p.
xtp(x*)p = x*p = (xp)* - (x'p(x)p)* - x~(x)p, so that Xp' is an involutive algebra homomorphism. It follows that
x~(x*x) = x,(x)x'(x) 9 ~ §
4.2 The Order Relation
207
for every x C E , i.e. xp is positive. Since x~p(p)p = pp = p , !
it follows that xp is non-degenerate. b) Let E be the C*-direct sum (or product) of the family (E~)~c/ of unital simple C*-algebra. For every c E I let E~ be the canonical image of E~ in E . Kerx' is a closed ideal of E (Proposition 2.2.4.19)and E / K e r x ! is one-dimensional. For every ~ C I , since E~ is simple, either E~ C Ker x ! or E~ f3 Ker x' = {0}. Since E / K e r x' is isomorphic to the C*-direct sum (or product) of the family (E~/Kerx')~el, it follows that there is a unique A E I such that E~ n Kerx' -- (0}. Moreover, E~ is one-dimensional. Then
p-=
(6~,~,)~e~ e 7:'
and Xp
X!
The last assertion follows from Proposition 4.2.7.20.
I
208
4. C * - A l g e b r a s
4.3 S u p p l e m e n t a r y R e s u l t s on C * - A l g e b r a s The ordering of a C*-algebra does not give rise to a lattice, even in the simplest n o n - c o m m u t a t i v e case of two-by-two matrices. However the situation improves when the order is complete (as in the case of W*-algebras) and when attention is restricted to orthogonal projections. Then we have a complete lattice. If, in addition, the C*-algebra is unital and if p is an orthogonal projection, then 1-p
is the complementary orthogonal projection - that is to say pA ( 1 - p ) = 0
and p V (1 - p )
= 1. (But observe that the orthogonal projections do not
form a Boolean algebra because of the failure of distributivity.) If the order is complete, then the functional calculus can be extended significantly for the selfnormal elements, with Borel functions in lieu of continuous functions, and every element of the C*-algebra has a left support and a right support. Even more is true: every self-normal element has an integral representation (by means of a spectral measure), which degenerates to a sum in the case of compact operators on Hilbert spaces. The entire theory takes on some of the features of integration theory in this case. For example, the above mentioned supports coincide in the case of self-normal operators. In the final part of this section, we discuss the hereditary C*-subalgebra and the simple C*-algebras.
In this section, E denotes a C*-algebra.
4.3.1 T h e E x t e r i o r Proposition
Multiplication
4.3.1.1
(x',p) c E ' x P r E ~
I]px'(1-
p)+
(1-
p)x'p[I < IIx'l 9
Take x E E # . By Proposition 4.1.2.23, Ilpx(1 - p) + (1 -
p)xpll = s u p { l l p x ( 1 - p) l, I1(1 - p)xpl[}~ 1.
Hence
I(x, px'(1 - p) + (1 - p ) x ' p ) I - I((1 - p)xp § px(1 - p),x')l <_ I1~'11, Ilpx'(1 - p) + (1 - p ) x ' p l I <_ IIx'll .
m
4.3 Supplementary Results on C*-Algebras
Proposition 4.3.1.2
209
Let F be an involutive E- mo d u l e and (x,a) C E x
R e F . If xa
--
ax
,
then
I x l a - alxl.
By Proposition 4.2.5.3, Ixl E E ( x ) . Thus the assertion follows from Proposition 2.3.6.3. m Proposition 4.3.1.3
( 0 ) Take (x, x') C E+ • E'+. If XX t --
Xt X
,
then
xx' e E ~ ,
Ilxx'll- x'(x).
By the Fuglede-Putnam Theorem and Proposition 2.3.6.4, 1
1
t
1
t
1
!
xx' = x ~ x ~ x = x~x x2 ~ E+ ,
iixx'll Proposition 4.3.1.4
1)
( 0 )
(x,x') ~ E+ x E~ ~
Ilxx'll ~ < :c'(x)l x'll Ilxll.
Given y c E ,
< x'(x)x'(y~y*) < x'(x)llx'll Ilxll Ilyll ~
(Proposition 2.3.4.6 c)) and so Ilxx'll ~ ~< x'(x)llx'll Ilxll 9 Corollary 4.3.1.5 valent:
a) x'(z) = o.
( 0 )
i
Given (x, x') C E+ x E'+, the following are equi-
210
4. C*-Algebras
b)
xz'=O.
c)
x'x=O.
d)
xx' x = O .
a =~ b follows from Proposition 4.3.1.4. b :=> c. x ' x = (xz')* = O.
c =~ d is trivial. d =~ a. If ~ is an approximate unit of E (Theorem 4.2.8.2), then x'(x 2) - l i m x ' ( x y x ) - l i m ( y , x x ' x ) = O. y,~
y,~
By Proposition 4.2.4.12,
z'(z) C o r o l l a r y 4.3.1.6
=o.
m
( 0 ) Let A be a n o n e m p t y downward directed set of E+
and ~ its lower section filter. Take x' C E'+. T h e n the following are equivalent:
a)
infx'(x) - O.
xEA
b) limxx' = O . x,~
c) l i m x ' x = 0 . x,~
a =~ b follows from Proposition 4.3.1.4 (and Corollary 4.2.1.18). b ~ c is trivial. c ~ a. By Theorem 4.2.8.2, E is quasiunital, so the assertion follows from Proposition 2.2.7.20. m D e f i n i t i o n 4.3.1.7
( 0 )
Given an E - m o d u l e
F, F
denotes the set of
a E F such that the m a p s E
>F~
x,
> xa~
E
>F,
x,
>ax
are order continuous. Proposition
4.3.1.8
E~ C
E'
Since E' is a vector subspace of E ' , the assertion follows from Corollary 4.3.1.6 a ~ b & c. m
4.3 Supplementary Results on C*-Algebras
Proposition
4.3.1.9
211
Let ~ be an approximate unit of E and F an (invo-
lutive) E - m o d u l e such that lim llxall - lim Ilaxll = Ilall
for ~w,-y a z F . Th~n k i~ an (in~olut~w) E-~ub~odul~ of F . Take a C F . If x E E , then
II(xa)yll = I]x(ay)II < Ilxl[ IlaYll,
Ily(ax) ll = II(ya)x I ~ [lyall Ilxll for every y C E . Thus the maps
E
>F,
y,
>(xa)y,
E
>F,
y,
>y(ax)
are order continuous. By Proposition 2.2.7.17, we may assume t h a t E and F are unital. Let x be an invertible element of E , A a downward directed set of E with infimum 0, and ~b the lower section filter of A. Then x * A x , xAx* are downward directed (Corollary 4.2.2.3) and have infimum 0 (Proposition 4.2.2.16). Given y C E ,
Ily(xa)l = I(x*)-l(x*yx)a]l
<_ l (x*)-~ l ll(x*yx)all,
[l(ax)yl[ = Ita(xyx*)(x*)-'ll <~ lla(xyx*)ll
II(x*)-~ll,
so that
lim Ily(xa)II = lira 11(ax)yll y,r
y,~
-
-
O.
Hence xa, ax E F . Since every element of E is the differnece of two invertible elements of E , /~ is an E - s u b m o d u l e of F . It is easy to see that if F is an involutive E - m o d u l e , then /~ is an involutive set of F . Proposition 4.3.1.10
m
Let E be unital, U the topological group of unitary
elements of E (Proposition 2.3.1.8, Theorem 2.2.~.1~), and .T the topological group of involutive isometries E r -+ E' (Example 2.2.2.~). Given x c U , define "~. E !
~E !,
xl t
) XXlX * .
212
4. C*-Algebras
a) I f x 9 U and u" E
>E ,
y~
>x * y x ,
then
b) 5 c)
5* 9 .T f o r every x 9 U .
The map
U
>~',
x,
>5
is a continuous group h o m o m o r p h i s m .
d) 5(~-0(E)) = To(E) f o r every x 9 U . e)
I f IK = ~ then { x e U I S = identity map} - {x e U I x ' 9 E' :=~ x x ' = x ' x } -
= {~ e U l~' e ~0(E) ~
zx'-
~'~}.
a) Take (y,x') E E • E'. Tken (~y, ~') = (~*yz, ~') = (y, x~'x*) = (y, ~ ' ) ,
so that u ' - 5. b) follows from a) and Proposition 4.1.1.24 (and Corollary 1.3.4.7, Proposition 2.3.2.22 d)). c) follows from a) and Proposition 4.1.1.24 (and Corollary 1.3.4.3 and 1.3.4.5). d) Take x'C T0(E) and y'E E+ with !
y <Sx'. Then x*y'x < x ~
(Proposition 2.3.6.4). By b) and Proposition 2.3.5.4 a ==> b, 9* y'~ = I1~*y'xllx' - Ily'll~',
4.3 Supplementary Results on C*-Algebras
213
and so
y'= Ily'llxx'x* -lly'll~x'. By b) and Proposition 2.3.5.4 b =~ a, 5x' e T0(E), so that ~(T0(E)) C t o ( E ) ) . By c), x(T0(E)) = T0(E). e) Take x E U with XXt--XtX
for ever x' E T0(E). Then ~X t =X
t
for every x' C ~'0(E). Since the map
Ek ---+ Ek,
z',
~ ~x'
is continuous, ~X t ---X i
for every x' C T(E) by Proposition 2.3.5.10 b). By Corollary 4.1.2.7 d), ~X t -~X t
for every x' E E ' . Hence { x e U I x ' C To(E) ==> x x ' -- x ' x } C {x C U I~ = identity map}.
The other inclusions are trivial. P r o p o s i t i o n 4.3.1.11
Take x', y' E E+' and p C Pr E . I f x' < py'p ,
then x' -- p x ' p .
Suppose that E is unital. Then 0 < (1 - p)x'(1 - p) < (1 - p)py'p(1 - p) -- 0 (Proposition 2.3.6.4),
214
4. C*-Algebras
(1 - p)x'(1 - p ) = 0 , (1 -
p)x'
=
x'(1
-
p) =
0
(Corollary 4.3.1.5 d :=> b & c), X ~ = p x t : x~p, x' = p x l p .
If, on the other hand, E has no unit, let ]K • E be the unital C*-algebra associated to E (Theorem 4.1.1.13). We use the notation from Proposition 4.1.2.27 (Theorem 4.2.8.2). By Proposition 4.1.2.27 e), j), O < _ X 'l _< (PY'P)x -- PY',P.
By the above considerations and Proposition 4.1.2.27 c),j), X '1
= px',p = ( p x ' p ) , ,
x' = Xlo' = (px'p)ao = p x ' p .
E x a m p l e 4.3.1.12
Take n C IN and let s
Example 4.1.1.7. Given a E s
I
n) be the C*-algebra defined in
'~) , define n
i,j=l
and identify C(]K n) with C(IKn) ' using the map
c ( ~ '~)
~ C(lKn) ',
a~, >'~
(Example 4.2.3.2). Then ax
--
ax
~
xa
--
xa
for all a, x E f~(IK n).
Given y E s
i,j= l
k=l
j,k= l
i--1
i,j-1
k=l
i,k= l
j--1
Thus a~'~ = ~x,
x~'h - x~.
I
4.3 Supplementary Results on C*-Algebras
215
4.3.2 O r d e r C o m p l e t e C * - A l g e b r a s Proposition
4.3.2.1 ( 0 )
Let S , T
be locally compact spaces, ~a" S ~ T
a continuous map, and x a Borel (Baire) function on T . Then x o ~ is a Borel (Baire) function on S .
Let U be an exact set of T. Then there is a y E C(T) such that
u : {y # 0 ) . It follows from --1
(u) = {y o ~ # o)
that ~ (U) is exact. We deduce that ~ ( A ) is a Borel (Baire) set of S whenever A is a Borel (Baire) set of T . Hence x o ~ is a Borel (Baire) function on S.
I T h e o r e m 4.3.2.2
( 0 )
Let E be an order a-complete (unital) Gelfand
C*-algebra. Take x E E . Let B be the C*-algebra of bounded Borel functions f
on a(x) such that
lim f ( a ) = 0
t~--+0
if E has no unit. ~v
a)
For each f C B , there is a unique f E E such that
{x' e
o ~(x') # f(x')}
is meager. Furthermore N
II/ll _< IIfi]-
~(f) c f(~(x)),
b)
The map B
>E,
is a homomorphism of involutive (unital) algebras such that for any sequence (f,~)nCIN in B+, E
t3
A/n= As . nEIN
nEIN
216
c)
4. C*-Algebras
Given e > O, there is a finite family ((cr~,p~))~e, in a(x) x P r E
such
that ~, A E I , ~ :/: A ==~ p~p~ = 0
eEI
d)
If f is a lower semicontinuous real function in B then fo~<_f.
a) Since the Gelfand transform on E is an isometry of C*-algebras (Corollary 4.1.2.5), Co(a(E)) is order a-complete. By Theorem 2.4.1.3 e) and Corollary 2.4.1.7 a), the map
o(E) is continuous and converges to 0 at infinity, if a(E) is not compact. By Proposition 4.3.2.1, f o ~ is a Baire function on a(E) converging to 0 at infinity, if a(E) is not compact. By Proposition 1.7.2.13 e =~ b, there is a unique g C C(a(E)) such that {f o ~ r g} is meager. Since a(E) is a Baire space,
g E Co(a(E)). Let f be an element of E with I - g- Then IIfll : r ( f ) <_ ]]fll (Theorem 4.1.1.16 b)) and
or(f) C f ( a ( E ) ) U {0} C f o ~ ( a ( E ) ) -
f(cr(x))
(Corollary 2.4.1.7 a), Proposition 1.7.2.13). b) It is easy to see that the map
B
>Co(a(E)),
f,
>f
is a homomorphism of involutive (unital) algebras. It follows that the map B-----+E,
f,
>f
is a homomorphism of involutive (unital) algebras. Put
4.3 Supplementary Results on C*-Algebras
217
B
/:= AA, nEIN
g'a(E)
>]K
x',
> inff.(x'). n E IN
From
{
nEIN
}I
/(~~~~
/) { ~ nEIN
}
it follows that
is meager (Proposition 1.7.2.13 d ~ c). Since a ( E ) is a Baire space,
I=AA. nEIN
Thus E
/=Afo ncIN
c) Let ~ be the a - a l g e b r a of all Borel sets B of a(x) such t h a t 0 ~ B and j = a(x) -~ IK the inclusion map. There is a finite family ((a~,At))te~ in a(x) • ~ with
~,)~ e l ,
~--/: A==~ At O A x - O ,
IIJ ~~
Given ~ c I , put PL " - - e A ~ .
By b), p~ E P r E
for every ~ C I . By a) and b)
~, A E I , ~ :fi A ::::::vp,p:~ = O,
x -- ~ tel
oL~pt
~llJ~~eA~~I
218
4. C*-Algebras
d) Take x' E or(E). Put
A:={fo~=?} and let ~ be the trace of the neighbourhood filter of x' on A. Since f o ~ is lower semicontinuous,
f(z') = limf(y') = lim f o ~(y') > f o ~(x'). y' ,~ y' ,~ -Hence
fo~f. Definition 4.3.2.3
( 0 )
m
E is called C-order complete, (C-order or-
complete) if either and E is order complete (order a-complete)
a)
IK = e
b)
IK = IR and E is order complete (order a-complete).
or
( 0 ) Assume that every x c S n E is contained in an order a-complete Gelfand C*-subalgebra of E (this holds whenever E is Corder a-complete). Then, given x E Sn E and ~ > O, there is a finite family ((c~,p~)),e , in or(x) x Pr E such that
C o r o l l a r y 4.3.2.4
If IK = e (IK = IR), then the vector subspace of E generated by Pr E is dense in E (in R e E ) . Let F be an order a-complete Gelfand C*-subalgebra of E containing x. By Theorem 4.3.2.2 c), there is a finite family ((c~,p,)),c t in a(x) • Pr F , such that
Obviously Pr F C Pr E . The last assertion follows. Suppose that E is e - o r d e r or-complete. By Corollary 4.1.2.3, every selfnormal element of E is contained in a maximal Gelfand C*-subalgebra of E and by Corollary 4.2.2.20, every maximal Gelfand C*-subalgebra of E is order or-complete, m
4.3 Supplementary Results on C*-Algebras
Corollary 4.3.2.5
219
( 0 ) Let E be (~-order a-complete. Take x C S n E .
Let j : a(x) --+ IK be the inclusion map and 13 the C*-algebra of bounded Borel functions f on a(x) such that
lim f(a) = 0
~---~0
whenever E is not unital.
a)
There is a unique homomorphism of involutive algebras u:13 --+ E such that uj = x and
so
A( Sn)
nEIN
nEIN
for every decreasing sequence (fn)ne~ in 13+.
b) u f = f ( x ) for every f C 13 M C(a(x)). Given f E 13, we define f ( x ) := u f .
c)
If f E 13 and if G is an order a-faithful C*-subalgebra of E (e.g. a maximal Gelfand C*-subalgebra of E ) containing x , then
Ilf(x)l[ ~ I[f[l~,
a(f(x)) : f(a(x)),
{x} ~ c {f(~)}~,
f(z) c c .
and
Cfo~ is meager whenever G is a Gelfand C*-algebra.
d) If there is a p C P r E such that x C p e p , then f (x) C p e p for every fc13.
If f C B and g is a bounded Borel function on f (a(x)) with
lim 9 ( a ) - - 0
o~--+0
whenever E is not unital, then 9(f(x)) = g o f(x)
220
f)
4. C*-Algebras
If f E 13 and { f 7~ 0} has n o n e m p t y interior with respect to a ( x ) , then f (x) 7~ O .
g) I f 0 is not an isolated point of a ( x ) , then there is a sequence (P,~),~eIN in {x}C M Pr E\{O} such that PmPn = 0 f o r all distinct m, n E IN and x = L
Xpn 9
n E IN
h)
I f IK =(~ and n c IN, then:
hi)
there is a bounded Borel function f e o(~) ~
h2)
on a ( x ) such that f(~)~ = ~.
f (x)n = x .
h3) x unitary ::v f ( x ) unitary. i) I f x C R e E ,
F is an E - m o d u l e , and a E F such that ax = 0 (resp.
xa = O) and such that the map E
)F,
y~
> ay (resp. ya)
is order a - c o n t i n u o u s then a f (x) = 0 (resp. f (x)a = O) for every f E 13 such that O c a(x) ==~ lim f (t) = O. t--+ O
a &: b & c & d. Let F be a maximal Gelfand C*-subalgebra of E containing x (and p in Case d)) (Corollary 4.1.2.3). By Corollary 4.2.2.20, F is an order a-faithful set of E and it is C-order a-complete. By Theorem 4.3.2.2 a),b), there is a homomorphism of involutive algebras u : 1 3 ~ F , such that u f = f(x) for every f E C(a(x)) N 13,
4.3 Supplementary Results on C*-Algebras
221
for every decreasing sequence (fn)~e~ in /3+, and or(uf ) c
f(a(x)),
for every f E/3. Let v :/3 --+ E be a homomorphism of involutive algebras with the property described in a), and uf--vfEG, .T':=
{x} ~ C { u f } c,
fEB G Gelfand ~
{ y__ G ~} f(x) =/= f o
is meager
Let (f~)nE~ be a decreasing (increasing) sequence in ~" with infimum (supremum) f in /3. Then uf = vf E G and
n E IN
by Proposition 4.2.2.17. Hence, by Proposition 1.7.2.13 e =v c, f E jr. Since c(o(~)) n u c y
(Theorem 4.1.3.1, Corollary 4.1.3.5), we get 9v = B. In order to prove f ( x ) E pEp in Case d), we may assume f real and [/] < 1. Then If(x)] ~ p and we get f (x) -- p f (x)p E p e p by Proposition 4.2.7.1 a :=~ c. e) We first remark that, by b) and c), g(f(x)) and g o f ( x ) are welldefined. Let /30 be the C*-algebra of bounded Borel functions g on f ( a ( x ) ) such that lim g(a) : 0
c~--+O
whenever E is not unital and let
222
4. C*-Algebras
I g(f(x)) = g o f ( x ) } .
G := {g 9 Since the inclusion map
f(a(x))
> IK
belongs to G and since the maps
/3o
~E,
g,
~g(f(x)),
/30
>E,
g,
>go f(x)
are homomorphisms of involutive algebras,
Uo c~c(o(f(~))) c ~. Let (gn)neIN be a decreasing sequence in G and let g be its infimum in /30. Then E
E
g(f(x)) = A
gn(f(x))-- A
n E IN
g" o f ( x ) = g
o f(x)
n E IN
by a), so that g 9 ~. It follows that /30 - - ~ 9
f) First assume that f is positive. There is a g 9 C(a(x))\{O) such that O_
g(x) < f (x) and by Theorem 4.1.3.1 b),
IIg(x)ll-Igll # o. Hence
g(~) r 0, so that
f (x) # O . Now let f be arbitrary. Put
4.3 Supplementary Results on C*-Algebras
>IK,
g: f(a(x))
oz,
223
>lc~l.
By e) and the above considerations,
g ( f ( z ) ) = g o f (x) =/: o. Hence
f (x) ~ o. g) We put r0 := Ilxll.
Since 0 is not an isolated point of a ( x ) , there is a decreasing sequence (r~)nE~ of real numbers with infimum 0 such that for every n E IN, a ( x ) n An ~: 0,
where An := {c~ E l K I r n < Ic~l < T n _ l } . Given n E IN, put
By a) and c), ( P n ) n ~ is a sequence in {x} c n Pr E such that PmPn = 0
for all distinct m, n E IN. By c), n
x - E
xpk
k=l
for every n E IN, so x -~ ~
xpn 9
n E IN
By f),
pn#O for every n E IN.
Tn
224
4. C*-Algebras
h) We set 1
f " a(x) where 0 9
>if2,
re i~ ,
9
>r - e ~
7r, 7r]. By e), f(x) n = x. If x is unitary, then by c) and Corollary
4.1.2.13, we get successively
f (x) 9 Un E. i) Put
Bo'-{geBIOca(x)==~
lim g(t)=O}
t ---~0 tea(~)
G = {g e Bo l ag(x) = O (resp. g(x)a = O) }. Let (gn)n~IN be a decreasing (increasing) sequence in G with infimum (supremum) g in Bo. By a), ag(x) = lim ag,(x) = 0 n----~ o o
(resp. g(x)a = lim gn(x)a = O) n----~OO
so that g C G. Since BoNC(a(x)) c G (Theorem 4.1.3.1 f)), it follows B0 C {7, i.e. f E G . Corollary
4.3.2.6
( 0 ) If E
is C-order a-complete then P r E
is upward
directed and its upper section filter is an approximate unit of E . 1 and ~q 1 Take p, q E Pr E . By Proposition 4.2.8.1, ~p haven an upper bound X
in E+# . Let r be the supremum of ( x } ~
rCPrE.
\
] nEIN
By Corollary 4.3.2.5 a),b),c),
From
1
~p _< r ,
1
~q _< r
(Corollary 4.2.7.2), we deduce p<_r,
q
(Corollary 4.2.7.6 h :=v a). Hence Pr E is upwards directed.
4.3 Supplementary Results on C*-Algebras
225
Take a E E and ~ > 0. By Theorem 4.2.8.2, there is an x E E+ such that I[x[[ < 1 and such that C
E
I i a y - all < 5' IlYa- all < 5"
y e E + , ilYll < 1 , ~ ___y ~ We may assume that
C
(1-
< 5
Let F be a maximal Gelfand C*-subalgebra of E containing x (Corollary 4.1.2.3). Define c~ - - inf { 1 - [Ix[I ~ } 2 ' 3(1 + flail) ' /,
f'a(x)
>IK,
a,
~~ 1 0
if a > if a < 5 ,
(
[
g- a(x)
>IK,
a,
>~ 0
(
if a > 5
a
ifa<_5,
and (Corollary 4.3.2.5 b))
p - - f (x), Then p e P r F ,
y := g(x).
[[y[[ < 5, and
+ y (Corollary 4.3.2.5 a),c)). Take q c Pr E with q > p. Then
x < Ilxllq + y,
I1 tlxllq + yll -_- I!~11 + Ilyll < 1,
so that C
Ila(llxllq + y) - all < 5 ,
II(llxllq + y)a - all < ~.
We get
C
s
C
[[qa- all ~ [[(lixil(q + y ) a - all + [[(1 -Ilxii)qa[I + Ilyall E"
C
E
C
Hence the upper section filter of Pr E is an approximate unit of E .
I
226
4. C*-Algebras
C o r o l l a r y 4.3.2.7
Let IK =qJ. Let E be unital and order o.-complete, x a
unitary element of E , and a E IR. Then there is a unique element y E R e E such that
~(y) c [~, ~ + 2~],
e, = ~,
~:(~)(y) = 0.
We have v = f(x), w h e re
f : o.(x)
eiO I
) 0 e
]0~, O/+ 271"].
In particular, the set of unitary elements is path-connected.
(Corollary 4.1.2.13 a =v b). Hence f is well-defined. Put y : - f (x) ,
g:[a,a+27r]
) q;,
~ ,
) e iz.
By Corollary 4.3.2.5 c), o.(y) = o . ( f ( x ) ) C f(o.(x)) C [a, a + 27r1.
Hence y C Re E (Corollary 4.1.2.13 b ::, a). Since E o.(x) ~
g o f(/3)=/3,
and e~ (y) o f - O, we get, by Collary 4.3.2.5 e), e ~y = g(y) = g ( f ( x ) ) = g o f ( x ) = x ,
4.3 Supplementary Results on C*-Algebras
227
We now prove the uniqueness. Take y E Re E with
~(y) c [ ~ , , + 2~],
r
= ~,
~:(~)(y) = o.
Given fl E [a, a + 27r], ( f o g - 27re~(Y))(/3) = / 3 , where g is the function defined above. By Corollary 4.3.2.5 e), y = (f o g-
27re~(V)(y) = f ( x ) .
27:e~(V))(y) = f ( g ( y ) ) -
The last assertion follows from the fact t h a t [0,1]
a,
>e ~iy
is a continuous path in the set of unitary elements of E connecting 1 to x. I C o r o l l a r y 4.3.2.8
Let IK = (~. Let E
be unital and order a-complete, x
an invertible element of E , and a E ]Ft. Then there are uniquely determined
y , z c R e E such that
~(y) c [ ~ , ~ + 2~],
~ = r
~:(~)(y) = o.
If, in addition, x is normal, then yz= zy,
x = e ~+iy,
z + iy= f (x),
where f:a(x)
>r
a,
;logaEIR•
By Theorem 4.2.6.9, Ix[ is invertible and there is a unique Y0 C U n E such that x = yolxl.
By Corollary 4.3.2.7, there is a unique element y C Re E such that
~(y) c [~, ~ + 2~],
~'~ = yo,
~2(~)(y) = o,
and by Proposition 4.2.4.5 a),b), there is a unique element z C Re E such that IX I =
ez .
228
4. C*-Algebras
If x is normal, then
y01xl = I~ly0 (Proposition 4.2.6.12). Thus yz = zy ,
x
:
e z+iy
(Corollary 4.3.2.5 c), Proposition 2.2.3.7 a)). By Corollary 4.3.2.5 e), x
--
e f(z)
and so z + iy = f ( x ) .
C o r o l l a r y 4.3.2.9
Let IK = ~ . I f E
1
is unital and order a-complete, then
the set of invertible elementes of E is path-connected.
Let x be an invertible element of E . By Corollary 4.3.2.8, there are y, z c R e E such that X
:
eZYe z .
The map [0,1]~E,
a,
>e i~ye ~z
is a continuous path in the set of invertible elements of E connecting 1 with
x.
m
L e m m a 4.3.2.10
( 0 )
Given n c IN, define 2n--1
An'-
2 k - 1 2k [ 2,~ '2,~ [ '
U k=l
1 ,o[0,1 ]
n j=l
a)
Given n, k C IN and a C [0,1] such that k-1 2 n
k
<~<
2n '
--
we have that
A(~)=
k-1
2-
4.3 Supplementary Results on C*-Algebras
b)
229
For every n E IN and a e [0,1], 1
o <_ ~ c)
f~(~) < 2 ~
(f~)ne~ is an increasing sequence of positive real functions on [0, 1] con-
verging uniformly to the identity map of [0, 1]. a) We prove the assertion by induction on n . The assertion holds for n = 1. Assume it holds up to n 2j-2 2n Then k c { 2 j - l , 2 j }
1 and take j c IN with j-1 2n-1 -
2n
and
2jL - - 1 (0~) -~-
If k = 2 j -
2j
j 2n_1
2
2n
1, then a ~ An and therefore 2j - 2
k-1 n
If k = 2 j , then a C An and therefore 1
fn (a)
2j-2
- - f n - 1 (0~) -+- 2 n - -
2n
1 Jr- 2---~ - -
2j-1 2n
k-1 2n I
b) and c) follw from a). C o r o l l a r y 4.3.2.11
( 0 ) Let E be ~-order a-complete and take x E E#+.
Given n E IN, put
u
2k[ ,2
n
,
kEIN
:(x)
p~ .-~o (x). (Corollary 4.3.2.5 b)). Then (Pn)nC~ is a sequence in P r E
and
is an increasing sequence in E+ converging to its supremum x . In particular, every element of Re E' is positive whenever it is positive on Pr E . The first assertion follows immediately from L e m m a 4.3.2.10 and Corollary 4.3.2.5 a),b). The final assertion follows from Corollary 4.2.2.10.
I
230
4. C*-Algebras
C o r o l l a r y 4.3.2.12
If E is (~-order e-complete and p C P r E \ { 0 } ,
then
the following are equivalent: a) p is a minimal element of P r E \ { 0 } . b)
If F is an order a-complete Gelfand C*-subalgebra of E containing p, then p is a minimal element of Pr F \ { 0 } .
c)
If F is a maximal Gelfand C*-subalgebra of E containing p, then p is a minimal element of Pr F \ { 0 } .
d)
If IK = r
(]K = IR) then pEp (p(Re E)p) is one-dimensional.
a =~ b is trivial.
b ==v c follows from Corollary 4.2.2.20. c ~ d. Take x C p(Re E ) p . Assume that a(x) contains at least two points. Let c~ be a point of IR lying between these two points. We define A := a ( x ) n ] - oc, c~[,
B := e(x) n [~, co[, :=
"=
By Corollary 4.3.2.5 a) ,b) ,c) ,d) (and Corollary 4.2.2.20),
XA, XB 9 (Pr F\{O}) N (pEp), X A --~ X B
-- p,
which contradicts the minimality of p. Hence e(x) contains exactly one point c~. We may assume that c~ -r 0. By Corollary 4.3.2.5 a) ,b) ,c) ,d) , o(~) 1 P = ~o(xl(X) = -(Yx , so that x = c~p and p e p (p(Re E)p) is thus one-dimensional. d ~ a. Take q E Pr E \ { 0 } with q < p. By Corollary 4.2.7.6 a =r f,
q = pqp C p(Re E ) p . By d),
q-p, hence p is a minimal element of Pr E \ { 0 } .
I
4.3 Supplementary Results on C*-Algebras
L e m m a 4.3.2.13
231
(Drewnowski, 1972) Let 3/t be a countable set of bounded
complex valued functions on 913(IN) such that for each # r # ( A U B) = #(A) + # ( B ) whenever A and B are disjoint subsets of IN. Then there is an infinite subset M of IN such that # ] ~ ( M ) is a measure whenever # E A/[. We first remark that
nEIN
for every # E Ad (Proposition 1.1.6.14 d => b). Given p C A/I, define ~'~(IN)
>r
A,
>#(A)-E#({n}). nEA
Let 91 be an uncountable set of infinite subsets of IN such that A N B is finite for distinct A, B E 92 (Lemma 1.1.2.17). Let Af be a finite subset of A4 and take s > 0. Let ~ be the set of A C 92 such that there are # C Af and B C A with
I~(B)I > c. Assume that ~ is infinite. Then there are a # C N ' , a sequence (An)net~ in 92 and a sequence (Bn)ne~ such that Am -J: An whenever m, n c IN are distinct and such that Bn C An and
I~(Bn)l > s for every n C IN. Given n c IN, put n--1
c ~ - - B~\ O B~. k=l
Then
(since ~ vanishes on the finite subsets of IN ), and so
I~(Cn)l = I~(C~)+ y'~ ~({k})l > c keC~
~ kEC~
Since (Cn)~e~ is a sequence of pairwise disjoint sets,
I~({k})l.
232
4. C*-Algebras
lira ~ ( C n ) -
n--->co
I~({k})l- 0
lira ~
n---~
kEC,~
(Proposition 1.1.6.14 d :=v b) and we deduce the contradiction O-lirn
[#(Cn)[ _> l i r n ( c -
E
[#({k})[) - c.
kECn
Hence if3 is finite. It follows that the set of those all A E 91 for which there are # E .hi and B C A with
~(B) ~ 0 is countable. Since 91 is uncountable, there is an M E 91 such that ~(B) = 0 for every # E .h/[ and B C M . It follows that # I ~ ( M ) is a measure for every
L e m - n a 4.3.2.14
(Nikodym's Boundedness Theorem, 1931) Let fit a a -
algebra of subsets of the set T and .M a set of bounded complex valued functions on fit such that #(AUB)-#(A)+#(B) whenever # E .A4 and A , B
are disjoint sets of fit. If {#(A) I # E All} is
bounded for each # E .h/t, then {#(A) [ # E .A4, A E fit} is bounded. Assume that {#(A) I # E M , A E fit} is not bounded. We use induction to construct a sequence (#n)nE~ in /%4 and a disjoint sequence (An)nEIN in fit such that for each n E IN: n-1
1) I,~(A~)I > n + E sup I,(A~)I, k - 1 #EAz[
\
k--1
/
Take n E ]iN and assume the sequences have been constructed up to the ( n - 1 ) s t term. By 2), there are Pn E 1~4 and B E fit such that .. \k----1
4.3 Supplementary Results on C*-Algebras
233
and
n-1 I#n(B)l > n + ~ sup I~(Ak)l + sup k=l
n-1 T\
/ Ak
k--1
then define
An:=B. Then 1) and 2) are trivially fulfilled. Otherwise, put
Then 2) is fulfilled and
I ( o-1 )
I#,,.(A.,)I = #,-,. T\ U Ak
- #,,.(B)
k---1
> I . ~ ( B ) I - "n
T\
Ak
:>
k=l
>_ I#~(B)l- sup ~ T\
A~
> n+
sup I~(Ak)l.
This completes the inductive construction. Note that ]#(A,~)I < oo
nEIN whenever # E M
(Proposition 1.1.6.14 d =~ b). Given n E IN, define
Then vn is bounded and
~,,~(AUB)=v,~(A)+vn(B)
234
4. C*-Algebras
for every n C IN and disjoint subsets A, B of IN. By Lemma 4.3.2.13, there is an infinite subset M of ]iN such that u n l ~ ( M ) is a measure whenever n E IN. There is a strictly increasing sequence (Pn),~er~ in M such t h a t oo
k:pn+l
whenever n c IN. Thus, by 1), n-1
_> Ipp,,(A,,,,)l- ~
]#p~(Av~)]-
k=l
> p,~ -I~,,o({p~ I k 9 ~ , k > n})l = oo
oo --
Pn
--
k=n+l
k=n+l
Hence sup
> sup
ttEM
nEIN
--
(X3
and this is a contradiction. T h e o r e m 4.3.2.15
Let IK = q~ (IK = IR). Let E be unital and (F-order
a-complete. If A is a subset of E' (of R e E ' ) for which {x'(p) l x' e A} is ! bounded whenever p C Pr E , then A is bounded. In particular, if (Xn),~eiN iS a ~ q ~ n c ~ in E' ~ c h that for ~a~h p e Pr E , ( ~ ' ( p ) ) ~ co~v~g~, th~n t h ~ is an x' C E' such that lim x',~(x) - x' (x) n--+ oo
lim x'~(x) = x ' ( r e x ) ) n---+ (x)
whenever x E E . By Banach-Steinhaus Theorem (Theorem 1.4.1.2), it is sufficient to prove that { x ' ( x ) ] x ' e A} is bounded for every x C E . Hence by Theorem 4.2.2.9 a), it suffices to show that { x ' ( x ) [ x ' e A} is bounded whenever x C E+# . So take x E E+# and let ~ be the a - a l g e b r a of Borel sets of a ( x ) . By hypothesis (and Corollary 4.3.2.5 a),b)) {x'(eB(x)) I x' e A} is bounded whenever B E ~ . By Nikodym's Boundedness Theorem (Lemma 4.3.2.14), c~ := suP{Ix'(es(X))l l x' C A , B ~ ~3} < c~.
4.3 Supplementary Results on C*-Algebras
235
Given n E IN, define
By Corollary 4.3.2.11,
converges to x. Since
k--1
k--1
whenever x ~ E A and n E IN, it follows
x,x, limlx
1
~A~(x)
~-~
)
<
k--1
whenever x' E A and therefore
sup I~'(~)1 < ~ .
x~EA
We now prove the last assertion. By the first assertion, (x~)nc~ is equicontinuous. Since the vector subspace of E generated by P r E is dense in E (in Re E) (Corollary 4.3.2.4) (X'n(X))ne~ converges for every z E E (x E Re E). Define x''E
)r
x,
!
) limxn(X )
(x''E
>lR,
x,
n----~o o
~ limx'(rex)). n----~(x)
Thus x' has the required properties (Proposition 1.2.1.7). Definition 4.3.2.16
I
( 0 ) Let T be a set, 9 a a-algebra on T , and E a
unital C*-algebra. A n E - v a l u e d spectral m e a s u r e on T. is a map
#:~s
~PrE
such that:
a) A, B E ~ , A N B = ~ ~ # ( d U B) = # ( A ) + # ( B ) , # ( A ) # ( B ) = O . b)
c)
,(T)=
1.
/f ( ( A n ) ~
is a decreasing sequence in ~ with empty intersection, then
0 is the infimum of (#(An)),c~ in E .
236
4. C*-Algebras
If # is a s p e c t r a l m e a s u r e on T , t h e n x ! o # is a m e a s u r e on %7 for every x! E E ~ . Proposition
4.3.2.17
( 0 ) Let T be a set, %7 a a-algebra on T , is %7-measurable},
U : = {x E e ~ ( T ) I x
E a unital C*-algebra, and # an E-valued spectral measure on %7. Then there is a unique continuous linear map B
~E,
x,
fxd,
such that eAd# = #(A) for every A E %7. This map is a homomorphism of involutive unital algebras and
for every decreasing sequence (x,~),,e~ in B+. I] x E 13, then a( f xd#) J
au(x)
f(/xd#)=
/(f
z(T)
and o x)d#
for any bounded Borel function on x ( T ) . Let jv be the set of step functions on T with respect to %7. ~" is a dense involutive s u b a l g e b r a of 13 a n d there is a u n i q u e linear m a p
u'J=
~E,
x,
~ f zdp, J
such t h a t
e m d # - #(A) for every A E ~s
4.3 Supplementary Results on C*-Algebras
237
Since the unital C*-subalgebra of E generated by #(~) is a Gelfand C*algebra (Corollary 4.1.2.3), we may assume that E is commutative. Take x E 9r . There is a finite family ((a~,A~))~e, in x(T) • ~ such that
t,A c 7s ~:/: A ~
A~MAx = 0 ,
tel
Now
II~xll = II ~
~eA~ II ~< sup I~1 - II~ll
tCI
tel
as can be seen using the Gelfand transform. Hence u is continuous and it can be extended uniquely to a continuous linear map
v'B
~ E,
x,
~ ]'xd#.
Take x, y E .~. There is a finite family ((at, fit, A~))~cl in IK • IK x ~2 such that ~,A C I , ~ - A ==~ A~MA)~ = 0 ,
tEI
LCI
Then
(ux)* -
a~p(A~)
atp(A~) = u
= tEI
= E t,AEI
-6~eA, = ux*, ~EI
a~/~p(At)p(A~)- E a ~ 3 t p ( A t ) - u ( E a t / ~ e A ~ ) --u(xy), tEI
~(~) = ,(T)=
tel
1.
Hence u is a homomorphism of involutive unital algebras. By continuity, v is an involutive unital algebra homomorphism.
238
4. C*-Algebras
To prove that
we may assume that
AXn--0.
nEIN
Take ~ > 0. Given n E IN, put A~ := {xn > c}. Then
f f 0 <_]xnd# <_] (CeT + ]lxllieA,,)d#
= ~1 + Iix, il#(An)
for every n E IN (Corollary 4.2.1.4). Since (An)ne~ is a decreasing sequence in with empty intersection,
A IIx~ll,(A~) - 0. nEIN
Any lower bound of
(fx~d#)
n E IN is therefore smaller than e l . Hence it must
be negative (Corollary 4.2.2.2). Thus
A [x d, = o. n E IN
J
We now prove the last assertion. The relation
is easy to see. Since the map
B
~E,
y,
~/yd#
is a homomorphism of involutive unital algebras
may extend this relation to any bounded Borel function f on ~B(x).
I
4.3 Supplementary Results on C*-Algebras
239
[
P r o p o s i t i o n 4.3.2.18 t 0 ) Let E be a unital C*-algebra. Take x E S n E . Let j" a(x) --+ IK be the inclusion map, ~ the a-algebra of Borel sets of a ( x ) , and
B := { f e e ~ 1 7 6
i~ a Borel function}.
Then any two E-valued spectral measures #, L, on ~ coincide whenever f jd# =
f jd..
Since the maps
f,
f >
if@,
13
>E,
B
>E, f, >/fd~,
are homomorphisms of involutive algebras (Proposition 4.3.2.17),
f fd.=
f ( x ) = f fdu
for all f e C(a(x)) (Theorem 4.1.3.1 a)). Put 9~ := {A e ~ }p(A) = u(A)}. Furthermore, let K be a compact set of a(x). There is a decreasing sequence (f~)~e~ in C(a(x))+ such that B
A A =e~ncIN
Thus
nEIN
nEIN
(Proposition 4.3.2.17). Hence ffl contains the compact sets of ~8. Since the union and the intersection of every monotone sequence in ffl belong to ffl, we see that
~-~ T h e o r e m 4.3.2.19 ( 0 ) Let E be C-order a-complete and unital. Take x E Sn E . Let j " a(x) --+ IK be the inclusion map, ~3 the a-algebra of Borel sets of a(x), and B the C*-algebra of bounded Borel functions on a(x).
2,40
4. C*-Algebras
There is a unique E-valued spectral measure # on ~ (called the spectral measure of x ) such that
a)
(the spectral decomposition o l x ) .
f jd#- x For each f r B,
b)
f f d# - f (x) -1
(Corollary 4.3.2.5 b)) and # o f is the spectral measure of f (x). c) ~ ( ~ ) c
d)
, ( ~ ) ~ = {x} ~
If U is a nonempty open set of a(x), then It(U) ~ O.
e) For each f E 13 and a E ]K, aCcr(f(x)) iff # ( 7
(U~(a))l ~ 0 for
\
/
every c > O. a) The uniqueness follows from Proposition 4.3.2.18. We set #:~3----+PrE,
A,
>CA(X).
By Corollary 4.3.2.5 a), # is a spectral measure and by Propositim. 4.3.2.17, the map
B~E,
f~
~/fd#
is a homomorphism of involutive unital algebras. Hence
for any decreasing sequence (f~)ne~ in /3+. Take c > 0. There is a finite family ((a~,A~))~e, in a(x)• f13 such that
~, A C I, t # A ~
I
j-
Then
A~AA~ =O,
II
a~eA, < -~.
4.3 Supplementary Results on C*-Algebras
241
Hence
iJ
d # - x.
b) The relation
i f d# - f (x) follows from a) and Corollary 4.3.2.5 a). The other assertion follows from
CTI(A)(X)--# (71(d)) ,
e A ( f ( x ) ) -~e, A o f ( x ) -
where A is a Borel set (Corollary 4.3.2.5 e)). c) By Corollary 4.3.2.5 c),
{x) ~ c By Corollary 4.1.4.2 b) and Corollary 4.2.2.19, p(~)cc is an order faithful C*subalgebra of E . Since it contains # ( ~ ) , it contains f(x) whenever f E B. In particular,
{x} c , ( B ) ~ ,
SO (Proposition 2.1.1.17 b),d),e)), and
,(~)
,(~)~ = (x) ~.
d) There is an f C C(o(x))+ such that f < e v and
Ilfll
= 1. Thus
0 < f(x) <_#(U). -1
e) By b), # o f is the spectral measure of f ( x ) . If a C a ( f ( x ) ) , then by
d),
242
4. C*-Algebras
()1 ( u y ( ~ ) ) ) = . o ,:
(u~(~))~
0
for every e > 0. If a ~ a ( f ( x ) ) , then there is an e > 0 such that
uy(~) n o(I(~))=0 Hence
. Definition 4.3.2.20 x E SnE
(u~(~))
:.
o : ( u y ( ~ ) ) : o.
m
( 0 ) Let E be C-order a-complete and unital. Take
and let # be the spectral measure of x . A n a t o m of x is an a E
a ( x ) , such that #({a}) ~ O. z is called a t o m l e s s if it has no atoms, z is called a t o m i c if for every Borel set A
of a ( x ) ,
#(A)
is the supremum of
(,({~}))~ C o r o l l a r y 4.3.2.21 and let f
Let E be C-order a-complete and unital. Take x E Sn E
be a bounded Borel function on a ( x ) . If x is atomic, then f ( x ) is
atomic too. -1
Let # be the spectral measure of x. By Theorem 4.3.2.19 b), # o f
is
the spectral measure of f ( x ) . Let A be a Borel set of IK. Since x is atomic, -1
-1
# ( f (A)) is the supremum of (#({a}))
_,
and # ( f (/3)) is the supremum
aE f (A) -1
of (p({a}))
_,
for any /3 E IK. Hence p o f (A) is the supremum of (# o
-1
f
({fl}))BEA
and f ( x ) is atomic.
P r o p o s i t i o n 4.3.2.22
I
Let E be C-order complete and unital. Take x E Sn E
and let p be the spectral measure of x . Then x is atomic iff 1 is the supremum -1
(l t ({a}))a~a(x) 9
Assume that 1 is the supremum of (~tl({a}))ae~(x). Let A be a Borel set -1
of a(x) and let y , z be the suprema of (it({a}))~eA and (ttl({a}))~E~(x)\A, respectively (Corollary 4.3.2.6). Then y __ it(A),
z <_ i t ( a ( x ) \ A )
and 1 = y + z <_ it(A) + i t ( a ( x ) \ A ) = it(a(x)) = 1.
Hence #(A) = y and x is atomic. The reverse implication is trivial.
I
4.3 Supplementary Results on C*-Algebras
243
4.3.3 T h e Carrier D e f i n i t i o n 4.3.3.1
( 0 ) Let F be an E - m o d u l e .
Take a 9 F
and put
(G:={x 9 A right (left) carrier of
a is an orthogonal p r o j e c t i o n p of E
ap = a , p G = { O} , A carrier of
such that
(pa = a , G p = { O} ) .
a is an orthogonal p r o j e c t i o n of E ,
which is at the s a m e t i m e a
right a n d a left carrier of a .
0 is a right (left) carrier of a iff a = 0. Let T be a locally compact space. Put E := Co(T) and F := Co(T) (resp. F := A4b(T)). An element a 9 F has a right carrier iff {a -r 0} (if Supp a) is relatively compact and open, in which case the characteristic function of this set is the carrier of a.
Proposition 4.3.3.2
( 0 )
Let F
be an E - m o d u l e .
Take a C F , p E P r E
and p u t
A:={qEPrE[aq=a
(resp. qa = a) } .
a)
qcA,
b)
I f p is a right (left) carrier of a , then it is the s m a l l e s t e l e m e n t of A .
c)
There is at m o s t one right (left) carrier of a .
d)
I f x, y C Re E such that Ix I _< ]Yl, a n d if p, q are the right carriers of x
rCPrE,
q<_r=~rCA.
and y , respectively, then p <_ q.
a) We have ar = aqr = aq = a
(ra = rqa = qa = a)
(Corollary 4.2.7.6 a =v c (resp. a :=~ d)), so r E A. b) If q E A, then a(q - p) = aq - ap = a - a = O
O -= p(q - p) = pq - p
( (q - p ) a = qa - p a = a - a -- O) ,
(O = (q - p ) p = qp - p) ,
244
4. C*-Algebras
Pq=P
(qP=P)
p
by Corollary 4.2.7.6 c :=~ a
(resp. d =~ a).
c) follows from b). d) We may assume y C E # . By Proposition 4.2.7.1 f :=v a, ]Yl < q, so that Ixl _ q. By Proposition 4.2.7.1 a =~ f, xq = x . By b), p < q. Proposition 4.3.3.3
( 0 )
m
Let F be an involutive E - m o d u l e . Take a C F
and let p be the right (left) carrier of a. Then p is the left (right) carrier of a*.
We have pa* = (ap)* = a *
(a*p= (pa)* -- a*).
Take x c E with xa* -- O
(a*x = O).
ax*=0
(x*a = 0),
pz*=o
(x*p = o),
Then
so that
xp = 0
(p~ = 0).
Hence p is the left (right) carrier of a*. Remark.
m
This proposition allows us to deduce the properties of the left carrier
from those of the right carrier. Proposition
4.3.3.4
Let F
be an E - m o d u l e .
right carrier of a. Take x C E with ax = x a .
a)
px = p x p .
Take a C F . Let p be the
4.3 S u p p l e m e n t a r y Results on C * - A l g e b r a s
b)
If F
is an i n v o l u t i v e E - m o d u l e
a E Re F ,
and
245
then
px = xp.
c)
If F = E
aEReE,
and
then
{a} c
{p}C.
a) We have a(xp - px) = axp - apx = xap - ax = xa - ax
= O,
so that 0 = p(xp - px) = pxp - px,
px = pxp.
b) and c). We have ax* = x * a .
Hence by a), px*=px*p.
Thus (x;)*
= p~* -
;x*p
= (;xp)*
-
(;~)* ,
x p -- p x .
P r o p o s i t i o n 4.3.3.5 Take a E F
a) {x E b)
If F-
( 0 )
Let E
E lax E
= O} = ( 1 - p ) E . a n d a is i n v e r t i b l e t h e n p - 1 .
a) Take x E E with ax=O.
Then
be u n i t a l a n d F
a n d let p be the r i g h t c a r r i e r o f a .
I a unital
E-module.
24 6
4. C*-Algebras
x = (1 - p ) x + p x = (1 - p ) x E (1 -- p ) E . Hence 0} C ( 1 - p ) E .
{x E E [ a x -
Take y E E . T h e n a(1 - p)y -- (a - ap)y = O, so t h a t (1 - p ) E
C {x E E l a x = 0}.
b) follows from p = a - l a p - a - l a - 1.
Proposition
4.3.3.6
a unital E - m o d u l e .
Suppose E
I
is unital and q~-order complete.
Let
F
be
Then every element of F has both a right carrier and a
left carrier.
Take a E F . P u t
A " : {p E P r E l a p = a} .
Then E
1 E A . Take p , q E A . T h e n
((pqp)n)ne~
is a decreasing sequence in
(Corollary 4.2.2.5) and its infimum r in E+ belongs to P r E
and is the
infimum of p and q in E+ (Corollary 4.2.7.13). We have ar = lim a(pqp) n
= a.
n ---+ o o
Hence r E A and A is downward directed. Let ~ be the lower section filter of A and p the infimum of A in E + . By Corollary 4.2.7.10, p E Pr E . Thus ap = lim a q q,15
Hence p is the smallest element of A . Take x E E # with ax=O.
Then
a.
4.3 Supplementary Results on C*-Algebras
a(zx*)
-
0 =
247
Oa.
By the F u g l e d e - P u t n a m Theorem a(xx*) ! = O~a = 0
for every n E IN. ( ( x x * ) ! ) , ~
is an increasing sequence in E+# (Corollary
4.2.7.2). Let q be its supremum. By Proposition 4.2.4.2 f), q E Pr E . Thus
aq = lim a(xx*) ! = 0, n---+oo
so t h a t
a(1 - q) = a - aq = a,
1-qEA,
p~l-q,
xx* < q ~ _ l - p ,
x x * (1 - p) = x ~ *
(Proposition 4.2.7.1 d => f),
xx*p -- 0,
x'p= 0 (Remark a) of Definition 4.1.1.1),
px=O. Hence p is the right carrier of a. The proof of the existence of a left carrier of a is similar.
Proposition 4.3.3.7 then p C {x'x} cc .
( 0 )
Take x C E . I f p
is the right carrier of x ,
248
4. C*-Algebras
Take y c {x*x}C. T h e n
x * x ( p y - yp) = x * x p y - x * x y p - - x * x y - yx*xp = yx*x - yx*x = 0,
x(py
-
yp) -
o
(Remark a) of Definition 4.1.1.1), o = p(py
-
yp) = py - pyp,
py -- pyp. Since y* E {x'x} ~, we deduce that (yp)* = p~* = py*p = (pyp)*,
yp = pyp = py ,
Theorem 4.3.3.8
( 0 )
Take (x,p) E E x P r E .
Then the following are
equivalent: a)
b)
p is the right carrier of x . (resp. c)) p e { x ' x } cc and if F is a (maximal) Gelfand C*-subalgebra of E containing { x ' x , p} , then F
= e~ r) ,
where F
A.-
{x~xx#0}. F
a =:v b. By Proposition 4.3.3.7, p C {x'x} co. It is obvious that ~ = 1 on A. Take f E Co(a(F)) with f - 0 on A. Then there is a y C F with F
~=f.
We see successively that
4.3 Supplementary Results on C*-Algebras
A
F
A
249
F
x * x y = x * x f -- O,
x * x y -- O,
xy-O
(Remark a) of Definition 4.1.1.1), O,
py=
F
F
~f = ~ = o . F
Since f is arbitrary, ~ = 0 on a ( F ) \ A .
Hence
F =
b ==> c is trivial. c =~ a. Since { x * x , p }
is commutative, there is a maximal Gelfand C*-
subalgebra F of E containing { x * x , p }
(Corollary 4.1.2.3). We have successi-
vely that F
F
F
F
x* x p = x* x ~ = x* x ,
x*xp - x*x,
xp=
x
(Remark c) of Definition 4.1.1.1). Take y E E with x y - O.
Then x * x y y * = 0,
y y * x * x -- O,
250
4. C*-Algebras
yy* p -- p y y * .
Hence there is a maximal Gelfand C*-subalgebra F of E containing {x*x, yy*,p} (Corollary 4.1.2.3). We have that F
F
F
yy* x* x = yy*x* x = 0 h
F
and so yy* vanishes on A. It follows that A
F
F
F
pyy, = ~&',
- 0,
pyy* = 0 ,
py = 0
(Remark b) of Definition 4.1.1.1). Thus p is the right carrier of x.
Corollary 4.3.3.9
( 0 )
A self-normal element of a C*-algebra possesses
a right carrier iff it possesses a left carrier and in this case the two carriers coincide.
This follows immediately from Theorem 4.3.3.8.
Corollary 4.3.3.10
( 0 ) Given ~ > 0 and ( x , p ) E E x P r E ,
m the following
are equivalent:
a) p is the right carrier of x . b)
p is the carrier of Ixl ~
c) p is the left carrier of x*. a r
c follows from Proposition 4.3.3.3.
a ~=~ b. By the Remark a) of Definition 4.1.1.1, xy - 0 ~
x*xy - 0
for every y C E . It follows that p is the right carrier of x iff it is the carrier of x * x . Take ~,'7 e IR+\{0}. By Proposition 4.2.4.2 b),
4.3 Supplementary Results on C*-Algebras
E( x ~) = E(Ix ~),
251
{IxlO) ~ - {Ixl~} ~.
If F is a Gelfand C*-subalgebra of E containing Ix[ z and p, then it contains [zl~ and A
F
A
F
{l~l e # 0} = {Ixl, # 0}. By Theorem 4.3.3.8, p is the carrier of x*x iff it is the carrier of {xl~ . C o r o l l a r y 4.3.3.11
I
Take x C E and let
( 0 )
~(x*x) f "= eo(x,x)\{0} . If f
is continuous or if E is unital and G-order cT-complete, then f ( x * x )
(Corollary ~.3.2.5 b)) is the right carrier of x . We have
f ( x * x ) E Pr E and {x'x} c C { f ( x * x ) } ~ (Corollary 4.3.2.5 a),c)). Thus
f ( x * x ) e { f ( x * x ) } cc C {x'x} c~ Let F be a maximal Gelfand C*-subalgebra of E containing {x'x} ~ (Corollaries 4.1.2.3 and 4.1.4.2 c)). Then f ( x * x ) e F and
f (x*x) :/: f o x%-'x is meager (Corollary 4.3.2.5 c)). Put A'=
x*xr
.
Then A
F
f o x*x -It follows that F
By Theorem 4.3.3.8 c =a a, f ( x * x ) is the right carrier of x.
I
252
4. C*-Algebras
C o r o l l a r y 4.3.3.12
(
0
\
)
Let E be unital and C-order a-complete, F
an order a-faithful unital C*-subalgebra of E , and take x C F .
Then the
left (right) carrier of x with respect to E belongs to F and is the left (right) carrier of x with respect to F . Define ~(~'~) f "= %(~-x)\{0} 9 By Corollary 4.3.3.11, f ( x * x ) is the right carrier with respect to E and by Corollary 4.3.2.5 c), it belongs to F . C o r o l l a r y 4.3.3.13
I
Assume that E is unital and r
a-complete. If x
is the supremum of a subset A of E+ , then the carrier of x is the supremum in Pr E of the carriers of the element of A . We may assume that x E E # . Given y E E+# , let py denote its carrier (Corollary 4.3.3.11). By Proposition 4.2.7.1 d r
e,
y
qx - x ,
Px~_q (Proposition 4.3.3.2 b)). Hence Px is the supremum of (Py)~A.
I
4.3 Supplementary Results on C*-Algebras
C o r o l l a r y 4.3.3.14
253
( 0 ) Let E be unital and q~-order a - c o m p l e t e . Take
x 9 Sn E . Let p be the carrier of x , f a bounded B o r e l f u n c t i o n on a ( x ) such that 0 9 a(x) ~
f(O) = O,
and q the carrier of f ( x ) . P u t 1K
g := e~\{0 } . Then g o f ( x ) = q <- p -~ g ( x ) .
Let F be a maximal Gelfand C*-subalgebra of E containing x (Corollary 4.1.2.3 d)). Then f ( x ) , g(x), g o f ( x ) 9 F and the sets
ego
,
goI(x) #gofo
are me~.ger (Corollary 4.3.2.5 c)). Since F
F
g o f o "2<_ g ~ "2,
we see that F
A
F
g f(~) < g(x),
g o f ( x ) <- g ( x ) .
By Corollary 4.3.3.11, m
q -- g ( f ( x ) * f ( x ) ) -- g ( f ( x ) ) < g(x) = g(x*x) = p .
P r o p o s i t i o n 4.3.3.15
Let F
be an involutive E - m o d u l e ,
a,b E R e F ,
and
p, q the carriers of a and b, respectively. I f pq = 0 then p + q is the carrier of a + b. I f x C R e E
and p , q
are the carriers of x + and x - ,
then pq = 0 and p + q is the carrier of x .
By Corollary 4.2.7.5, p + q C Pr E . From aq = apq = O ,
bp = bqp = O
respectively,
253
3. C*-Algebras
it follows (a + b)(p + q) = ap + bq = a + b.
Let y E E
with (a + b)y = O.
Then ay = p ( a + b)y = O.
It follows py=
0,
qy=O,
(p + q)y = O.
Hence p + q is the carrier of a + b. In order to prove the final assertion, we remark that from X+X -
=0
it follows px- =0,
pq = O.
I
By the first assertion, p + q is the carrier of x. Proposition 4.3.3.16
We a s s u m e
lK = ~
complete. Take x C E # 9 If _( ( x ' x ) - )1 _ , ~ m u m p in E ,
or IK :
(resp.
_
then p is the right (left) carrier of x .
By Proposition 4.2.4.2 b),
for every n E IN. Hence
IR and E
order a -
~-) new ) h a s
a supre-
4.3 Supplementary Results on C*-Algebras
255
by Proposition 4.2.2.17. Let F be a Gelfand C*-subalgebra of E , containing {x'x, p}. It is easy to see t h a t F
~--
e~ F) ,
where
A'=
x*x#O
.
By Theorem 4.3.3.8 b =~ a, p is the right carrier of x. Example 4.3.3.17
Let
7
~
be a non-zew, non-invertible element of the C*-algebra s
2) (Example
,4.1.1. '7), i.e. a5-/~7
= o.
(
OL
tg~ =
5 m
Ot
and 0 E IR with
_ =l lei0 OL
OL
e iO cos ~ sin ~ ]
cos2
sin 2
e -i~ cos ~ sin
J
is the left (right) carrier of
7 Use Example 4.1.2.19.
5 II
256
4. C*-Algebras
Example 4.3.3.18
Take n 9 IN and let s
Example 4.1.1.7. Given a 9 s
~) be the C*-algebra defined in
, define n
~ Z~(lZ~)
~IK,
z,
~ ~_~zijaj~ i,j=l
and identify s
n) with s s
' using the map n)
~s
',
a,
(Example ~.2.3.2). Then, for every a 9 s
)~d the left (right) carrier of a
and the left (right) carrier of "5 coincide.
The assertion follows immediately from Example 4.3.1.12. Proposition 4.3.3.19
( 0 )
I
Take x 9 E . If x*x 9 P r E (xx* 9 P r E ) ,
then x*x is the right (xx* is the left) carrier of x .
Now XX*X
--- X
(Corollary 4.1.2.22 a =v c, b =v c) and for each y 9 E, xy = 0 ::::=vx*xy = 0, yx - 0 ====vyxx* - O.
T h e o r e m 4.3.3.20
( 0 ) (Kadison, 1951) Take x 9 E # and put ?.t "--- X ' X ,
V "-- XX*
Then the following are equivalent:
a) x is an extreme point of E # b) y 9 o# c) (x, 0) is an extreme point of E If these equivalent conditions apply, then:
d) u, v C P r E ,
x=xu-vx.
I
4.3 S u p p l e m e n t a r y Results on C * - A l g e b r a s
e)
257
u a n d v are the right a n d the left c a r r i e r o f x , respectively,
E
is unital,
and uv = vu,
f)
l = u + v - uv.
x is an e x t r e m e p o i n t of E ( x )
and 1 e E(x).
We have 9 ~-
~*~-
vx,
ll~ll- il~*xll- ll~ll: <_ 1.
a ::~ d. Define Z :'--U
1 3
Then
IIzll = Ilu ll
,lu,l
1
(Proposition 4.2.4.2 b)), so that
~(z)
[0,1]
(Corollary 4.2.1.16 a :=v c). By Corollary 4.2.4.14, there is some y c E with x--yz,
Ilyll _< Iillli u~-~
-
~ I1'11
-Ilull ~' <__]
(Proposition 4.2.4.2 b)). Since
Ilxzll ~ Ilxll Ilzl ~ ~, x z C E # . Define f "o'(z)
> IK,
~,
> 2o~-ol 2
Then
112z- z ~ l l - IIf(z)ll- Ilfll ~ 1 (Theorem 4.1.3.1 b)), so that
Ix(2- 1
-
z)ll - Ilyz(2 91
-
x(2-1-z)
z)ll ~ Ilyl1112z- z~ll ~ EE #
1,
258
~,. C * - A l g e b r a s
We have
X"-~1 ( x ( 2 - 1
- z) + xz)
Since x is an extreme point of E # , it follows successively t h a t X--XZ~
1
it
=
X *X
u E Pr E
vCPrE,
--
X *X Z
--- itit-3
4
-~- U -~
(Corollary 4.2.4.3 a ~ c),
x-xu-vx
(Corollary4.1.2.22).
a ::a b. We may assume t h a t Ilyll <_ 1. P u t z : = (1 - v ) u ( 1 - u ) .
Then
(Corollary 4.2.1.17 a =~ c). We show that x + z C E # I1~ + zll ~ - II(x* + z * ) ( x + z)ll = I1~*~ + ~*z • ~*x + z*zll.
By d), x * z = x'v(1 - v)y(1 - u) - 0,
z*x = ( x ' z ) * - 0 ,
x * x z * z = x*x~(1 - ~ ) y * ( l - ~)z = 0 ,
z * z x * x - z*(1 - v ) y ( 1 - u ) u x * x
- O,
so t h a t I1~ + zll ~ - I I z * x
+ z*zll.
The C * - s u b a l g e b r a of E generated by { x ' x , z ' z } being a Gelfand C*-algebra (Corollary 4.1.2.3), it follows that
4.3 Supplementary Results on C*-Algebras
259
x • zl ~ = IIx*~ § z*zll - sup{I x*xll, IIz*zll} ~ 1, by the Gelfand t r a n s f o r m (Corollary 4.1.2.5). So x 9 z E E # . Since x is an extreme point of E # and x = ~1 ( ~ + z) + l ( x -
z),
we deduce t h a t x -x+z,
z-0. b ~ d. We have successively t h a t 0 - x*(1 - v)x(1 - u) - x*(x - vx)(1 - u) =
= x*(x
xu)(1
u) = x ' x ( 1
a ( u ) C {0, 1}
u E PrE
u) 2 = u(1
u) 2 ,
(Theorem 2.1.3.4 a ) ) ,
(Proposition 4.1.2.21 c ~ a ) ,
v E Pr E , x -
xu-
vx
(Corollary 4.1.2.22).
b =~ a. Take y , z E E # and c~,/~ El0, 1[ with c~+~-l,
x-ay+~z.
Then Iiy*yli-ilyll 2-<1,
IIz * z i i = i i z i I 2 < 1
and so y*y~
1,
z*z~_ 1
(Corollary 4.2.1.16 a => b). It follows by d) t h a t - ~ = ~x*~
= ~(~u* + Zz*)(~u + Zz)~ =
= u(c~2y *y + c~(y* z + z* y) + ~2z* z ) u ~ u(c~2y *y + c ~ ( y * y + z* z) + ~2z* z ) u ~
260
4. C*-Algebras
+
+
+/~21)u =
+ ~)21u = u 2 = u
(Corollary 4.2.2.4, Corollary 4.2.2.3). Thus u(y* z + z* y)~, = ~ ( y * y + z* z ) u ,
II(y - z)~ll ~ = II~(y* - z*)(y - z ) ~ l l -
II~(y*y + z * z - y * z - z*y)~ll = o,
(y - z ) u = O.
A similar a r g u m e n t shows t h a t v(y-
z) = o .
We conclude t h a t y-
z -- (1 - v ) ( y -
z)(1 - u) = O,
x--y=z.
b ::~ c follows from b :~ a. c =v a is trivial. b & d :=~ e . . B y
d) and Proposition 4.3.3.19, u and v are the right and
the left carrier of x , respectively. Let ~ be an approximate unit of E ( T h e o r e m 4.2.8.2). By b), y = vy + uy-
vyu
for every y C E , so t h a t lim
~-
v + u -
vu.
n - - ~ 0(:)
By Proposition 2.2.7.25 a), E is unital and 1 =u+v-vu.
By d), u v - (vu)* - u* -t- v* - 1" - u + v - 1 - v u .
f) It is obvious that x is an extreme point of E ( x ) . Corollary 4.3.3.21
( 0 )
If E
f o r e v e r y x C N o E (x C R e E ) "
By e), 1 C E ( x ) .
m
is u n i t a l t h e n the f o l l o w i n g are equivalent
4.3 Supplementary Results on C*-Algebras
a)
x is an extreme point o r e # (of R e E # ) .
b)
x is an extreme point of E ( x ) # (of Re E ( x ) # ).
c)
x E UnE.
261
a =v b is trivial. b ~ c. P u t U :~
X*X
=
XX*
.
By Theorem 4.3.3.20 a =~ b & d & e, 1-u=(1-u)
e=(1-u)l(1-u)=0,
c =~ a follows from T h e o r e m 4.3.3.20 b =~ a. Corollary 4.3.3.22
If E is unital, then the following are equivalent for every
xcReE: a)
x is an extreme point of E#+ .
b)
1 - 2x is an extreme point of R e E #
c)
x is an extreme point of E ( x 1) #-t-
d)
1-2xEUnE.
e)
xCPrE. a r
"
b. By Corollary 4.2.1.16 a w b, the map
R~E#
~E+~,
E+#
)ReE #,
1
~, ~ ( 1 - ~ )
is bijective. T h e m a p xk
>l-2x
is its inverse. a ::v c is trivial. c => d. By a ~ b, 1 - 2x is an extreme point of Re E ( x , 1) # . By Corollary 4.3.3.21 a :==>e, 1 - 2x C Un E . d r
e follows from P r o p o s i t i o n 4.1.2.25.
d :=v b follows from Corollary 4.3.3.21 c ==v a.
I
262
4. C*-Algebras
Example 4.3.3.23
Let n C IN and E E {IR,C, IH}. Then Un En,n is the set
of extreme elements of (En,n) # Let u be an extreme element of (En,n) # U :--- X ' X ,
[Ctij]i,jElNn : = 1 -
u,
Define
?3 : ~ XX* ,
[~ij]i,jEINn : = 1 -
v.
Take p, q C INn and put y ::
[~ip~jq]i,jEINn.
Then
By Theorem 4.3.3.20 a =~ b, 0=(1-v)y(1-u)=
~ ~iVSkaakj]
----[/~ipO~qj]i,jEiNn .
i,jEINn
k=l
Hence ~ipOlqj :
0
for all i, j, p, q E INn. Assume flip -r 0 for some i, p C INn. Then Olqj -- 0 for all
q,j C INn, i.e. u = 1. By Proposition 2.1.2.13, v = 1, so that x ~ Un En,n. If flip = 0 for all i,p C INn, then v = 1 and, as above, u = 1 and x C UnEn,n. The converse implication follows from Theorem 4.3.3.20 b =~ a.
Remark.
I
The left and the right shift of t~2 are not unitary elements of s
but they are extreme points of/2(t~2) # (Example 5.3.1.20). This shows that not all C*-algebras have the above property of En,n. It will be shown (Remark of Corollary 6.3.6.5), that for every finite-dimensional C*-algebra E , the set of extreme points of E # .
UnE
is
4.3 Supplementary Results on C*-Algebras
263
4.3.4 H e r e d i t a r y C * - S u b a l g e b r a s Definition 4.3.4.1
( 0 ) A subset F of E is called h e r e d i t a r y if for every
x E E+ , if x ~_ y for some y c F , then x E F .
{0} and E are hereditary C*-subalgebras of E . The intersection of any nonempty family of hereditary C*-subalgebras of E is hereditary. I f A is a subset of E , then the smallest hereditary C*-subalgebra of E containing A is called the hereditary C*-subalgebra generated by A .
If E is order complete (order a-complete) then every hereditary C*subalgebra of E is order complete (order a-complete). E x a m p l e 4.3.4.2
( 0 ) Given p E P r E , p e p is a hereditary C*-subalgebra
orE.
Take x E E+ and y C E with x < pyp.
Then we get successively that 0 <_ (1 - p)x(1 - p) <_ (1 - p)pyp(1 - p) = 0
(Corollary 4.2.2.3), IIx 89
1
x~(1 - p ) = 0 ,
x(1 - p) - 0,
x - xp - p x , x - pxp C p E p .
P r o p o s i t i o n 4.3.4.3 ditary.
I
A convex subcone of E+ is a face of E+ iff it is here-
264
4. C*-Algebras
Let F be a convex subcone of E + . First suppose that F is a face and take (x, y) E E+ x F with x ~ y. Then x,y-xEE+,
x+(y-x)=ycF.
Since F is a face of E + , x C F . Hence F is hereditary. Now suppose that F is hereditary and take x, y C E+ with x + y c F . It follows from
x~x+y,
y~_x+y
that x , y C F . Hence F is a face of E+.
I
If F is a hereditary C*-subalgebra of E+ , then a convex
C o r o l l a r y 4.3.4.4
subcone of F+ is a face of F+ iff it is a face of E+. In particular, given pE+p is a face of E+. 1
pCPrE,
( 0 ) Let ~ be the set of closed left ideals and 05 the
P r o p o s i t i o n 4.3.4.5
set of hereditary C*-subalgebras of E . a)
Given F C ~, F
F* C 05. Define ~ '~
b)
>05,
>F n F * .
Given G e ~), {x 9 E Ix*x e G} 9 ~. Define r
c)
Given h , F 2 G ~,
d)
Given F e ~, r
~iY,
G,
>{ z e E l x * z e G } .
F . Given G e 05,
R~(r e)
F,
c G c ~(r
Every closed ideal of E is a hereditary C*-subalgebra of E . a) It is obvious that F N F* is a C*-subalgebra of E . Take
(z, y) e E+ • (F n F*) with x < y. By Proposition 4.2.6.1, there is a sequence (x,~),~c~ in F+# such that
4.3 Supplementary Results on C*-Algebras
lim y x n
=
265
y.
n--+ oo
It follows successively that 0 _< (1 - x . ) x ( 1 - x . ) _< (1 - x.)y(1 - x . ) (Corollary 4.2.2.3), x-~ - x ~ x ,
=
- x,~)
x 89
=
I1(~ -
x,)x(1 -
~.)ll
_
(Corollaries 4.2.1.17 a =~ c, 4.2.1.18), 1
1
x~-
limx~xnEF, n--~ oo
1
1
x = x~x~ E F,
x E FVIF*,
FNF*
b) Take x, Y C r
C r
Then
(x + y)*(x + v) = ~*x + v*y + x*v + u*x <_ 2(x*~ + u'u) c a (Corollary 4.2.2.4). Hence x + y e r
and r
is thus a vector subspace
of E . Take (x, y) e E x r
Then
(xy)* (xy) = y'x* xy _< II~ll~y*y (Corollaries 4.2.1.18, 4.2.2.3). Thus x y C r
and so r
is a left ideal of
E . Since the map E .... ) E ,
is continuous, r
is closed. Thus r
x~
) x*x
c ~.
c) Assume that 9~(F1)+ C F2 and take x C F1. By Proposition 4.2.6.1, there is a sequence (xn)ne~ in F1 N E+# such t h a t
266
4. C*-Algebras
lim XXn
x.
n---+ (x)
Now, given n
IN,
x~ ~ ~(F~)+
F2.
Thus x x , C F2
for every n C IN and x
=
lim xxn C F2, n----~ o o
F~ c F2. The other implications are trivial. d) Take x e G. Then xx*, x*x E G, and so
9 x e ~(r
Take y E ~(r Re~(r
Then y2 C G and so y c G. Since every element of
is the linear combination of two elements of ~(r
(Theorem
4.2.2.9 a)), it follows that
By the above considerations, Re~(r
C
~(F)
C
~(r
Thus, by c), r
C
Cr
~(~(F)) = P.
e) By Corollary 4.2.6.2, every closed ideal of E is involutive. The assertion thus follows from a). II
4.3 Supplementary Results on C*-Algebras
P r o p o s i t i o n 4.3.4.6
267
( 0 ) I f F is a C*-subalgebra of E , then the following
are equivalent:
a)
F is hereditary.
b)
x c E + , y c F+ =:v y x y C F ,
I f IK = ~ , then the above assertions are equivalent to:
c)
Given x C E and y, z C F , yxz EF.
a :=~ b & c. By Proposition 4.3.4.5 b),d), there is a left ideal G of E such that Re (GN G*) c F C G N G * . We have that y*, z C G and so yxzcG,
(yxz)* - z* x* y* C G .
If x E E + , y E E +
then y x y C Re (G N G*) C F .
If IK = ~ , then yxz CGAG*-
F.
b :=> a. Take x C E+ and y E F with x < y. Take z E F+. Then 0_< ( 1 - z ) x ( 1 - z ) _ _
(1-z)y(1-z)
(Corollary 4.2.2.3), so that, by Corollary 4.2.1.18,
IIx~l z~ll~- al/1-z/x/l
z/ll ~ la/l z/~l-z~ll
= I1~1
z~ll~
Let ~ be the canonical approximate unit of F (Theorem 4.2.8.2). Then O_
II ~
II~ _
z) 2
II
268
4. C*-Algebras
Thus x~1 = l i m x 89z = lim z x 89 z,~
z,~
1
1
x = x89189= l i m z x ~ x ~ z - l i m z x z E F . z ,~
z ,;~
Hence F is hereditary.
I
c =~ b is trivial. Corollary 4.3.4.7
A s s u m e IK = C .
I f x C E + , then x E x
is the hereditary
C*-subalgebra of E generated by x .
It is easy to see t h a t x E x
is a C*-subalgebra of E . By Proposition 4.3.4.6
b =~ a, it is hereditary. Let ~ be an approximate unit of E (Theorem 4.2.8.2). Then x 2 = limxyx C xEx, y,~d
so that
z-
(z 2) ~ c z E z .
Let F be a h e r e d i t a r y C*-subalgebra of E containing x. By Proposition 4.3.4.6 a =~ c, xExc
F,
and so xEx C F.
Hence x E x Proposition
is the hereditary C*-subalgebra of E generated by x. 4.3.4.8
Let F
be a hereditary C*-subalgebra of E
I and take
x E E + . I f E is unital and if for each c > O there is a y C F+ such that
x < y+~l, then z c F .
4.3 Supplementary Results on C*-Algebras
269
Take c > 0. By hypothesis, there is a y~ E F+ such t h a t x_
2.
Hence 0 _~ (YE + e l ) - l x ( y E + c1) -1 _~ 1 (Corollary 4.2.2.3) and + cl)-lx(Y~ +
1
(Corollary 4.2.1.17 b ==~ a). It follows from (y~ + el) - y~ = E1 that 1 - Ye(Ye + c1) -1 - c(ye -~- c1) -1 , so that
Hence x89 = limx 89
1
+ gl) -1 = limo(y ~ + e l ) - l y ~ x ~
1
1
X -- X~X~ -- limo(YE + c1)-lyexye(y~ + e l ) -1 e :>O
Since (y~ + c l ) - l y ~ = y~(y~ + ~1) -1 e F+ (Corollary 4.2.1.12), it follows by Proposition 4.3.4.6 a :=v b, t h a t (y~ + cl)-ly~xyE(ye + ~1) -1 E F for every c > 0. Hence z C F .
m
270
4. C*-Algebras
4.3.4.9 Assume E unital. Let F be a unital C*-subalgebra of E and G a hereditary C*-subalgebra of E such that G c F and such that every positive linear form on E which vanishes on G vanishes on all of F . Then R e F C G .
Proposition
Take z E F+ and c > 0. P u t K ' := {x' E
T(F) Ix'(x) >_c}.
K ' is a compact set of T(F) (Proposition 2.3.5.9 a)). For each x' E K ' , there is a y~, E G+ such that 9 '(y~,) > 0.
Given x' E K ' , define
Ux, "= {y' E K' i y'(y~,) > 0}. Then
(Uz')x'Eg'
is an open covering of K ' . Since K' is compact, there is a
finite subset A p of K ' such that
K'C
Uux,. x~EA ~
We put
Y'-- E
Yx,.
x'EA'
Since K ' is compact, a "-
inf x'(y) > 0. x tE K ~
Let z'--
tlxll y. C~
Take x' E T ( F ) . If x'(x) >_ c, then x' E K ' so that
x'(z + c1) :> x'(z) = IIx]lx'(y) >_ Iixi] _> x'(x). O~
If x'(x) < c, then
x'(x) < c <_ x'(z + e l ) . Hence x a). By Proposition 4.3.4.8, x E G . Hence F+ C G and therefore Re F C G. (Theorem 4.2.2.9 a)).
I
.4.3 Supplementary Results on C*-Algebras
P r o p o s i t i o n 4.3.4.10
271
Assume IK = ~ . Let F be a hereditary C*-subalgebra
of E and G a closed ideal of F . Then there is a closed ideal H of E such that G=FNH.
Given subsets A, B of E , define A B := {xy l (x ,y) 9 A • B } ,
A := the vector subspace of E generated by A . Further, let H := E G E .
Step 1
H is a closed ideal of E containing G
It is obvious that E G E
is an ideal of E , so H is a closed ideal of E .
Using the approximate unit of E (Theorem 4.2.8.2), we see that H contains G. Step2
F N H C FHF, G C G3
This follows from the existence of approximate units (Theorem 4.2.8.2). Step3
FEF c F
The assertion follows from Proposition 4.3.4.6 a ==> c. Step 4
FEG3EF C G
By Step 3, FEG3EF = (FEG)G(GEF) c FGF c a.
Step5
G=FNH
By Steps 2 and 4, FEGEF C FEG3EF C G,
so, by Step 2
272
4. C*-Algebras
F N H c FHF c ~FEGEt~ c G.
By Step 1, GcFMH,
so that G=FMH.
I
[
P r o p o s i t i o n 4.3.4.11 ( 0 ) Let E be r a-complete. Take x c S n E and let F be the hereditary C*-subalgebra of E generated by x. If f is a bounded Borel function on a(x) such that 0 e a ( x ) =:=V lim f ( a ) = O, aea(~)
then f (x) E F .
We may assume that 0 E a ( x ) , since otherwise 1 C E(x) C F and this implies that F=E.
First take I to be real and positive. Then there is a g e C(a(x))+ such that
f_
g(o)=o.
By Theorem 4.1.3.1 d), g(x) e E(x) C F and by Corollary 4.3.2.5 a),b),
o < f(x) < g(x). Since F is hereditary, f ( x ) C F . If f is real, then f ( x ) -- f + ( x ) - f - ( x ) e F ,
by the above result. If ] K - (I?, then f ( x ) = (re f ) ( x ) + i(im f ) ( x ) E F
for an arbitrary f . Remark.
We cannot replace the condition
I
4.3 Supplementary Results on C*-Algebras
O e a(x) ~
273
lim f (a) = O c~ ---} o
by the weaker one f(0) = 0 as shown by taking E = g ~ , x'IN
>IK,
n:
1
~--, n
and f := %(x)\{0} "
Proposition 4.3.4.12
A s s u m e E unital. Let F be a hereditary C*-subalgebra
of E and y' C F ~ . Then there is a unique x' E E'+ such that 9 'JE = y',
Il~'JJ = Ily'll.
If ~ is an approximate unit of F , then x' (x) = lim y'(y*yxy*y) y,~
for every x C Re E (Proposition ~.3.~.6 a => b). The existence follows from Corollary 4.2.8.9. We now prove the last assertion (which implies the uniqueness). Now limx'(1 - y ' y ) - x'(1) - limx'(y*y) = IIx'll - limy'(y*y) Y,qd
y,q~
y,~
Ix'll - IlY'II = 0
(Proposition 2.3.4.10 a)). By Corollary 4.2.1.17 and Proposition 4.2.4.2 e), for every y C F # , 0< (1-y'y)2<
1-y*y
so that 0 < x'((1 - y,y)2) < x'(1 - y ' y ) . It follows that
limx'((1y,~
y* y)2) - O .
Now rx'(x) - ~ ' ( ~ * y ~ * y )
l < I~'(~ - u*y~)f + r~'(y*yx - y * ~ y * y )
r=
274
4. C*-Algebras
= Ix'((1 - y * y ) x ) l § I x ' ( y * y x ( 1 - y * y ) ) l -
1
< x'((1 - y*y)2)89
1
!
+ x'(y*yxx*y*y)~x ((1 - y,y)2)
(Proposition 2.3.4.6 c)). By the above considerations, lim Ix'(x) - y'(y* yxy* y)l = O, y,~
x' (x) = lim y'(y*yxy*y).
I
Y,~
Take p E P r E such that p e p is one-dimensional (this happens if IK = C, E is order a-complete, and p is a minimal element of
C o r o l l a r y 4.3.4.13 Pr E \ { 0 } ) . a)
There is exactly one x' E T(E) such that ~'(p) = 1,
and this x' belongs to To(E). b) pxt p = x ~. c)
x' is order continuous. By Corollary 4.3.2.12 a =~ d, p e p is one-dimensional if IK = C, E is
lI~-order a-complete, and p is a minimal element of Pr E \ { 0 } . a) Let y' be the unique element of T(pEp). By Example 4.3.4.2, p e p is a hereditary C*-subalgebra of E , so by Proposition 4.3.4.12, there is a unique x' E T(E) extending y' (which is equivalent to x'(p) = 1) and this x' belongs to ~ 0 ( E ) .
b) The map
E
>IK,
x:
",x'(pxp)
belongs to T(E). By a), it is x'. c) Let A be a downward directed set of E with infimum 0. Then pAp is a downward directed set of pEp with infimum 0. Hence by b), inf x'(x)
xEA
and x r is order continuous.
= inf xEA
x'(pxp) = 0 I
4.3 Supplementary Results on C*-Algebras
Proposition
4.3.4.14
( 0 )
275
Let F be an E - m o d u l e . Then f o r every a c
F , {x e E l x a = x*a = 0} (resp. {x e E l a x = ax* = 0}) is a hereditary C*-subalgebra of E .
Put G := {x E E l x a = x*a = 0} (resp. G := {x e E l a x = ax* = 0}). It is easy to see that G is a C*-subalgebra of E . Take x C E and y , z c G . Then (yxz)a = (yx)(za) = 0
(a(yx~) = (ay)(xz) = 0),
(yxz)*a = z* x*y* a = 0
(a(yxz)* = az* x*y* = 0),
i.e. y x z C G . By Proposition 4.3.4.6 b =v a, G is a hereditary C*-subalgebra of E .
I
276
4. C*-Algebras
4.3.5 S i m p l e C * - a l g e b r a s Definition 4.3.5.1
( 0 ) E is called simple if {0} and E are its only
closed ideals. P r o p o s i t i o n 4.3.5.2
Every hereditary C*-subalgebra of a simple complex
C*-algebra is simple. Let E be a simple complex C*-algebra and F a hereditary C*-subalgebra of E . Further let G be a closed ideal of F . By Proposition 4.3.4.10, there is a closed ideal H of E such that
G=FnH. Since E is simple, H is {0} or E . Hence a
is {0} or F and thus F is
simple as well.
P r o p o s i t i o n 4.3.5.3
m
( 0 ) The following are equivalent for every real C*-
algebra E : a)
E is simple.
b)
E is a simple purely real C*-algebra. a =~ b. Let F be a closed non-zero ideal of E . Then F x F is a closed o
ideal of E , so t h a t F • F - E . Let z C E . There is an (x,y) C F • F such that (z, 0) = ( x , y ) . We get z = x C F , i.e. E = F . Hence E is simple. By Theorem 4.1.1.8 b), E is a purely real C*-algebra. b ~
a. Assume /~ is not simple. Then there is a closed ideal F of /~,
{0} # F ~-/~. P u t G := {x c E l(0, x) c F } . Then G is a closed ideal of E . Assume that G -r {0}. Then G = E and {0}x E C F . It follows that E • {0} = i({0} x E) C F ,
- (E • { 0 } ) + ({0} • E) c F , which is a contradiction. Hence G = {0}. Put
4.3 Supplementary Results on C*-Algebras
H
:= {x e
277
E I 3 y e E , (z, y) e F } .
Take x e H and y , z e E such t h a t (x,y), (x,z) e F . Then
(o, (y - z)) = (x, ~) - (x, z) e F ,
y-zcG,
y=z, by the above considerations. Hence, for every x C H , there is a unique x' C E such that (x, x') E F . Let x , y E H , z C E , and a,/~ E JR. Then
( ~ + Zy, ~x' + Zy') = ~(x, x') + ~(y, y') ~ F ,
(~z,x'z) = (x, ~')(z, 0) e p , (z~, ~x') = (z, O)(x, ~') e F, r
- x ) = - i ( x , ~') e :'~ ,
(z*,-z'*) = (z,z')* c F. Hence H is an involutive ideal of E , the map H
)H,
x,
is linear, and X , 11 - - -
--X
,
X *!
(~y)'= ~'y,
---
--X
I*
,
(y~)'= yx'
for all (x,y) E H • E . Take x c H and a , ~ E IR. Then
II~x + Z~'ll ~ = II(~x* + Z x ' * ) ( ~ + Zr
= II(~z* - Z ~ * ' ) ( ~ + Zx')ll =
278
4. C*-Algebras
= Ii.~x*z + ~ Z ( x * ~ ' = ii~
x*'z) - 9~*'~'11 =
*~ + . Z ( x * ~ ' - (~*~)') - z : ( ~ * z ' ) l l =
= i i ~ z *~ + ~ ( x * x ' = I~x*x
+ ~*xll
By taking c~ = 0 and ~ = 1, we get
- x*~') - z ~ * ~ " l l =
= (~ + ~)llxll ~ 9
IIx'II-
Ilxll. Hence H is closed and so
H=E. Given c~,/~ E IR and x E E , put
(~ + i~)x := ~ x + fix'. By the above considerations II(~ + i~)xll = I~ + i~l IIxll 9 For all c~,/~,7,5 C IR and x , y C E ,
((o~ + iZ) + ('7 + iS))x = ((~ + 7) + i(Z + 5))x = (o~ + 7)x + (Z + 5)x' = = ( ~ z + Zz') + (Tx + ~ ' ) = (~ + iZ)x + (7 + i~)x,
(c~ + i~)((7 + iS)x) - (~ + i~)(Tx + 5x') - (~Tx + (~Sx' + ~Tx' + - (o~7- ~5)x + (o~5 + ~ 7 ) x ' -
~Sx"
-
((o~ + i~)(7 + iS))x,
(~ + ig)(xy) = . x y + 9xy' - x ( ( ~ + iZ)~), ((~ + iZ)~)* - ( ~
+ Z~')* - ~ *
- Zx*' - ( ~ -
iZ)x*.
Hence E endowed with the map
is an involutive complex Banach algebra and so a complex C*-algebra. This o
contradicts b), so E is simple.
Remark. 2.3.1.43 d).
I
The last part of the proof can also be deduced from Proposition
4.3 Supplementary Results on C*-Algebras
279
P r o p o s i t i o n 4.3.5.4 ( 0 ) Assume that I K - - ( ~ and let F denote the underlying real C*-algebra of E (Theorem ~.1.1.8 a)). Then any closed ideal of F is a closed ideal of E . Hence if E is simple, F is simple too.
Let G be a closed ideal of F and take x C G. Further, let F be an approximate unit of E . Then ix - l i m i(yx) - l i m ( i y ) x e G. Y,F
Y,F
Thus G is a vector subspace of E and therefore a closed ideal of E .
I
P r o p o s i t i o n 4.3.5.5 ( 0 ) Let (E~)~I and (F~)x~L be families of nondegenerate simple C*-algebras, and E and F the C*-direct products(sums) of these families, respectively. Given ~ c I and )~ E L , define
E~ := {x e E ] ~ ' e I\{~} ~
x~, = 0}
N
F~ := {y e F IA' e L\{A} ~
y~, = 0}
for every L C I and )~ E L. If u : E ~ F is an isomorphism of C*-algebras, then there is a bijective map ~ : I --+ L such that F~(~) = ~(E~) for every ~ E I .
Take ~ e I . There i s a A E L such that u(E~)AF~ 7(= {0}. Since E~ is a closed ideal of E , u(E~) a closed ideal of F and u(E~)M F~ a closed ideal of F~. Since F~ is simple, F~ C u(E~). Repeating the above argument for u - l , we get E~ C u-l(F~), so that u(E~) = F~. We put ~ ( ~ ) : = A. The map ~a : I --+ L is obviously injective. By symmetry, it is bijective.
I
P r o p o s i t i o n 4.3.5.6 ( 0 ) Let (E~)~c, be a family of non-degenerate simple complex C*-algebras, E the C*-direct product (sum) of it, .~ the set of real C*-subalgebras F of E such that the map
is an isomorphism of complex C*-algebras, and $2 the set of pairs (~p, (u~)~e,) such that ~a is an involution of I and for every ~ E I
uL :E~
> E~(~)
280
4. C*-Algebras
is an isomorphism of real C*-algebras, which is conjugate linear and such that U-t I =
Ucp(L ) .
Put E~ "= {x E E I~ ~ I ~
So~ ~ y
~ = (~, ( ~ ) ~ , )
e ~.
x~( o - u~x~}
Th~n E~ ~ ~: fo~ ~v~ry ~ ~ ~
and th~ map
is bijective. Let H be the set of conjugate involutions on E . If we set E(u) := {x E E l u x = x} for every u E H , then by Proposition 2.3.1.43 a),b), E(u) E .T for every u E H and the m a p H
>~',
u,
~E(~)
is bijective. Given w = (~, (u~)~e,) E .(2, define u~" E - - ~ E ,
x,
> (u~(0x~(~))Lel.
Then u~ E / 4 and
Hence we have only to show t h a t the map ~Q
~H,
w. ~ >u~
is bijective. Take u C H . By Proposition 4.3.5.4, for every ~ E I , the underlying real C * - a l g e b r a of E~ is simple. For every ~ 6 I , put
E~ . - {x e E IA e I\{~} ~
x~ - 0 } .
By Proposition 4.3.5.5, there is a bijective m a p ~ " I ~ I such that E~(~) - u(E~) for every ~ E I .
We denote for every ~ E I ,
by u~ 9 E~ -+ E~(~), the m a p
obtained from u by identifying EL and Ev(L) with EL and Ev(~), respectively. Since u 2 is the identity map, it follows that ~ is an involution and u~-1 - uv(~) for every ~ c I . Hence a; := (~, (u~)~eI) C f2 and u~ = u. Thus the m a p [2
~ l/l,
w ~----)"u,,
is surjective. It is easy to see t h a t this m a p is injective, so it is bijective.
I
4.3 Supplementary Results on C*-Algebras
Corollary 4.3.5.7
281
( 0 ) The following are equivalent for every simple com-
plex C*-algebra E "
a) E is isomorphic to the direct C*-product E • E . b) E has a conjugate involution. a =~ b follows from Proposition 4.3.5.6. b =~ a follows from Example 2.3.2.35 f). Proposition 4.3.5.8
I
( 0 ) Let E be a simple non-degenerate complex C*-
algebra, Y the class of real C*-algebras F , such that F and E are isomorphic, ~ a subset of .~ such that every element of ~ is isomorphic to exactly one element of ~ , Eo the underlying real C*-algebra of E , and 74 "= ~ U {Eo} . We assume .~ ~ O. Let further I be a set and 79 the set of maps p" 74 --+ ~3(I) such that:
1) F, G 9 74 , F C G =~ p(F) N p(G) = O . 2) there is a bijective map p(Eo) -+ I \ I p , where
I, := U p(F). FET-t
Given p 9 P
and ~ 9 Ip, let Fp,~ be the element F of 74 with ~ 9 p(F) , Fp
the C*-direct product (sum) of the family (Fp,~)~eip, E the C*-direct product (sum) of the family (E)~ei, and ~ the class of real C*-algebras g such that o
F and E are isomorphic.
a) For every element F of .T, there is a p 9 P such that F is isomorphic to Fp.
b) p 9 7:'::~ Fp E.Y'. c) The following are equivalent for all p, q 9 P "
C1) Fp
and Fq are isomorphic.
c2) r 9 7-/=~ Cardp(F) = Card q ( r ) .
d)
We define a map ~" IN x IN -+ IN inductively by
r ) -z p(1, n ) = 1 (n z+ n1 +T cos2
282
4. C*-Algebras
n-1
~(m,n) = 1 + E
~(m-
1, n -
k)
k=O
for all n C IN and m C I N \ { 1 } . Let ~ be a subset of J~ such that every element of 2F is isomorphic to exactly one element of ~ . If m:=CardGEIN,
n'=CardlEIN,
then Card 6 - ~ ( m , n ) . a) We may assume that F is a real C*-subalgebra of E such t h a t the map o
F
> E,
>x+iy
(x,y),
is an isomorphism of complex C*-algebras (Proposition 2.3.1.43 c)). By Proposition 4.3.5.6, there is an involution ~ of I and a family (ut)teI of conjugate involution E --4 E with u~-1 = u,o(t) for every ~ C I such that
F = {x C E I~ C I ~
zv(~) - utxt}.
Take c c I such that ~(~) - ~. By Proposition 2.3.1.43 a),
Hence there is a unique Gt c G and an isomorphism vt "G~ -4 Ft. We define the map p" 7 / - 4 ~ ( I ) as follows: for each G E ~ , we put
p(G) "- {~ E I IG = G~} and take as p(Eo) a subset of
I\ U p(G)
such that
GE~
~(p(Eo)) - I\ U p(G). GET-I
Then p c P . For x C Fp and ~ c I , define VtXt
if
~e
Up(G) GEG
!
X t :z
Xt
if
~ c p(E0)
u~(~)x~(~)
if
~eI\Ip
4.3 S u p p l e m e n t a r y Results on C * - A l g e b r a s
and put N
11, " G
!
) E,
X l
) (xr
I .
u is an injective homomorphism of real C*-algebras. Take x E Fp. Then 9 U
p(a)
~
~y,
= ~,v,x, -
I ~,,
v,x, -
GE6 !
E p(Eo) ~
!
x~(,) = uv2(,)xv2(,) = u~x~,
E I \ I p ==:, u~x~ - u~u~(~)x~(~) = x~(~)
~(~).
N
Hence u x E F , u ( F p ) C F . Take y E F . Then Y~o(~) -- uty~
for every c E I . For c E Ip, put
Xr
Then x "-(X~)LE I
E
5
:----
{
if LE U p(G)
v~-~y~
GEG
y~
if ~ E p ( E o ) .
and for every ~ E I \ I p , !
x~ - u~(~)x~(~) = ue(~)y~(~) - y~.
Hence u x = y , F C u ( F p ) , and F is isomorphic to Fp. b) Since 9c -7(=~, there is an isomorphism of complex C*-algebras u'E
>ExE
(Corollary 4.3.5.7 b ::v a). Put 7rl"ExE
>E,
(x,y),
>x
~2"ExE
>E,
(x,y),
~y.
For every LEp(E0) put u~'--u and for every ~E U GEO o
U~ " Fp,t
~E
be an isomorphism of complex C*-algebra. Let
~/\G
~;(E0)
p(G) let
283
284
4. C*-Algebras
be a bijection. Given x 9 Fp and t 9 Ip define
u~x~
in ~ 9 U p ( a )
7rlux~
in ~ 9 ~(E0)
GE~
x[ :=
7r2ux~o(L) in ~ 9 I \ I p . Then the map o
...
Fp
>E,
!
z,
>(x~)~el,
is an isomorphism of complex C*-algebras, so C*-algebras, so Fp E ~'. cl ~ c2. By Proposition 4.3.5.3 a ~ b, all C*-algebras of G are simple and none is isomorphic to E0. By Proposition 4.3.5.4, E0 is simple. Let u :
Ep ~ Eq be an isomorphism. By Proposition 4.3.5.5, there is a bijective map ~p : Ip ~ Iq such that Fp,~ and Fq,~o(~) are isomorphic for every ~ c Ip. Let f e 7-l, ~ e p ( F ) , and G e 7-/ such that ~(~) 9 q(G). Then Fp,~ = F and Fq,~o(~) = G , so that F and G are isomorphic. It follows that F = G,
~(~) 9 q(F), r
C q(F). By symmetry, r
= q(F). Hence
Card p ( F ) = Card q ( F ) . c2 => cl is obvious. d) follows from a),b), and c).
I
Let E, F be C*-algebras, G a C*-subalgebra of E x F and Eo a simple C*-subalgebra of E such that:
P r o p o s i t i o n 4.3.5.9
1) for every x 9 Eo, there is a y 9 F 2)
with (x,y) 9 G .
th~r~ i~ a~ xo e Eo\{0} ~ith (~o, O) e a .
Then Eo x {0} C G . Define g := {x 9 E o l ( x , 0) 9 G}. Take x C H and y 9 Eo. By 1), t h e r e i s a
z 9F
such that (y,z) E G. We
get (xy, 0 ) = (x, 0)(y, z) e G ,
(y~, 0)= (y, z)(z, 0) e v , and so xy, yx C H . Hence H is an ideal of Eo, which is obviously closed. By 2), g 7~ {0}. Since E0 is simple, H = E0, i.e. E0 x {0} C G .
I
4.3 Supplementary Results on C*-Algebras
C o r o l l a r y 4.3.5.10 product, and F
285
Let (Et)tE I be a family of C*-algebras, E its C*-direct
a C*-subalgebra of E . For each ~ 9 I , let F~ be a simple
C*-subalgebra of E~, p~ " E -----+ E~ ,
x l
> x~ ,
Ea,
x,
and q~ " {x 9 F ] x~ 9 F~}
~ n
~ (xa)~ei\{~}.
XCI\{L}
If, for every t E I , the map q~ is not injective and F~ c p~(F) , then F contains the C*-direct sum of the family (F~)~ei.
By Proposition 4.3.5.9,
H {0}cr for every ~ C I and the assertion now follows. C o r o l l a r y 4.3.5.11
I
Let (E~)+ei be a finite family of simple C*-algebras, E
its C*-directed product, and F a closed ideal of E . Given J C I , define p~ " E
~ H E~, tEJ
If EL = p{d(F) for every t C I ,
~ ,
~ (~)~.
then there is a g C I such that pjI F is
injective and
p (F) - H eCJ
We prove the assertion by induction on Card I . If pI\{~}tF is not injective for every ~ E I , then by Corollary 4.3.5.10, F=I-IE~. LEI
So take ~ C I with p/\{~)l F injective. By the induction hypothesis, there is a J C I\{~} such that pjIpI\{L}(F) is injective and RJ(RI\{~}(F)) -- H EL, tEJ
then p j I F is injective and LEJ
286
4. C*-Algebras
4.3.6 S u p p l e m e n t a r y R e s u l t s C o n c e r n i n g C o m p l e x i f i c a t i o n P r o p o s i t i o n 4.3.6.1
( 0 ) Let E be a real C*-algebra and
~.E
,~,
~,
~(x,O).
Given x' E E ' , define
~r
x").~
(~,y),
~ ~'(x) + i~'(y).
a)
z' e E' ~ x' E (E)', ~'z' = z', z'* = z'*
b)
The following are equivalent for every x' E E ' '
bl)
x' E E l _ .
b2)
x'E (E)~.
o
c) x, y E E , 0 E ~ ~ II(cos O)x + (sin 0)yll < Ix*x + y*yll89 < II(x, y) l.
I1~'11- IIx'll.
d)
x' E E' =,
e)
The following are equivalent for every x' E E ' '
el)
x'EReE'. o
e2) x' E R e ( E ) ' . If these conditions are fulfilled then z '~+ - x-' +
~-
- x-'-
f)
The identity map E --+ E x E is an isomorphism of real Banach spaces.
g)
If ~ is an approximate unit of E then for every x' E E', x' E E + ~
limx'(x*x)= x,~'
Iix'l .
o
a) The relation x' E (E)' is easy to check. For every x E E ,
Thus
4.3 Supplementary Results on C*-Algebras
287
Take (x, y) c / ~ . Then x3" ((x, y)) - ~((~, y) ,) __ ~ ( ( ~ * , - y * ) )
_.. x r (x,) - ix'(y,)
_
= ~'(x*) + i~'(y*) - x'*(~) + iz'*(y) - ~,*((~, y)).
Hence X I
---
X I*
"
b) By a) and Proposition 4.1.1.27 c), o !
x' ~ (E)+ ~
x ' - ~'x' c E+. o
c) Take x' e T(E). By b), x' e (E)~_ and by a), ~'x' e T(E). By Propoo
sition 4.1.1.27 d) (and Theorem 4.2.8.2), x' e T ( E ) . We get I1(~, y)ll ~ = I1(~, y)*(~, y ) l l -
II(x*x + y ' y , ~ * y -
y*~)ll >__
> I~'((x*x + y * y , x * y - y*x))l- I~'(x*x + y'y) + i~'(~*y- y * x ) l -
= ~'(~*x + y'y). By Corollary 4.2.8.5 c) and Corollary 4.2.2.7, ]](x, y)[[2 > iix, x 4- y,yll > I[(cos fl)x -4- (sin O)yll 2
d) We have ]l~l] = 1 , s o ][~'[[ = 1 and by a),
IIx'll- II~'x'll < IIx'll. Take (x, y) C E . For some ~9 E JR, ]x")((x, y ) ) ] - x~((x, y))e -ia -- xh((cos ~ - i sin O)(x, y)) --- x'(((cos O)x 4- (sin tg)y), (cos ~9)y- (sin tg)x)) --- x'((cos ~9)x 4- (sin O)y)4- ix'((cos O ) y - (sin O)x) -= x'((eos O)x + (sin O)y) < Ifx'll II(cos 0)~ + (sin O)y][ < IIx'll II(x, y)ll,
288
,4. C*-Algebras
where the last inequality follows from c). Hence
IIx'll < IIx'll,
IIx'll = IIx'll.
e) By a) and Proposition 4.1.1.27 b), o
x' C Re (E)' ~
x ' - ~'x' C Re E'.
Assume x' c Re E'. Then x ' + , x '- C E'+ and 9 '-
~'+ - ~'-,
Ilz'll =
I~'+11 + II~'-II
(Theorem 4.2.8.13). By b) and d), ~
o
9 '+, x'- e (E)~
IIx~ll- IIx'll = IIx'+ll + IIx'-II = IIx'+l1-4-IIx'-II 9
Since X t =
X !+
_X
!-
it follows by Theorem 4.2.8.13, x%+ _ x ~ ,
X t
--
xt-
.
f) follows from c) and Proposition 4.1.1.27 a). g) Suppose lim x' (x'x) x,~
IIx'll 9
By d), limx'((x*x, 0)) = l i m x ' ( x * x ) -
I1~'11-
I~'11.
By Proposition 4.1.2.27 i), o
x ' C (E)+ so that by b2 =~ bl, !
x' E E + . The reverse implication follows from Proposition 2.3.4.10 a).
9
4.3 Supplementary Results on C*-Algebras
Corollary 4.3.6.2
( 0 )
Let E be a real C*-algebra. For (x', y') C
289
E'
define
(x', y')"/~ -----+r
(x, y), o
~x'(x)-
~
y ' ( y ) + i(x'(y) + y'(x)).
o
a)
For every (x', y') e E ' , (x', y') e (E)' and
b)
If we endow
E'
with the norm o
E'
~ ~§ ,
(x', y') ,
~ JJ(x', ~')ll
then the map o
E'
) (E)',
(x', y'),
) (x', y')
is an isometry of involutive complex Banach spaces and the identity map o
E' • E' -+ E'
c)
is an isomorphism of real Banach spaces.
,.:,., Take (x', y') e E' . The following are equivalent: o
c~) (~,, y,)e (E)~.
c2)
(x',y') e R e ( E ) ' ,
z' e E~+, and II(x',y')ll-
llx'll -
o
Ca) (x', y') e Re (E)'
and
x'(~*x + y'y) + ~ ' ( V x fo~ ~ r y
(~, y) e
x'y) e
~§
~. o
c4) ~ e [-1,1] ~ (~', ~y') e (E)~.
d)
If A is a subset of El+ with infimum O, then 0 is the infimum of
o
e)
Given (x,y) c E and (x',y') c E' ,
(x, y)(~', y,) = "(xx' - yy', ~y' + yx';, Ix', y'~)Ix, yl - ~(x'x - y'y, x'~ + y'xT.
290
f)
4. C*-Algebras
Take x' e E and let p be its right (left) carrier. Then (p, O) is the right (left) carrier of (x', 0). o
g)
We identify
E' o
(~", v")(E)'
with (E)' using the isometry of b) and put
~r
(x'. v')'
~ z"(~') - y"(y') + i(*"(y') + v"(*'))
o
for every (x", y") 9 E" . Then:
g~)
(x".y")e
E" ~ ( ~ " . v") e ( E)"
g2) T h e m a p
,':", E"
o
~ (E)".
(~,,. y,,)
.
~ (z".
r
is an isomorphism of involutive complex vector spaces.
g~)
We have
(.'. y')(~", y") - (.'." - y'y". z'y" + y'z"{.
(x". r
r
= (z';.' - y"y'. z"y' + y"~'{
o
for all (x', y') 9 E'
o
and (x", y") 9 E"
(Definition 2.2.7.8).
a) By Proposition 4.3.6.1 d), II(x', y')l -
I(~'~,o~ + i(y', o)ll < II(x', o)ll + II(y', o)ll - I~'ll + Ily'l[ 9 o
Let x C E # . T h e n (x, 0) C E # , s o I1(x', y')ll >_ l( x', y')((x, 0)) I - x'(x) + iy'(x)l > sup{Ix'(x)l , ly'(x)l}.
Since x is arbitrary, ]](x', y')I[ -> sup{ [Ix']l, ]IY'I]}. b) By Proposition 4.3.6.1 f) and Proposition 2.3.1.39, the map E' ---+ (E)',
(~', y'),
~ (~', V')
4.3 Supplementary Results on C*-Algebras
291
is involutive, linear and bijective. The assertion now follows from a). o
c) Assume that (x', y') 9 Re (E)'. Then x'* - x', y'* - - y ' and for any
(x, y ) c ~,
(x', r
y)*(z, y)) = (x', r
y)) =
= (x,,y')((x*x + y*y,z*y-
= x'(~*x + y ' y ) + y'(y*x - x ' y ) + i ( x ' ( ~ * y -
y*~))
-
y*~) + y'(x*~ + y ' y ) ) -
= ~,(~,~ + y'y)+ y'(y*~_ ~,y). This proves the equivalence of c,) and c3) as well as Cl) ::~ X' 9 E~_. Let ~ be an approximate unit of E (Theorem 4.2.8.2). By Proposition 4.1.2.27 i) and Proposition 4.1.1.27 d), x' e E+ ~
x' E Re E' and
(x,,
IIx'll-
limx'(x*x)
tl(x,, y, )ll- lim (x', x,~
0))
Moreover, lim (x', y')((x*x, 0)) -- lim(x'(x*x) + iy'(x*x)) -- lim x'(x*x). z,~ x,~ x,;~ Hence if (x', y') e (E)~_, then x' e E+ and
IIx'll-
limx'(x*x) = lim(x',y')((x*x,O)) = x,~ z,;~
(x',y')
.
If x ' C E~_ and II(x',y')ll- [Ix'll, then [[(x',y'~)l[ = I[x'][ = l i m x ' ( x * x ) = lim(x',y'~)((x'x,O))
and we get (x', y ' ) e (E)~_. o
o
C4. Take (x, y) E E+. By Proposition 4.2.2.15 c), (x, ay) C E + . By Corollary 4.2.2.10, C1 =:~
( ( x , y ) , ( x ' , ~ y ' ) ) = x ' ( x ) - ~ y ' ( y ) - ( ( ~ , ~ y ) , (~',y')) e ~ + ,
292
4. C*-Algebras
~
o
(x', ~v')e (E)+. c4 ~ cl is trivial. d) Take ( x ' , y ' ) e
Uound for { ~
(/~)' such that (x',y') belongs to (/~)~_ and is a lower
I z' ~ ~}. ~y c1 ~ c~, 0 ( x r ( zr
for every z' E A. Hence x' = 0 and by cl =:~ c2, y' = 0. e) Take (u, v) C /~ . Then
: ((ux-~..~+v~).
(~,.~,~-~) -
= ~ ' ( ~ - ~y) - y'(~y + w ) + i(x'(~y + w ) + y ' ( ~ - vy)) -
= (~,xx')-
(v, w ' ) - (u, v v ' ) -
(~,~y')+
+i( (~, w ' ) + (v, ~ ' ) + (~, zv') - (v, vv') ) =
= (~, ~ x ' - vv') - (v, zy' + vx') + i((~, xv' + yx') + (~, ~ ' -
=
(u, v), ~(xx' - yy', xy' + y x
vy')) -
,
which proves the first equality. The second one can be proved similarly. f) By e),
~(p,
o) = (x,p, o'--'~- (x,, o'--~
Take (x, y) C/~ with (x,,O)(x,y) - o Then by e),
(ix. y)(x,. 0--~ = 0)
4.3 Supplementary Results on C*-Algebras
ix'x, x ' ~ = ~ ( x ,
y) - o
293
9 ~', y~'7 = (~, y)(~', o) - o
,
and so, by b), x'x = x'y = 0
(xx' = yx' = 0).
We get (p, o)(~, y) = ( p x , ; y ) = o
((z, y)(p, o) = (x;, yp) = o ) .
Hence (p, 0) is the right (left) carrier of (x~~,0~. gl) and g2) follow from b) and Proposition 2.3.1.39. ga) can be proved similary to e). Corollary 4.3.6.3 ( 0 ) x, y E No E , such that
Let E
m
be a real C*-algebra.
Take x ' E
E'
and
xx ~= x'y. Then X* X t -- x~y * "
By Corollary 4.3.6.2 e),
(x, o)~.~:
(x~, o) - (x,~,~) = (x,, o~ (~, o)
Hence, using the same argument and the Fuglede-Putnam Theorem (Theorem 4.1.4.1), (~*x', o) = (~*, o)(~', o) - (~, o ) * ~
-
= (x,, o)(y, o)* = (~', o)(y*, o) = (x'y*, o).
By Corollary 4.3.6.2 b), x*x ~ -- x~y * .
Theorem
4.3.6.4 ( 0 )
m
Let E be a real C*-algebra, F
an involutive E -
submodule of E ' , and (using the notation of Corollary 4.3.6.2), take
9=
(x',r
~
294
4. C*-Algebras
a)
F is an involutive E-submodule of
b)
If the map
E
>F',
(~) x,
'
>(z,-)IF o
is an isometry, then E x {0} is a closed set of E~ and the map
>(?)',
z,
>(z,.)l?
is an isometry of involutive complex Banach spaces. a) follows from Corollary 4.3.6.2 b),e). o
b) Let ~ be a filter on E x {0} converging in E ~ to (x0, Y0). Then for aEF, (Xo, a) + i(yo, a) = ((Xo, Yo), (a, 0)) = lim((x, O) ' (a, 0)) = lim(x a) C I R . z,~ x,~ ' o
Hence (Y0, a) = 0 and y0 - 0. Therefore E • {0} is a closed set of E ~ . Define
p-~:
>~+,
z,
>lllFII.
Then p is a norm on /~ smaller than the initial norm of /~ such that
p(z*)-p(z) for every z C E . o
Take c E F # and z C
/~ with p(z) _< 1. For all z0 E E # , we have
ZoC C F # , so
Il- Il ~
1
and therefore cz C F # . o
Take zl, z2 C E with p(zl) < 1, p(z2) < 1. Then, by the above consideration, we have for c E F , [(C, ZlZ2)[- [
o
by Ep. The map Ep --+/~, x ~-+ x , being an involutive algebra homomorphism, it follows from Corollary 4.1.1.20, that [[-[[ < p. Hence [[-[[ - p and the map
is an isometry of involutive complex Banach spaces.
I
4.3 Supplementary Results on C*-Algebras
Example 4.3.6.5
295
If we set [11] 1 1
v [11111
then I1~11- Ilvll = Ilu*~ + ~*vll 1 = I1(~, v)ll
(Proposition 2. 3.1.4 0 and Example 4.1.1.7).
We have (Example 1.2.2.8)
ll ll 1( ~
Ilvll2 - ~
1 + 1 + 1 + 1 + V/(1 + 1 + 1 + 1) 2 -
1( l + l + l + l + v / ( l + l + l + l ) ,u+v,v=2(~+~)=4[1
411 - II 2
2-411-112
)
-- 4 ,
) =4,
o]
0 1
'
I1(~, ~)I ~ = = -- 2 + 2 + 2 + 2 + v / ( 2 + 2 + 2 + 2)~ - 41(1 + i/~ - (1 - i)~l ~
2
= 4
"
This example shows that the inequality of Proposition 4.3.6.1 c) is sharp (see also Example 4.1.1.28). Remark.
Proposition 4.3.6.6
( 0 ) Let E , F be real C*-algebras, u" E --+ F an involutive algebra homomorphism, and consider
o
If we identify (E)' with then
~., E'
o
and (F)' with
~., F'
(~)'((z', y')) - (~'x', ~'y') /or any (~', y ' ) 9 ~ .
as in Corollary ~.3.6.2 b),
296
4. C*-Algebras
By Proposition 2.3.1.41, u is an involutive algebra homomorphism and o
therefore continuous (Corollary 4.1.1.20). For (x, y) C E,
((x, y), (~)'((x', y'))) = (~((~, y)), (~', y')) = ((~z, u~), (~', y')) = = (~x, x') - (~y, y') + i((ux, y') + (~y, ~')) -
= (z, u'z') - (y, ~'y') + i((z, u'y') + (y, ~'~')) = ((x, y), (r
~'y')),
so that
(~)'((x', y')) = (~'~', ~'y').
m
3.4 W*-Algebras
4.4
297
W*-Algebras
W*-algebras are C*-algebras E which are duals of involutive E-submodules of E'. They represent the highpoint of the theory of involutive Banach algebras. They are order complete, which has important consequences as we have seen in the previous section. Moreover, every element of a W*-algebra (and every element of its predual) has a polar representation and for every self-normal element x, the functional calculus may be constructed with respect to the elements of the bidual of C(a(x)) (this construction will be carried out in Subsection 6.3.3). But their most important feature is the fact that the order continuous positive linear forms are exactly the positive elements of the predual. Throughout this section, E is always a complex (resp. real) C*algebra and F is always a closed vector subspace of E' (resp. an involutive E-submodule of E') such that the map E .... ) F ' ,
x,
)(x,.)lF
is an isometry. This is precisely the definition of a W*-algebra (see Definition 4.4.1.1). A great part of the results in this section were proved by S. Sakai. 4.4.1 G e n e r a l P r o p e r t i e s
Definition 4.4.1.1
( 0 )
(Sakai, 1956)A complex (real) W*-algebra is
a complex (real) C*-algebra E for which there is a closed vector subspace (an involutive E-submodule) F of E t such that the map E
~F',
x,
~(x,.)lF
is an isometry. F is called the predual of E .
Each finite-dimensional C*-algebra is a W*-algebra. In particular, each s n) (n E IN) is a W*-algebra (Example 4.1.1.7). g~~ is a W*-algebra for every set T (Example 1.2.2.3 d)). We shall show (Corollary 4.4.4.4) that F appearing in the above definition is unique.
Theorem
4.4.1.2
( 0 ) Assume IK = lit. For (a,b)C F , define
298
4. C*-Algebras
(a, b~j"
E --+ r
(x, y ) ,
) (x, a) - (y, b} + i((y, a) + (x, b))
and put
b'~)[(a,b)9F}.
9= {(a,
and the map
>(~)',
z,
>(z,.)IF
is an isometry of involutive complex Banach spaces. In particular, E is a complex W*-algebra with F as predual.
m
The assertion follows from T h e o r e m 4.3.6.4. Proposition
4.4.1.3
( 0 ) Re E and E+ are closed sets of E F , Re E # and
E~+ are compact sets of EF , F is an involutive set of E ' , and the map EF
~ EF ,
x,
>x*
is continuous.
First assume ]K = IR. Take a 9 F . T h e n (x*, a) = (x, a*) for every x 9 E , so t h a t the map EF
is continuous. It follows that R e E Step 1
ReE
> EF ,
x,
> x*
is closed in E F .
is a closed set of EF
By the above, v)e may assume IK = r Let x be a point of adherence of Re E # in EF. Let ~ be the trace of the neighbourhood filter of x in EF on Re E # . Take a E cr(imx). Then a + / 3 e ~r(/31 + im x) and
lly + i91 ii ~ = II(y - i91)(y + iZl)ll = liy ~ + Z~ 111 <__ Ilyll ~ + Z~ < 1 + Z~
4.4 W*-Algebras
299
for every y E Re E # and fl E lR. Hence la +/3[ < r(fll + im x) = [[/31 + im xil _< <_ ]Ix + ifll[] <_ lim inf [lY + ifll[] <_ V/1 +/3 2 y,~
for every fl C ]R (Theorem 4.1.1.16 b), Proposition 2.3.2.7, Proposition 1.2.6.6). We deduce successively that a 2 + 2 a f t + f12 __<1 +/3 2 , a 2 + 2aft <_ 1, a=0,
imx=0,
xCReE,
xCReE #,
(Theorem 4.1.1.16 c), Proposition 1.2.6.6). Thus R e E # is a closed set of EF. By Corollary 1.3.7.5, Re E is a closed set of EF. Step2
E+ is a closed set of EF
It follows from E+# C R e E
#N(I+ReE
#) c E + #
(Proposition 4.2.1.16 b ~ a), that E +# = R e E # M (1 + R e E #) By Step 1, E+# is a closed set of E r . Hence by Corollary 1.3.7.5, E+ is a closed set of EF. Step 3
R e E # and E+# are compact sets of EF
The assertion follows from the preceding steps and the Alaoglu-Bourbaki Theorem. IflK =G Step 4
EF# - + E r ,
then the map x~+rex(imx)
is continuous at 0. Let ~ be an ultrafilter on EF# converging to 0. Put x :-- lim r e ( ~ ) , n-+oo
y := lim im(~) n---~oo
300
4. C*-Algebras
(Alaoglu-Bourbaki Theorem). Then x+iy=
lim ~ = 0 . n----~ o o
By Step 1, x , y E R e E . Thus x=y=O,
which proves the assertion. Step 5
The map E F ~ E F ,
We may assume ]K = r
x ~-+ x* is continuous
By Step 4, the map E#F ---+ E F ,
x,
> x*
is continuous at 0. By Proposition 1.3.7.11 b =~ a, the map EF
>EF ,
x,
> x*
is continuous. Step 6
F is an involutive set of E'
Take a E F . Given x E E ,
(z, a*> = (x-~,a> for every x E E . By Step 5, the map
Er
>IK,
z,
>(z,a*)
is continuous. By Proposition 1.3.7.7 c =~ a, a* E F . Hence F is an involutive set of E ' . Remark.
I F is an involutive Banach space and an ordered Banach space.
C o r o l l a r y 4.4.1.4
( 0 )
The map
E~F',
x,
~(z,-)IF
is an i s o m e t r y of involutive B a n a c h spaces.
C o r o l l a r y 4.4.1.5
I
( 0 ) I f x E ( R e E ) \ E + , then there is an a E F+ with (z, a) < O .
4.4 W*-Algebras
301
By Proposition 4.4.1.3 and Proposition 2.3.2.25, there is a b E Re F such that sup (b, y) = 0 < (b, x). yEE+
Put a :-- -b.
Then (y, a) = - ( y , b) > 0 for every y E E+. By Corollary 4.2.2.10, a is positive. Hence a E F+ and (x, a) = - ( x , b) < 0. Corollary4.4.1.6 a)
( 0 ) Let I K - - I R ( ] K = r
I and x E R e E
(xEE).
x is positive iff (x, a} E IR+ for every a E F+.
b)
x=0
iff (x,a) = O for all a E F+ .
a) If x is positive, then (x, a / E JR+ for every a E F+ (Theorem 4.2.2.1 a =~ d). Now suppose that (x, a) E IR+ for every a E F+. Then (re x, a) = re(x, a) E JR+ ( = im<x, a) = 0) for every a E E+ (Corollary 2.3.1.23 a :=> c) and so rexEE+ (Corollary 4.4.1.5). Hence x E E+. b) follows from a).
(imx=0)
I
302
4. C*-Algebras
C o r o l l a r y 4.4.1.7
E~+ is the closure of F+ in E~E .
Let A' be the closure of F+ in E ~ . It is obviously a convex cone contained in E~_. Assume there is an x' C E~+\A ' . By Proposition 2.3.2.25, there is an x E Re E with sup y ' ( ~ ) = o <
~'(~).
yr E A r
Then - x C E+, by Corollary 4.4.1.5). Hence 0 < ~'(-~) = -~'(x) < 0
by Theorem 4.2.2.1 a =v d. The contradiction obtained implies that E~_ - A'. I T h e o r e m 4.4.1.8 a) b)
(0)
F is order complete and an order faithful set of E ' . Every norm bounded upward directed nonempty set A of E+ has a supremum x and the upper section filter of A converges to :r in E F .
c) d)
E is C-order complete and a is order continuous for every a C F . Every subset of E which is closcd in EF is order faithful and order complete. Every involutive subalgebra of E which is closed in EF is a W * algebra.
e) f)
E+, Re E # , and E#+ are order complete and order faithful sets of E . For every x 9
and c > O, there is a finite family ((c~,p~))~c, in
a(x) x Pr E such that
g) If IK - C
(IK = IR), then the vector subspace of E generated by Pr E is
dense in E (in Re E . ) .
h)
E has a unit.
i) Pr E is an order complete lattice and an order faithful set of E . j)
Every element of E has a left and a right carrier.
4.4 W*-Algebras
k)
303
Every m a x i m a l Gelfand C*-subalgebra of E is order complete.
l) For every x 9 R e E in E
(x 9 E + ) , there is an increasing sequence (Xn),cIN
converging to its s u p r e m u m x such that f o r every n 9 IN there is
a finite f a m i l y ( ~ , P~)~et in IR x Pr E (/n 1R+ x Pr E ) with Xn = ~2 ~ ol~p~ .
a) follows from Proposition 1.7.2.5. b) follows from Proposition 4.4.1.3 and Proposition 1.7.2.6. c) By Corollary 4.2.1.18, every upper bounded set of E+ is bounded in norm and so the assertion follows from b) and Theorem 4.4.1.2. d) The first assertion follows from b) and c). In order to prove the second assertion, let G be an involutive subalgebra of E which is closed in E F . By Theorem 1.3.5.12 c), G' may be identified with E ' / G ~ . F / F • G ~ is a closed vector subspace of E ' / G ~ and, in the real case, it is a G-submodule of E ' / G ~ . By Corollary 1.3.5.5, the map G
~(F/FnG~
',
x,
~<x,.)I(F/FnG
~)
is an isometry. Hence G is a W*-algebra. e) follows from d) and Proposition 4.4.1.3. f) and g) follow from c) and Corollary 4.3.2.4. h) By Theorem 4.4.1.2, we may assume IK - ~ . By c) and Corollary 4.3.2.6, P r E is upward directed. By b), P r E has a supremum x in E+# . Then p _< x and so p x - xp - p
for every p 9 Pr E (Corollary 4.2.7.3 a =v d). By g), x is a unit of E . i) follows from c) and Proposition 4.2.7.17. j) By c) and h), E is unital and order complete. Thus by Corollary 4.3.3.11, every element of E has a right and a left carrier. k) follows from c), Theorem 4.4.1.2, and Corollary 4.2.2.20 a). l) follows from c),f), and Corollary 4.3.2.11. C o r o l l a r y 4.4.1.9
i~ ~(E).
F#+ is dense in (E~+#)E and {a 9 F+
I
I Ilall-
1} is dense
304
4. C*-Algebras
Take x' E E~_\{0}. By Corollary 4.4.1.7, there is a filter ~ on F+, such that jF(~) converges to x' in E ~ . Then lim a,~
Ilall-
lim(1 a)a,~ '
(1 ' x ' ) -
IIx'll
(Theorem 4.4.1.8 h), Corollary 2.3.4.7). Hence 1
1
lim a,~ ~ - - ~ [ a = ~ x in E~:. C o r o l l a r y 4.4.1.10
I ( 0 ) If E is Gel]and, then a(E) is a compact hyper-
Stonian space. By Theorem 4.4.1.8 h), E has a unit, a(E) is thus compact (Theorem 2.4.1.3 c)). By Theorem 4.4.1.8 c), E is order complete and so or(E) is a Stone space (Proposition 1.7.2.13 d ~ a). Let U be a nonempty open set of a(E). There is an x E R e E \ { 0 } with Supp Y C U (Urysohn's Theorem). By Corollary 4.4.1.6 b), there is an a E F with (z, a) r O .
Let # be the Radon measure on a(E) for which
f ~d# = (y, a) for every y E E . Then
f ~d# = (x, a) :/: O. Thus U A Supp p -r O. By Theorem 4.4.1.8 c), a E E" and so # E C(T)'. Hence T is a hyper-Stonian space. I C o r o l l a r y 4.4.1.11 ( 0 ) Take x E S n E and let j " a(x) ~ IK be the inclusion map. Let f8 be the a-algebra of Borel sets of a(x) and 13 the C*algebra of bounded Borel functions on or(x).
4.4 W*-Algebras
a)
305
There is a unique E-valued spectral measure # on ~3 such that jdp=x (in fact # is the spectral measure of x from Theorem 4.3.2.19).
b) For every f C B and every order a-faithful C*-subalgebra G of E containing x , f (x) = /
f d# ,
IIf(x)ll ~ Ilfll, and
a(f(x)) : f(a(x)),
{x} ~ : {f(x)} ~,
f(x) e G ,
{cA
f (x) ~ f o
is meager whenever G is a Gelfand C*-algebra.
c) If f C B and g a bounded Borel function on f ( a ( x ) ) , then g(f(x)) = g o f(x) .
d) #(~3)c #(if3)c= {x) ~. e)
a o # is a measure for every a C F and
f fd(ao#)-(f fdp, / for every f C B .
f) x is atomic iff a o # is atomic for every a C F . g) If 0 is a nonisolated point of a ( x ) , then there is a sequence (Pn)nElN in {x}C N Pr E\{O} such that PmPn = 0 for all distinct m, n E IN and x
--
~ nEIN
xpn
9
306
4. C*-Algebras
By Theorem 4.4.1.8 c),h), E is unital and (13-order complete. Thus the assertions follow from Theorem 4.3.2.19, Corollary 4.3.2.5, and Corollary 4.4.1.6. m
C o r o l l a r y 4.4.1.12 Let A be a subset of IF[ and f function on A . Take x, y E Re E with a(x) C A ,
a(y) c A ,
x <_ y,
an increasing Borel
xy - y x .
Then f (x) < f (y) .
Let G be a maximal Gelfand C*-subalgebra of E containing x and y (Corollary 4.1.2.3). Then G
~<~
G
(Proposition 4.2.1.9) and so G
G
fo~
< fo~'.
Since f1~) # f ~
,
f (y) # f o ~
are meager (Corollary 4.3.2.5 c)), it follows that :~) < f(x)
is meager. Since ~(G) is a Baire space, G
G
f(x) _< f (y), f (x) <_ f (y)
(Proposition 4.2.1.9).
m
C o r o l l a r y 4.4.1.13 If C is a hereditary cone (hereditary C*-subalgebra) of E which is closed in EF, then, given any x E C N E+, (x c C N Re E ) , the carrier of x belongs to C (Theorem ~.~.1.8 j)).
4.4 W*-Algebras
307
First assume x 6 E+. Define j:o(x)
>IK,
a,
>a, ifa>O
f : ~(x)
)IK,
oe,
>{ if a = O ,
and for each n 6 IN, define
A : ~(x)
)IK,
a,
if a >- - !n
>{1 0
if a <
•n .
Then fn <_ n j ,
o(x) f~ __~ fn+l __~ Ca(x)
SO that f~ (x) <_ nj (x) = n x ,
h(x) < A+l(X) < ~, A(~) c c for every ~ E IN (Corollary 4.3.2.5 a), Proposition 4.3.4.3). By Corollary 4.3.2.5
~),c),
n6IN
and by Corollary 4.3.3.11 (and Corollary 4.3.3.10 a ~ b, Corollary 4.3.3.9), f ( x ) is the carrier of x. By Theorem 4.4.1.8 b), ( f n ( X ) ) n ~ converges to f ( x ) in EF. Thus f (x) C C. Now assume x 6 R e E and let p and q be the carriers of x + and x - , respectively (Theorem 4.4.1.8 j)). By the above, p,q C C. By Proposition
4.3.3.15, p + q is the carrier of x. Hence the carrier of x belongs to C . I
C o r o l l a r y 4.4.1.14
Let ]K = C (IK = IR). Let (an)n~i be a sequence in F
(in Re F ) such that ((p, an) )ncIi converges whenever p C Pr E . Then there is an order continuous x' 6 E' (x' 6 Re E ' ) such that lira (x, a,,) = x'(x) 7%--+00
for every x C E .
308
4. C*-Algebras
By Theorem 4.3.2.15 (and Theorem 4.4.1.8 c)), there is a y' E E' such that (lira (x, an) : y'(rex))
lim (x, an) = y'(x) n--). oo
for every x C E . Put Xl :=
y'
yl
re
if IK = C if ]K = IR.
Then lim (x, an) - x'(x) for every x E E . By Theorem 4.4.1.8 c), an is order continuous for every n C IN. Since E is order complete (Theorem 4.4.1.8 c)), it follows from Proposition 1.7.2.11, that x' is order continuous. E x a m p l e 4.4.1.15 L~176
If ( T , ~ , # )
I
is a measure space, then the C*-algebra
(Example ~.1.1.5) is a W*-algebra iff L~176 is order complete. This
condition is fulfilled whenever # is a-bounded or a Radon measure on a locally compact space. By Example 4.2.3.8, the order relation in the C*-algebra L~176 coincides with "~he usual order relation on L ~ ( # ) . If L~176 is a W*-algebra, then by Theorem 4.4.1.8 c), L~r
is order
complete. If L~176 is order complete, then it is isometric to the dual of L:(#). Thus L~(#) is a W*-algebra. The proof of the last assertion is standard and may be found in most books on integration theory.
I
4.4 W*-Algebras
4.4.2
F
as a n E - s u b m o d u l e
Proposition 4.4.2.1
309
of E'
( 0 ) pE(1-p)
+ (1-p)Ep
is a closed set of EF for
every p C P r E . First assume IK = 1R. Take a C F . T h e n
(px(1 - p) + (1 - p)xp, a) = (x, (1 - p)ap + pa(1 - p)) for every x C E , so t h a t the m a p
EF
) EF ,
x,
) p x ( 1 - - p) + (1-- p)xp
is continuous. Take x E pE(1 - p) + (1 - p ) E p . T h e n there are y, z C F such that
x = py(1 - p) + (1 - p)zp. It follows
px(1 - p) + (1. - p)xp = py(1 - p) + (1 - p)zp = x . Hence
pE(1 - p ) + (1 - p ) E p
= {x e E I x = px(1 - p ) + (1 - p ) x p } .
By the above, pE(1 - p ) + (1 - p ) E p
is a closed set of E F .
Now assume ]K = C . T h r o u g h o u t this proof, we consider E endowed with the topology of E F . We note t h a t p e p is a C * - s u b a l g e b r a of E for which p is a unit. We denote by G the corresponding unital C*-algebra. Let x be a point of adherence of p E # ( 1 -
p). There is a filter ~ on E
with E #E~,
limpy(1-p)-x. y,~
Given y c E # and c~ C ~ , IIpy(1 - p) + c~pl]2 = ]](py(1 - p) + o~p)( (1 - p)y*p + -sp) I[ =
= ]]py(1 - p ) y * p + lal2piI _< Iipy(1 - p ) y * p l ] + lai21ipiI _< 1 + Ion]~ , so t h a t
310
4. C*-Algebras
Ilpxp + ~pll = IIp(x + ~p)pll ~ IIx + ~pll lim inf Ilpy(1 - p ) + ~Pll ~ V/1 + I~12 y,~
(Proposition 1.2.6.6) for every a C q~. Take c~ E IR. P u t y'=rex
(y--imx),
and take t3 E a c ( p y p ) . T h e n fl E ]R (Theorem 4.1.2.1, Proposition 2.3.1.29) c~ + ~ = cra(py p + olp) ,
IlPYP + ~P[I ~ IIpxp + c~Pl <~ v/1 + c~2
(llpyp § ~pll <~ Ilpxp + i~pll <~ j1
+
c~2)
(Proposition 2.3.2.7), so t h a t Ic~ +/31 <_ ra(pyp + ap) < Ilpyp + ~pll ~ v/1 + c~2 (Proposition 2.2.4.1). Since c~ is arbitrary, we deduce successively that
13-0, pyp = 0 (Theorem 4.1.1.16 c)), p x p - O. Similarly, (1 - p)x(1 - p ) = 0 . Now let x be a point of adherence of A " - E # V) (pE(1 - p) + (1 - p ) E p ) . Then there is an ultrafilter ~ on A converging to x. P u t xl := limy,;~py(1 - p) ,
x2 "= 1~,~(1 - p)yp
~.4 W*-Algebras
311
( A l a o g l u - B o u r b a k i Theorem). T h e n X :
X 1 -~-X 2
and, by the above proof,
pxp = p x l p + px2p = O,
(1 - p)x(1 - p) = (1 - p)x 1(1 - p) + (1 - p)x2(1 - p) = O. Hence
x = ((1 - p ) + p ) x ( ( 1 - p ) + p ) = px(1 - p ) + (1 - p ) x p
e A.
A is therefore a closed set. By Corollary 1.3.7.5, pE(1 - p) + (1 - p ) E p is a closed set of E F .
m
Proposition 4.4.2.2
( 0 )
Given p E P r E ,
the set p e p
is closed in EF
and the map EF is continuous, p e p unit.
) EF ,
x J >pxp
is an order faithful set of E and a W*-algebra with p as
It is easy to see t h a t the m a p
EF
) EF ,
x,
) pxp
is continuous and t h a t p e p is a closed set of EF if IK = 1R. So assume IK = r By P r o p o s i t i o n 4.2.7.1 c r
d,
E #n(p(ReE)p)={xeReEI-p_<x
> Ep
v : EF
) EF,
,
x,
~rez,
xl
)imx.
By Proposition 4.4.1.3, u and v are continuous. From
312
4. C*-Algebras
p E p = u 1(p(Re E ) p ) M v 1( p ( n e E ) p ) it follows t h a t p e p
is a closed set of E F .
Consider u :E v:E w:E
) E,
>E, >E,
x~
x,
) pxp,
> (1-p)x(1-p), p x ( 1 - p) + (1 - p ) x p .
xl
u, v, w are p r o j e c t i o n s in E a n d VW
=
WV
=
O,
I m v = (1 - p ) E ( 1 - p ) , I m w C p E ( 1 - p) + (1 - p ) E p as c a n easily be seen. Take x, y E E a n d p u t z : = p x ( 1 - p) + (1 - p ) y p . Then w z = p x ( 1 - p) + (1 - p ) y p
= z
a n d so z E I m w . p E ( 1 - p) + (1 - p ) E p C I m v , I m w = p E ( 1 - p) + (1 - p ) E p . B y t h e a b o v e p r o o f a n d P r o p o s i t i o n 4.4.2.1, I m v a n d I m w are closed sets of E F . H e n c e I m v + Im w is a closed set of FIR ( P r o p o s i t i o n 1.3.7.6). Since
Im u = pEp, Keru = Imv + Imw, it follows t h a t b ~
Imu
and Keru
are closed sets of E F . B y C o r o l l a r y 1.3.7.12
a, the map EF
~, EF ,
x J
~, p x p
is c o n t i n u o u s . T h e last a s s e r t i o n follwos from T h e o r e m 4.4.1.8 d).
4.4 W*-Algebras
Proposition
( 0 )
4.4.2.3
EF
313
The map
> EF ,
x,
> xp
( resp. px)
is continuous for every p C P r E . Let x be a p o i n t of a d h e r e n c e of E # N (pE(1 - p)) in E F . B y P r o p o s i t i o n 4.4.2.1, there are y, z E E such t h a t
x - py(1 - p) + (1 - p ) z p . Hence
px(1 - p) + (1 - p)xp = py(1 - p) + (1 - p)zp = x . Assume that (1 - p ) x p :/: O . Let cr be a real n u m b e r ,
Then
( P r o p o s i t i o n 4.1.2.23). O n the other h a n d ,
_< sup{ 1, c~ I(1 - p)xpll
} = ~11(1 -
p)xpll
for every y C E # ( P r o p o s i t i o n 4.1.2.23). Since x + a ( 1 - p ) x p a d h e r e n c e of E # A (pE(1 - p)) + a ( 1 - p)xp in E F , we deduce the c o n t r a d i c t i o n
is a p o i n t of
314
4. C*-Algebras
(Proposition 1.2.6.6). Hence (1 - p ) x p = 0,
x = p x ( 1 - p) 9 E # n (pE(1 - p ) ) ,
and E # n (pE(1 - p)) is a closed set of E F . By Corollary 1.3.7.5, p E ( 1 - p) is a closed set of E F . Consider u :E
v:E
>E,
;E,
x,
x,
>p x p ,
> (1-p)x(1-p),
w :E
> E,
z,
>p z ( 1 - p ) ,
t:E
>E,
x,
>(1-p)xp.
u, v, w, t are projections in E and UV
~
V?.t
--
?.tW
~
WU
--
VW
--
W'V
z
O.
By the above considerations and by Proposition 4.4.2.2, the sets Im u , Im v, Im w , Im t are closed in E F . From Kert - Imu + Imv + Imw it follows t h a t Ker t is closed in E F (Proposition 1.3.7.6). By Corollary 1.3.7.12 b ==~ a, the map EF
~ EF ,
x,
> (1-- p)xp
x l
> p x ( 1 -- p)
is continuous. Hence the m a p EF ~
EF ,
is also continuous. Since x p -- p x p + (1 - p ) x p ,
p x = p x p + p x ( 1 - p)
for every x E E , we deduce that the m a p EF ~
EF,
is continuous (Proposition 4.4.2.2).
x,
>xp
(resp. p x ) I
4.4 W*-Algebras
T h e o r e m 4.4.2.4
( 0 ) EF
315
Given x C E , the map > EF,
y'
>xy
(resp. yx)
is continuous. The assertion is easy to see if ]K = IR. So assume ]K - r
Let G be the
set of x C E for which the map E#F
>E p ,
y ~ > xy
(resp. yx)
is continuous. G is obviously a closed vector subspace of E . By Proposition 4.4.2.3, Pr E C G , so that G = E by Theorem 4.4.1.8 g). The assertion now follows from Proposition 1.3.7.11 b =~ a. C o r o l l a r y 4.4.2.5
( 0 )
I
Given A C E , A ~ is a closed set of E F .
Given x c E , put u~ : E F
> EF ,
y J
~ xy--
yx.
By Theorem 4.4.2.4, ux is continuous, so that Kerux is closed in E r for every x C E . The assertion now follows from AC= N K e r u s "
I
xEA
C o r o l l a r y 4.4.2.6
Let A be a commutative set of E and x a point of ad-
herence of A in EF (this is the case, for example, when there is a downward directed subset B of A with x as infimum). Then A U {x} is commutative. Since A c is closed in EF (Corollary 4.4.2.5) and contains A, we see that x E A c , i.e. A U {x} is commutative. The assertion in parentheses follows from Theorem 4.4.1.8 b). C o r o l l a r y 4.4.2.7
I ( 0 )
Every maximal Gelfand C*-subalgebra G of E
is closed in EF and so it is order faithful, order complete, and a W*-algebra and a(G) is a compact hyperstonian space. I f A is a downward directed set of G and x is its infimum in E , then x E G and
z' c ~(a) is meager.
~(x') < inf ~(x') yEA
316
4. C*-Algebras
By Corollary 4.1.4.2 b),c), G cc is a commutative C*-subalgebra of E containing G and by Corollary 4.4.2.5, G cc is closed in EF. If IK =([J, then G cc is a Gelfand C*-subalgebra of E . If IK = IR, then Gc~N Re E is a Gelfand C*-subalgebra of E . Moreover, in this case, the map EF ---+ EF ,
y ~ ~ y*
is continuous, so that Re E and G~Cn Re E are closed in EF. Since G is maximal, G - G ~ in the complex case and G = G~M Re E in the real case. Thus G is closed in EF in both cases. By Theorem 4.4.1.8 d), G is order faithful, order complete, and a W*-algebra. By Corollary 4.4.1.10, or(G) is a compact hyperstonian space. Since G is an order faithful set of E , x c G and by Proposition 1.7.2.13 a =~ c,
(
o
x ' E a(G)
o }
~'(x')< inf ~(x') yeA
is meager.
I
C o r o l l a r y 4.4.2.8 Let A be a commutative downward directed set of R e E , x its infimum, and f an increasing upper semicontinuous real function on a(x) U { U a(y)}. Then (f(Y))vem is a downward directed family in E with
yEA
i n ~ m ~ m f (~) .
Let G be a maximal Gelfand C*-subalgebra of E containing A (Corollary 4.1.2.3). By Corollary 4.4.2.7, a(G) is a compact hyperstonian space, x E G, and {x' c a(G)
~ ( x ' ) < infyEAy ( x t ) }
is a meager set of a(G). Since f is upper semicontinuous, {x' c a ( G ) f
o ~(x')< O infy efA
y'(x')} C
and
A
foy<_fo~
for every y C A. Moreover,
{A ) fox#fo~ is meager. From
y(x')}
{Xt E dr(G) ~(X')< infyEA
4.4 W*-Algebras
x' 9 ~(a)
A
A}
f(x)(x') < inf f(y)(x') yEA
{A
} t2{x'Ca(x) If~
C fox#fo~
317
C
it follows that
x' e a(G) f(x)(x') < inf f(y)(x') yEA
is a meager set of or(G). Let z be the infimum of {f(y) I Y 9 .4} in E . By Corollary 4.4.1.12,
(f(Y))veA is downward directed. By Corollary 4.4.2.7, z 9 G. Since f(x) <_ z, it follows that
{ft(X) 7s
C {x'E(:r(G ) f(x~)(x') < .infy f(y~)(x')} E A
Hence
is meager, f(z) = z,
m
and f(x) is the infimum of (f(Y))vcA.
Remark. The above relation need not hold if f is lower semicontinuous. Let x be an invertible element of E+ and consider
f.[0,
llxll]
IK,
a,
~~ 1
t Then
0
if a # 0 if a = 0 .
A in x = 0 (Corollary 4.2.2.2) while f(~x) = 1 for every n e IN. nc]N
C o r o l l a r y 4.4.2.9
( 0 ) The closure in EF of any involutive subalgebra of
E is a C*-subalgebra of E. Let G be an involutive subalgebra of E and G its closure in E F . Consider
f " EF
>EF , X:
; x*
318
4. C*-Algebras
By Proposition 4.4.1.3, f is continous. From -1
-1
G - - f (G) C f ( G ) , we see t h a t
-1 f(G),
GC w
i.e. G is an involutive set of E . Take x E E . Consider gl " EF
> EF ,
yl
)x+y,
g2" EF
) EF,
y,
) xy,
ga" EF
~ EF,
y,
) yx.
gl, g2, and g3 are continuous (Theorem 4.4.2.4). If x E G , then 3
3
G C ~ - 19i (G) C N -1 9i (--e) , i=1
i=1
so that e
3 C [r - 1 i--1
It follows that 3
GCN
-1
i=1
and so
3
a c
g~(U), i=1
for every x E G. Hence G is an involutive subalgebra of E . Being closed, it is a C*-subalgebra of E . Corollary 4.4.2.10
II ( 0 )
F is an involutive E - s u b m o d u l e of E' and the
maps EF
> FE ,
y ~ > ya
( resp.
ay) ,
FE
) FE ,
b,
(resp.
bx) ,
) xb
~,.~ W*-Algebras
319
are continuous for all a C F and x E E . Moreover, the map
E
~F',
x,
>(x,')lF
is an isometry of involutive E-modules. In particular, for IK = ~ , if we restrict the scalars of E to IR, then E becomes a real W*-algebra.
By Theorem 4.4.2.4, the map EF
~, IK ,
y,
~ (yx, a)
(resp. (xy, a) )
is continuous. Hence xa (resp. a x ) belong to F (Proposition 1.3.7.7 c => a) and the map EF
~ FE
y '
> ay
(resp. ya)
is continuous. By Proposition 4.4.1.3, F is an involutive E - s u b m o d u l e of E ' . Since the map FE
~ ]K ,
b,
~ (yx, b)
(resp. (xy, b) )
is continuous for every y C E , the map FE
>FE
b,
> xb
(resp. bx)
is continuous. By Proposition 2.2.7.5, jE is a homomorphism of E - m o d u l e s . It is easy to see that jE is involutive. Since the map E
~F',
z,
)(x,')lF
is an isometry of Banach spaces, it is an isometry of involutive E - m o d u l e s . I C o r o l l a r y 4.4.2.11
Take A C E and B C F . Put
CA := {a c F i x E A ~
ax = xa} ,
cB := {x C Ela C B ~
ax = x a } .
Then CA is a weakly closed set of F and CB is a closed set of E F . If B is involutive, then CB is a closed C*-subalgebra of E F .
320
4. C*-Algebras
Given a C F and x E E , define ua " EF
~ FE ,
y ~ > ay - y a ,
v~" FE
>FE,
b,
and ) bx-xb.
Since CA -
Kerv:,:,
CB =
xEA
Kerua, aEB
the first assertion follows from Corollary 4.4.2.10. The second assertion follows from the first one and Proposition 2.3.6.3. Corollary 4.4.2.12
( 0 )
I
I f we endow E # with the topology of u n i f o r m
convergence on the weakly compact convex sets of F ,
then, given a E F , the
map
E#
~IK,
z,
~(x*z,a)
is continuous. Take Xo C E # and e > O. Since E # is a compact convex set of E F , the set
A:={ax*ixcE is a weakly compact convex set of F
#}
(Corollary 4.4.2.10). Hence there is a
neighbourhood V of x0 in E # with c
I(x,b) - (xo, b)l < ~ ,
c
](x,a*xo) - (xo, a*x~))l < -~,
for every b c A and x c V. Hence I(x*x, a) - (x~)xo, a)l = I(x, ax*) - (Xo, ax~))l < < I(x, ax*) - (Xo, ax*) + I(Xo, ax*) - (Xo, axe) I = I(x, a x * ) -
-
(xo, ax*)] + I(x*,xoa) - (Xo, X o a ) ] c
c
= I(x, ax*) - (xo, ax*)l + I(x,a*x~) - (xo, a*x~)[ < -~ + -~ - c for every x C V. Thus the map E# is continuous.
) ]K,
x,
> (x*x,a) I
4.4 W*-Algebras
Corollary
4.4.2.13
321
( 0 ) Take x' E E~+. If x' ~_ a for some a C F , then
x' belongs to F . Take E # with the topology of uniform convergence on the weakly compact convex sets of F . Then
Ix'(x)l ~ ~ x ' ( x * x ) x ' ( 1 ) < (x*x,a)x'(1) for every x E E # (Proposition 2.3.4.6 c)), so lim x' ( x ) -- 0
x-+0
by Corollary 4.4.2.12. By Proposition 1.3.7.9 a =:~ d, x' c I m j F . Corollary 4.4.2.14
I
Given a E F , define ~d" E
>F ,
x,
>ax
(resp. xa)
(Corollary ~.~.2.10). a)
ff A is a nonempty downward directed set of F with infimum 0 and if is its lower setion filter, then lim~ - 0. a,~
b)
For every x E E , then map F
>F,
a,
>ax
(resp. xa)
is order continuous. a) First assume IK = ~ . By Proposition 4.4.1.3, E+# is a compact set of
EF and by Theorem 4.4.1.8 a), lim(x, a) - - 0 a,~:
for every z C E . Choose r > 0 and ao C A. By the above considerations and by Dini's Theorem, there is a b E A, b _< a0, such that C2
(x,a) <
1 + 16[[a0[[
for every a C A, a <_ b, and z E E+# . Thus
322
4. C*-Algebras
~.2
Ilaxll 2 < (x,a)l[all Ilx]l < -
-
6.2
Ila0[I <
1 + 1611a011
C
Ilax[ < -
i-6'
4'
for every a E A, a <_ b, and x E E+# , by Proposition 4.3.1.4 (and Corollary 4.2.8.3). It follows that c
c
c
Ilaxll = Ilaz + - az-II < Ilaz+ll + Ilaz-II < ~ + ~ = for every a c A, a < b, and x E Re E # (Theorem 4.2.2.9 a),b)). Furthermore, c
II~*all -
lazll -
Ila re x + ia im ~11 -< Ila
re
c
zll + Ila i n zll < ~ + ~ = c
for every a c A, a < b, and x C E # (Proposition 2.3.2.7). Hence lim~ = 0 a,~
Let now IK = IR. By Theorem 4.4.1.2, the complexification of E is a W * algebra and the assertion follows from the above considerations and Corollary 4.3.6.2 d) (and Corollary 4.4.2.13). b) follows from a).
I
P r o p o s i t i o n 4.4.2.15
Let 3 be the set of closed left ideals of EF
( 0 )
ordered by inclusion. Then Ep E :J for every p C Pr E and the map >3,
PrE
p~
> Ep
is an isomorphism of ordered sets. Take p C Pr E . By Proposition 4.4.2.3, the map u : Ev
> EF ,
x J
) x(1--
p)
is continuous. From
E p = Ker u it follows that Ep is closed in EF. Since it is obviously a left ideal, it belongs to 3. Take G c 3. Consider
H:=GnG*. G* is obviously a right ideal of E . By Proposition 4.4.1.3, G* is a closed set of EF and so H is closed in EF and a C*-subalgebra of E . Define
4.4 W*-Algebras
A:=
323
{ x E H 10 < z < 1}.
Let B be a totally ordered nonempty set of A. By Theorem 4.4.1.8 d), the supremum of B in E belongs to A. Hence A is inductively ordered and therefore possesses a maximal element p (Zorn's Lemma). By Proposition 4.2.7.8, pCPrE
and x p = x
for every x E A .
Take x C G # . Then x*x E A (Corollary 4.2.1.17 a =~ b) and we get successively that [Ix(1 - p)I12 = 11(1 - p)x*x(1 - p)[I = 0,
z(1 - p) = 0 ,
x = xp C E p ,
G# C E p ,
GcEp,
G-
Ep.
The map PrE
>3,
p,
is therefore surjective. Take p,q C P r E . If p < q, then p - - pq
(Corollary 4.2.7.6 a ==v c) and so Ep C E q .
Conversely, if Ep c E q ,
then pC Eq,
>Ep
324
4. C*-Algebras
p q - - p,
p<_q
(Corollary 4.2.7.6 c =~ a). In particular, Ep= Eq~p=q
and the map PrE
>3,
p,
>Ep
is an isomorphism of ordered sets. C o r o l l a r y 4.4.2.16
I
( 0 ) Let G be a closed ideal of EF and u" E
the quotient map. Then there is a p E Pr E with G=pE=Ep,
( 1 - p ) E ( 1 - p) is a W*-algebra with 1 - p
(1-p)E(1-p)
as unit, and the map
> E/G
,
x,
> ux
is an isometry of unital C*-algebras (Theorem 4.2.6.5).
By Proposition 4.4.2.15, there are p, q E Pr E with G = pE = Eq.
We deduce that p E Eq,
q E pE,
p=pq-q.
Put v:(1-p)E(1-p)
>E / G ,
x,
>u x .
Then x=px=O
for every x E Ker v. Thus v is injective. Let y E E . We have
E/G
~.~ W*-Algebras
325
y - (1 - p)y(1 - p) = py + yp - pyp, so that v((1 - p)y(1 - p ) ) = u y . Hence v is surjective and v(1 - p ) = u l . v is therefore an isometry of unital C*-algebras (Theorem 4.2.6.6). By Proposition 4.4.2.2, (1 - p)E(1 - p) is a W*-algebra. Corollary 4.4.2.17
m
( 0 ") Every element of F has a right and a left carrier.
Every element of Re F has a carrier. Take a C F . Consider G:={xeElax=0}
(G:={xeElxa=0}).
G is a closed right (left) ideal of EF (Corollary 4.4.2.10). By Proposition 4.4.2.15, there is a q C Pr E with a = qE
(a = Eq).
Put p:-l-q. Then
aq=O
(qa = 0),
ap= a
( p a = a).
so that
Since
pG=pqE=
{O}
( G p = E q p = {O}),
p is the right (left) carrier of a. The last assertion follows from the first one and from Proposition 4.3.3.3. m
326
4. C*-Algebras
C o r o l l a r y 4.4.2.18
Take a, b E F + , a <_ b. If p and q are the carriers of a
and b, respectively (Corollary ~.~.2.17), then p <_ q. We see successively that b(1 - q) - O,
0_< ( 1 - q , a ) _ < ( 1 - q , b ) - ( 1 ,
b(1-q))=O,
(1--q,a) =0,
a(1 - q) = 0 (Corollary 4.3.1.5 a :=v c), a=aq,
p<_q I
(Proposition 4.3.3.2 b)). P r o p o s i t i o n 4.4.2.19
Let ~ be the set of faces of E+ which are closed in
EF. Then pE+p E ~ for each p E Pr E and the map PrE
>~,
p~
>pE+p
is bijective. By Corollary 4.3.4.4, pE+p is a face of E+ whenever p C P r E
and by
Propositions 4.4.1.3 and 4.4.2.2, this face is closed in EF. It is obvious that the map PrE--+r
p ~ >pE+p
is injective. In order to show that this map is surjective, take C C ~. Choose p, q C C n P r E . By Theorem 4.4.1.8 i), pV q C P r E and pVq<_p+q. Since p § q E C , it follows from Proposition 4.3.4.3 that p V q C C. Hence the set C N P r E
is upward directed. Let ~ be its upper section filter. By
4.4 W*-Algebras
Theorem 4.4.1.8 b),i), ~ converges in EF to the supremum p of C N P r E
327
and
pc CAPrE.
Take x E C. By Corollary 4.4.1.13, the carrier q of x belongs to C . Thus q~_P,
xp = xqp = xq = x
(Corollary 4.2.7.6 a ==~ c). It follows that x = pxp C pE+p,
C c pE+p.
Take y c p E + p . Then
y <_ Ilyllp,
ycC,
pE+p c C ,
pE+p -- C .
Hence the map PrE
>q:,
p~
>pE+p
is surjective.
I
T h e o r e m 4.4.2.20 u : G -+ E
( 0 )
Let G
be a closed C*-subalgebra of E F . I f
is the inclusion map, then u ' ( F )
is an involutive G - s u b m o d u l e
of G' and the map
G is an isometry.
~(u'(F))',
x,
>(x,.>lu'(F)
328
4. C*-Algebras
By Theorem 1.3.5.12 c), the factorization E ' / G ~ ~ G r of u' through E ' / G ~ is an isometry. Its restriction
v" F / ( F n C ~
~ u'(F)
is also an isometry. By Proposition 2.2.7.7 (and Corollary 4.4.2.10), u'(F) is a G - s u b m o d u l e of G' and it is easy to see, t h a t it is an involutive G-submodule of G ' . Let q . F --~ F/(F n C ~
be the quotient map. By Corollary 1.3.5.5, the map
r
( F / ( F AG~ '
> G,
x,
~q'x
is an isometry. Put
V" G - - ~ (u'(F))',
x,
~(x,.)lu'(F).
Take x c G and a E F . Then (r
ov),a) -- (q'((~ax) ov),a) = ((px) ov, qa) =
= <~x, vqa)= <~x, u'a)= (x, u ' a ) = (ux, a ) = <x, a). Hence r
o v) = x ,
((~X) 0 V = ~ D - l x ,
~
= (r
v -1 ,
and the map v. a
is an isometry.
> (,;(e))',
:,:,
> (r
o x -~
I
4.4.2.21 ( 0 ) A real C*-algebra is a real W*-algebra iff its complexification is a complex W*-algebra.
Theorem
~.4 W*-Algebras
329
h 0 1
Let G be a real C*-algebra and H a closed vector subspace of (G)' such that the map >H',
z,
>(z,-)lH
is an isometry. Put I := {a C G ' 1 3 b E G', (a,b) E H } . I is obviously an involutive vector subspace of G' (Corollary 4.4.2.10). Take x E G and a E [0, [[x[[[. Then
o~ < Ilxll = II(x, o)11 = II<(x, o), >IHII. Hence there is an (a, b) C H # such that a < I((x, 0), (a, b)) I = ](x, a) + i(x, b)[. Then
((x,a)a + ( x , b ) b , - ( x , b } a + (x,a)b) - ((x,a) - i(x,b)))(a,b) e H , [l<x,a)a § <~, b>bll ~ [l(@, a)a § (z, b ) b , - ( z , b)a + <x, a>b)ll-= ((x,a} 2 +
1 2 1 (x,b>2)~ll(a,b)ll <_ ((x,a) 2 + (x,b})3
(Corollary 4.3.6.2 a)). Thus
[[(z,')[II]((x,a} 2 + (z,b)2) 89> > [(x, (x, a)a + (x, b)b)[ = (x, a) 2 + (x, b) 2 ,
II<x,->llII ~ ((x,a) 2 + (x,b)2) 89> a . Since a is arbitrary,
I1~11 ~ II<x, .> 1/11 ~ II <x,-> 11 : Ilxll, [I(z,-) [Ill = [Izll. Hence the map
330
4. C*-Algebras
G
~I',
x,
~(x,.}lI
is an isometry. Let (x,a) 9 G x I . There is a b 9 G' such that (a,b) 9 H . Then (xa, xb) - ( x , 0)(a, b) 9 H ,
(ax, bx) - (a, b)(x, O) e H
(Corollary 4.4.2.10), so that xa, ax C I . The closure of I in G' is an involutive E submodule of G', so that E is a real W*-algebra. The converse implication was proved in Theorem 4.4.1.2. I P r o p o s i t i o n 4.4.2.22
Suppose IK = ~ and let E denote the underlying real
C*-algebra of E (Theorem 4.1.1.8 a)). Put
x"E
~IR,
x,
~re(x,x')
.for every x' C E' and F "= {~dla E F } .
a) x' e (F)' for every x' e E' and the map
E'
~(E)',
z',
)z'
N
is an isometry of involutive E-modules.
b)
F is an involutive E-submodule of (E)' and the map E
>(F)',
x,
><-,x>IF
is an isometry of involutive E-modules.
c) E is a real W*-algebra with F as predual. d)
The complexification of E is a complex W*-algebra.
a) follows from Proposition 2.3.6.8. b) follows from a) and Proposition 1.3.6.26 c). c) follows from b). d) follows from c) and Theorem 4.4.2.21 (and Theorem 4.1.1.8 a)).
4.4 W*-Algebras
Proposition
4.4.2.23
331
( 0 ) For every x E E + , the space {YEEIy*y<X}F
is compact.
By Theorem 4.4.2.4, the map 1
9~ : EF
> EF ,
y t
> yx2
is continuous so that by the Theorem of Alaoglu-Bourbaki, ~p(E#)F is compact. We want to prove
{y ~ E ly*y ~ x} = ~a(E#) which proves the proposition. Take y C E # . Then y*y < 1 (Corollary 4.2.1.17 a ~ b), so that ,
!
i,
~
~-
1
(Corollary 4.2.2.3). Thus
~(E ~) c (y c E ly*y ~ x}. Now take y C E with y*y <_ x . For every n C IN put
A~ : : {~ e o(x) I ~ > 1 } , n
1
a-~ fn " a(X)
) ]K,
at
if a E An
)
0
if a C a ( x ) \ A n ,
Then for n C IN, Y n*Y n
: xny*yx~ < XnXXn
-
e A ~ ) (x) < -- 1
(Corollary 4.2.2.3, Corollary 4.3.2.5 a),b)) and Yn c E # . Hence y~(~)(x)
Put
: yXnX 1 -- y ~
e ~(E#)
.
332
4. C*-Algebras
o(x)
P := %(x)\{0}(x). By Theorem 4.3.3.8 a ==v b (and Theorem 4.4.1.8 j), Corollary 4.3.3.9), p is the carrier of x. By Proposition 4.3.3.2 d),
y*y = y*yp, so that ll~ - ypll ~ = II(~* - p y * ) ( y - yp)ll = Ily*y - p y * y - y * y p + py*ypll = 0 ,
y=yp. \
is an increasing sequence in E with p as supremum ) Since (A(~ e )(x)_n~iN (Theorem 4.3.2.5 a),b)), it follows that it converges to p in EF (Theorem 4.4.1.8 b)). Thus for every a E F ,
(y, a) = (yp, a) = (p, ay) -- n-~lim (eA(~)(X), ay) =
= n-~lim(YeA(~)(x),a). Hence (YeA(:)(x))n~XN converges to y in EF and y C ~(E #)
{~ e P r o p o s i t i o n 4.4.2.24 a)
E l~*y
_<~}
c ~(E#).
(Kaplansky) Let p,q C P r E .
(1-p) A(1-q)=l-pVq.
b) p - p A q is the right carrier of (1 - q)p. c) p V q - q
is the left carrier of ( 1 - q)p.
a) F r o m
1-pVq~_l-p,
1-pVq~_l-q,
it follows
1-pVq< From
(l-p)
A ( 1 - q).
m
4.4 W * - A l g e b r a s
1--(1-p)
A ( 1 - q ) _> l - ( 1 - - p ) = p ,
1- (l-p)
A ( 1 - q) _> 1 - (1-- q) = q
333
it follows pVq<
1 - (1--p) A ( l - q ) ,
(1-p) A(1--q)_< 1-pVq,
(1-p) A(1--q) =l--pVq. b) By Corollary 4.2.7.6 a ~ b, (1-q)p(p-pAq)=(1-q)(p--pAq)=
= p -- p A q -- qp + p A q = p -- qp = ( 1 - - q ) p .
Take x c E with (1 - q ) p x = O. Then px = qpx, SO
p x -- p q p x .
By induction px -
for every n E N. ( ( p q p ) n ) n e N
( p q p ) '~x
is a decreasing sequence in E+ with infimum
p A q (Corollary 4.2.7.13). By Theorem 4.4.1.8 c), lim ((pqp)~, a) -- (p A q, a) , n---+ (:x:)
so that ((p A q ) x , a )
= (p A q, x a ) = lira ( ( p q p ) " , x a ) n --+ ( X )
= n lim ((pqp)"x,a) -'-~ ( X )
334
4. C*-Algebras
for every a E F (Corollary 4.4.2.10). It follows ( (p - p A q)x, a) = (px, a) - ( (p A q)x, a) =
= (px, a) - l i m ((pqp)nx, a) -- (px, a) - (px, a) -- 0 for every a E F and therefore ( p - p A q)x = O. Hence p - p A q is the right carrier of (1 - q)p. c) P u t pX := l - p ,
qX := l - q .
By b), p• - p• A qX is the right carrier of (1 - qX)p• = q(1 - p) = ((1 - p)q)* Hence p• - p• A qZ is the left carrier of (1 - p)q (Proposition 4.3.3.3). By a), p•
l--p-(1--pVq)=pVq-p.
I Remark.
a) follows also from Corollary 4.2.7.13 and Proposition 4.2.7.14.
4.4 W*-Algebras
335
4.4.3 P o l a r R e p r e s e n t a t i o n T h e o r e m 4.4.3.1
( 0 ) For every x C E , there is a unique y C E with
1) x = y ] x ] ; 2) y*y is the carrier of Ix] and the right carrier of x ; 3) yy* is the carrier of ]x*] and the left carrier of x . (y, Ix[) is called the polar r e p r e s e n t a t i o n o f x .If x is invertible then the above definition coincides with the one given in Theorem ~.2.6.9.
Given n c IN, put Xn :=
(1
-l+x*x
,
n
1
yn'--X
1 +X*X
Then * YnYn --
1
1
( 1 ) - 2 ( 1 --1 + x*x n
)-2
x*x
1 + x*z
( 1 ) -- x*x
--1 + x*x n
-1 ,
so that []yn]] _< 1 for every n C IN. Let z be a point of adherence of (yn)ne~ in EF# (AlaogluBourbaki Theorem). Then zlx I is a point of adherence of (y~lxl)ne~ in EF (Theorem 4.4.2.4). Take e > 0. Since lim xn = Ix], n---~ OG
there is an no E IN with
liz~ -I~l Ji < for every n E IN, n > no. Thus
336
3. C*-Algebras
for every n C IN, n >_ no and hence IlzlxJ - xll <_ lim s u p l l y n l x l - xll <_ e n--~oo
(Proposition 1.2.6.6). Since e is arbitrary, we deduce that x = zlxl.
Let p (resp. q) be the carrier of I~1 (of I~*l) (Theorem 4.4.1.8 j)) and put y := q z p . By Corollary 4.3.3.10 b ~ a & c, p (resp. q) is the right (left) carrier of x. Then x = qx = q z l x I = qzplx I = y]x],
Ixlplxl - IxJ2 - x*x = I x l y * y l x l ,
[xl( p -
y ' y ) x I - O.
Since z c E # , we get successively that
IIz*z
I = :lzll ~ _< 1,
z'z<1 (Corollary 4.2.1.17 a ==> b), y*y = pz* qzp < pz* zp < p
(Corollary 4.2.2.3), O
(p-y'y) 89
< p
(Corollary 4.2.7.2),
Il(p- y*y)~lxl II~ -II I~l(p- y*y)lxl II- o, (p - y * y ) ~ l z l -
0,
4.4 W * - A l g e b r a s
(p-
y'y) 89 = (p-
337
y*y) 89 - 0
(Proposition 4.2.7.1 d ==~ f), y*y=p.
We have that yy*=
q z p z * q <_ q z z * q <_ q
(Corollary 4.2.2.3). By Corollary 4.1.2.22, y y * E Pr E and 9 x*yy* = xlxly*yy* = ~lzly* = ~ x * ,
so that Ix*l~y* = ix*l
(Proposition 4.2.7.1 d r
f, Corollary 4.2.7.2) q<
yy*
(Proposition 4.3.3.2 b)), yy*=q.
We now prove the uniqueness of y. Take z E E , so that x--
z[xl ,
z* z = p ,
zz* = q .
We get successively that z* ylxl = z*~ = z* ~i~i = p l x l , (p-
~*y)i~i = o,
(p - z* y ) p = O,
p - z * y = p - z * y y * y = p - z* y p = 0
(Corollary 4.1.2.22), z = zz*z = zp = zz*y = qy = yy*y = y
(Corollary 4.1.2.22). If x is invertible then the left and the right carrier of x are 1 (Proposition 4.3.3.5 b)). Thus y is unitary and the above polar representation of x coincides with the one defined in Theorem 4.2.6.9. m
338
4. C*-Algebras
Corollary 4.4.3.2
( 0 )
Take x E E
and let (y, lxl) be its polarrepresen-
tation.
a)
The following are equivalent: al ) x is normal.
a2)
y is normal and
ylzl = Ixjy.
a3)
y is normal and xy = y x .
If x is self-normal then y = f (x), where I "~(x)
~ lZ,
~ ,
~
~
0
b)
if o ~ # o ifa=O.
The following are equivalent: bl ) x is selfadjoint.
b2)
y is selfadjoint and ylzl = Ixly.
bs)
y is selfadjoint and xy = y x .
If these conditions are fulfilled then x § = y§
= ~§
~- = y-I~l = y x
al =v a2. We have Ix] = Ix*l, so that y*y = yy*,
i.e. y is normal. We deduce successively that
4.4 W*-Algebras
Ixl ~ = x x *
= ylxl~y
* ,
Ixl~y = ylxl~y*y = ylxl ~ ,
(Proposition 4.2.4.2 b)). a2 :=> a l . We have xx*
= yfxl~y * =
Ixl~yy * =
fxf~y*y
=
Ixl ~ = x * x .
a2 =~ a3. By Theorem 4.1.4.1, y*x = y*yJzl = Ixly*y = I~l,
9 y* = I~lyy * =
Ixly*y = I~1 = y * z ,
xy = yx.
a3 =~ a2. By Theorem 4.1.4.1, xy* = y ' x ,
so that y x * -- x * y ,
~l~l ~ = y ~ * x = ~ * x y = I~l~y,
and ~lxl = I~ly
by Proposition 4.2.4.2 b). Now we prove the final assertion in a). Put j:cr(x)----+lK,
g:~(~)
Then
,~,
a,
~'
>o~,
~1~1.
339
340
4. C*-Algebras
(Proposition 4.2.5.4) and fg=j
so that --
From Ifl 2 = %(x)\{0} :(~) it follows
f*(x)f(x)
=
]fl2(x)
= e~{o:~ (x)
:
e~{o)(Ixl)
9
By Corollary 4.3.3.14, f * ( x ) f ( x ) is the carrier of x and of Ixl so that (f(x), Ixl) is the polar representation of x. Hence y = f(x).
b) First suppose that x is selfadjoint. By a), y is normal and
We get x = ~* = : l z l ,
and so (y*, Ixl) is the polar representation of x. This implies that y* = y. Moreover,
and by the Fuglede-Putnam Theorem, (y§
= y§
~ = o,
(y-I~l)(y§
= y-y+lxl
~ = 0.
Hence x + = y§
x-
= y-Ix1.
4.4 W * - A l g e b r a s
341
It follows t h a t y+x- = y+y-lxl = 0,
y-~+ = y-y§
= 0,
so t h a t y+J~l = y §
§ + ~-) = y§247
y-lxl = y-(~+ + ~-) = y 9 9
The implication be =~ bl is trivial. The implication b3 =~ be follows from a) and the F u g l e d e - P u t n a m Theorem.
Proposition 4.4.3.3 ( 0 ) x*x C PrE,
Take
m a C F.
T h e r e is a n x E E
a x , x a E F+ , x*xa = axx* = a.
I f a is s e l f a d j o i n t ,
then we may take x
to be also s e l f a d j o i n t .
We may assume t h a t Ilall = 1. There is a y E E # with (a, y) = 1. By T h e o r e m 1.3.1.10 b), there is an extreme point x of E # with (a, x) = 1. By Theorem 4.3.3.20 a :=~ b & d, x ' x , (1 - x x * ) E ( 1
x x * E Pr E and
- x ' x ) = {0}.
Thus (1 - x* x ) a ( 1 - x x * ) = O,
a(1 - x x * ) = x* x a ( 1 - x x * ) .
Now 1= (x,a)=
(1, a x } <_ [[axll _< Ila[[ I[x[[ <_ 1,
1 = ( x , a ) = (1,xa} < I[xall <_ Ilxll I]all <_ 1, (1, ax) = Ilaxll ,
(1,xa) = Ilxal[
such that
342
4. C*-Algebras
and ax, x a E F+ (Corollary 4.3.6.1 g)). We deduce t h a t a x = (ax)* = x* a* ,
x a = (xa)* = a* x* ,
axx* = x* a*x* = x* x a ,
a(1-xx*)
=axx*(1-xx*)
=0,
x * x a = axx* = a .
If a is selfadjoint, then we may take the above y to be selfadjoint (Proposition 2.3.2.22 1)) and x to be an extreme point of Re E # . By Proposition 4.3.3.21 a ==>c, x is an extreme point of E # , so the above considerations apply.
I Proposition 4.4.3.4
(0)
Take ( y , a ) E E x F
with y*y E P r E , ya E F+,
and y*ya = a. Put
b := y a . Let q be the carrier of b (Corollary ~4.~.2.17) and p u t
x := y*q. Then x * x = q,
x b = a,
b = x*a,
Iiail=llbil.
We have yy* E Pr E (Corollary 4.1.2.22 a =~ b,) and y*b = y*ya = a,
yy*b = ya = b,
so t h a t q
4.4 W*-Algebras
343
(Proposition 4.3.3.2 b)), x*x - qyy*q - q
(Corollary 4.2.7.6 a =~ e), xb = y*qb = y*b = y*ya = a,
b = qb = x*xb = x*a.
We deduce that
IlaII = [Ixbll _~ Ilxll Ilbll ~ IIbll- IIx*all _< IIx*ll Ilall ~ Ilall, IJalJ : IJbll. T h e o r e m 4.4.3.5 a)
m
( 0 ) (Sakai, 1958) Let a E F .
There is a unique (x,b) E E x F+ such that x*x is the carrier of b and a:xb.
We set
lal : : b
(the modulus o f a ) ,
and call (x, b) the polar r e p r e s e n t a t i o n o f a.
b) xx* is the carrier of la*[ and let left carrier of a. c) I f s C E c and if (t, isl) is the polar representation of s (Theorem ~.~.3.1), then (tx, IsI la]) is the polar representation of s a . In particular, Isa I = ls ]la]. Step 1
Uniqueness in a)
By Theorem 4.4.1.2 (or Corollary 4.4.3.7), we may assume IK = ~ . Take (x, b), (y, c) e E x F+ with xb = y c ,
such that x*x (rasp. y ' y ) is the carrier of b (of c). We have, successively, that b = x*xb = x*yc - x*ycy*y,
344
4. C*-Algebras
b -- b y * y ,
x * x < y*y
( P r o p o s i t i o n 4.3.3.2 b)), x*x = y'y,
Ilxbll ~ Ilxll Ilbll ~< Ilbll <~ IIx*[I Ilxbll ~ Ilxbll, Ilbll = Ilxbll = Ilycli- I1~11Put p:=x*x,
u := r e ( x * y ) ,
v := i m ( x * y ) .
Then px* = x * x x * = x * ,
py* = y*yy* = y*
( C o r o l l a r y 4.1.2.22 a =~ c), so t h a t 1
,
1
p u = -2P(X y + y ' x ) = -~(x* y + y ' x ) -- u ,
1 p v -- ~ p ( x * y -
~p
-
(p~)*
1 y'x) = ~(x*y
-
u,
vp
-
- y'x) - v,
(pv)*
-
v.
Since (c, u ) , (c, v) e IR (Corollary 2.3.1.23 a =:~ b) and (c, u) + i(c, v) = ( c , x * y ) = (z*yc, 1) = (b, 1) -
Ilbll - Ilcll -
(Corollary 2.3.4.7), we see t h a t
(c, ~) - li~li, (c, p - u) = (c, p) - (c, u) - (,pc, 1) - Ilcll - (c, 1) - licil - 0.
From
i~li < il~*vll <__ I1~*11 liyll <
4.4 W * - A l g e b r a s
345
(Proposition 2.3.2.7), we deduce successively that u
(Corollary 4.2.2.3), p-u>_O,
(p-
u)c = 0
(Corollary 4.3.1.5 a ==~ b), o -
(p-
,~)p -
p-
~,
p--u,
x * y -- u § iv -- p + i v ,
1 >_ II~*yll ~ = lip + i v l t ~ = II(p - i ~ ) ( p + i~)11 = lip + ~ l l .
The C*-subalgebra G of E generated by p and v is a Gelfand C*-algebra (Corollary 4.1.2.3). Identifying G with C o ( o ( G ) ) and using the above relation, we get 1 _> [[p+ v2[[ - [[p[ + [[v[[2 -- 1 + [ v i i 2, since we may assume p ~-0. Hence Ilvll = 0,
v--O~
x*y -- u - - p ,
b-pc-
c,
346
4. C * - A l g e b r a s
(x -
x-
y = xx*x
-
y)b = xb-
y c = O,
yy*y = xp-
yp = (x - y)p = 0
(Corollary 4.1.2.22 a =~ c ), x=y.
Step 2
The existence in a) and b)
By Proposition 4.4.3.3, there is a z C E with
z*z
C
PrE,
az,
za C
F+,
and z* z a = a z z *
= a.
Put b := z a C F + ,
c := z*a* = (az)* = az C F+,
and let p (resp. q)denote the carrier of b (resp. c)(Corollary 4.4.2.17). Put x:=z*p,
y := zq.
By Proposition 4.4.3.4 (and Corollary 4.1.2.22) x*x = p,
xb = a,
y*y = q ,
Thus
xx* C
b = x'a,
Ilall
c=y*a*.
y c = a* ,
Pr E, xp = xx*x
= x
(Corollary 4.1.2.22 ), ap = xbp = xb = a,
pa* = a*,
( q a ) * = a * q = y c q = y c = a* ,
qa--a.
-Ilbll,
~.~ W * - A l g e b r a s
We deduce successively that x* c = p z c = p z z * a* = p a * = a * ,
xx*c
-
xa* = xbx* = ax* = apz -
y*y-
az -
c,
q ~ xx*
(Proposition 4.3.3.2 b)), (qx - x)b = qxb - xb = qa -
0 = (qx-
x)p = qxp-
a = O,
xp = qx-
x,
qxx* = xx*,
xx*
(Corollary 4.2.7.6 d :=> a), xx* = q.
Let w E E such that wa-
O.
Then wxla I -- wa
= O,
and we get 0 -
wxp
= wxx*x
wq = wxx*
= wx,
= 0
(Corollary 4.1.2.22). Hence q is the left carrier of a. Step 3
c)
E c is a C*-subalgebra of E (Corollary 4.1.4.2 b)) and IsIc E c,
34 7
348
4. C*-Algebras
1~1 lai = ial i~l e F§ ( P r o p o s i t i o n 4.2.4.2 d),g), C o r o l l a r y 4.4.2.10). For every r E E , we have successively, rt*tlsl =
~1~1 : I~1~ -
t*tlslr = t*trlsl,
(rt*t- t*tr)lsl 0 = (rt*t-
= 0,
t*tr)t*t = rt*t-
t*trt*t,
rt*t = t * t r t * t ,
r*t*t = t * t r * t * t ,
t*tr - t*trt*t - rt*t ,
so t h a t t*t C E c . It follows (tx)*(tx) = x*t*tx = t*tx*x e Pr E
(Corollary 4.2.7.4 a =~ b), (tx)*(tx)lsl lal-
t*tx*xlal
I~1 =
t*tlal
I~1 =
t*tlsl lal = Isl lal.
Take r C E with
~1~1 iai = 0. Then
o = (rl~l)~*x = r ~ * x l ~ l ,
0 = rx*xt*t = r(tx)*(tx).
Hence ( t x ) * ( t x ) tion of s a .
is the carrier of is] ]a] and (tx, Is] la]) is the polar representaI
4.4 W*-Algebras
Corollary 4.4.3.6
Take a E F and let (x,
lal)
349
be its polar representation.
Choose y C E such that xx* < y*y C P r E . (This condition is fufilled if y is unitary.) Then l y a l - lal and (yx, lal) is the polar representation of y a .
We put q "- xx*,
r := y*y.
x'q-x*,
qx = x
Then q C P r E ,
(Corollary 4.1.2.22), and qrq -- q
(Corollary 4.2.7.6 a =~ e). It follows (yx)* (yx) -- x*y*yx = x*rx = x*qrqx - x*qx = x * x .
Hence (yx)*(yx) is the carrier of [a I . Since ya = y~lal,
it follows that (yx, lal) is the polar representation of ya and l Y a i - lal.
Corollary 4.4.3.7
m
( 0 ) Let IK = IR. Take a C F , and let (x,b) be
the polar representation of a. Then (using the notation in Theorem ~.~.1.2)
((x, 0), (b, 0)) is the polar representation of (a, 0). By Proposition 4.3.6.1 a),b), ~
(a, 0 ) e (/~)',
o
(b, 0) e (E)+.
By Corollary 4.3.6.2 e),f), (a, 0) -- (x, 0)(b, 0) and (x, 0)*(x, 0) is the carrier of (b, 0). Hence ((x, 0), (b, 0)) is the polar representation of (a, 0). II
350
~. C*-Algebras
Proposition 4.4.3.8
( 0 ) Let G be a closed C*-subalgebra of EF and
u : G -+ E the inclusion map. By Theorem ~.~.2.20, u'(F) is an involutive G-submodule of G' and the map
G
>(u'(F))',
x,
>(x,.)lu'(F)
is an isometry (i.e. u'(F) is a predual of G ) .
a) a e F+ =~ u'a e u ' ( F ) + . b)
Take a E F and let p be its left (right) carrier. If p C G , then p is the left (right) carrier of u'a.
c) Take a E F and let (x, lal) be its polar representation. If x E G , then (x, u'lal) is the polar representation of u'a. a) follows from Corollary 4.2.2.11. b) Consider A:={qCPrE
I qa = a}
(A := {q C Pr E l a q = a}) .
p is the infimum of A (Proposition 4.3.3.2 b)). If p C G, then p is the infimum of ADPr G. By Proposition 4.3.3.2 b) and Corollary 4.4.2.17, p is the left (right) carrier of ua. c) x*x is the carrier of lal. By a), u'la I is positive and by b), x*x is the carrier of u'lal, i.e. (x, u'lal) is the polar representation of u'a (Proposition 2.2.7.6). m T h e o r e m 4.4.3.9
( 0 ) Take a C R e F and let (x,
lal)
be the polar repre-
sentation of a. Then x is selfadjoint and xa = ax,
a + = z+a
a-
= -x-a
xlal = lalx,
= az + = ~+lal
= -a~-
= ~-Ial
a + + a-
Ixl, x + , and x -
=
lal~ + e F+,
=
lal~-
e F§
= lal,
are the carriers of la], a + , and a - , respectively (and belong
therefore to Pr E).
~.~ W*-Algebras
Step 1
351
IK = C =~ x selfadjoint
Since a=a*, we see that X*X ~ XX*
(Theorem 4.4.3.5 b)). Let G be a maximal Gelfand C*-subalgebra of E containing x (Corollary 4.1.2.3). By Corollary 4.4.2.7, G is closed in EF and a(G) is a Stonian space. Let u : G --+ E be the inclusion map. By Proposition 4.4.3.8 e), u'a belongs to the predual u'(F) of G and (x, u'lal) is the polar representation of u'a. By Theorem 4.4.1.8 c), u'a is order continuous. By Example 1.7.2.14 c), x is selfadjoint. Step 2
]K = ]R =~ x selfadjoint
By Corollary 4.4.3.7, ((x, 0), (la], 0)) is the polar representation of (a, 0). Since (a, 0) is selfadjoint it follows from Step 1 that (x, 0) is selfadjoint, i.e. x is selfadjoint. Step 3
The other assertions
We deduce, successively, that xlal
x§
= a = a* =
= lalx §
x-lal
lalx,
= lalx-
(Corollary 4.2.8.16), x a = x + x ] a ] - x +]a]-]a[x + - [ a ] x x + - a x +, -x-a
-
-x-xlal
= ~-Ial-
lalx- = -la ~-
= -a~-
(Proposition 4.2.2.9), x a = lal = l a l * =
ax,
(jFa) + = (jFx]al) + = x+jF(lal) = jF(x+lal) = jF(x+a),
352
4. C * - A l g e b r a s
( j F a ) - - -- ( j F x l a l ) -
--
x-jF(lal)
=
jF(x-lal)
= --jF(x-a),
(Corollary 4.2.8.16, Proposition 2.2.7.5). By Proposition 4.2.2.9 and Theorem 4.4.3.5 b), Ixl is the carrier of lal and a and x +, x -
e PrE.
It follows
a+,a-CF, ~ + ~+ = ( ~ + ) ~
~
= ~ + ~ = ~+ ,
= -(x-)~a
= -~-~
= a-,
= x a = x21al = lal.
a + + a- = (x + - x-)a
Take y C E with ya + = 0
(ya- = 0).
Then yx+ a = ya + = 0
(-yx-a
= ya-
= O)
so that yx + = yx+lxl = 0
(yz- = -yx-Izl
= o).
Hence x + and x - are the carriers of a + and a - , respectively. Corollary 4.4.3.10
( 0 ) For every a C F, the map E
is o r d e r c o n t i n u o s ,
II
~F,
i.e. F = F
z,
)az
(resp.
za)
(Definition 4.3.1.7).
By Proposition 4.2.2.15 d) (and Corollary 4.3.6.2 e)), we may assume IK = r
Let A be a nonempty downward directed set of E with infimum 0 and let be its lower section filter. We need to prove that lim a x = lim x a = O . z,~
z,i~
Since a is a linear combination of four positive elements of F
(Theorem
4.4.3.9), we may assume a to be positive. By Theorem 4.4.1.8 c), inf (a, x) - 0
xcA
and the assertion follows from Corollary 4.3.1.6 a ==~ b & c.
II
~.~ W*-Algebras
C o r o l l a r y 4.4.3.11
353
Take a E F (resp. z E E ) and x, y E S n E xa = ay
such that
(resp. x z = z y ) .
Then for every bounded Borel function f f(x)a = af(y)
on a(x) U a ( y ) ,
(resp. f ( x ) z -- z f ( y ) ) .
Let B denote the C*-algebra of bounded Borel functions on a(x) U a(y) and put B := {f E B l f ( x ) a -
af(y)
(resp. f ( x ) z - z f ( y ) ) } .
Let (fn)nC~ be an increasing (decreasing) sequence in B with supremum (infimum) f in B. By Corollary 4.4.3.10, f (x)a = a f (y)
(by Theorem 4.4.1.8 c), (b, f ( x ) z ) = (zb, f ( x ) ) = l i m (zb, f,~(x)) =
= lira(b, fn(X)Z) = l i m ( b , z f n ( y ) } -- l i m ( b z , fn(Y)} - (bz, f(y)} = (b, z f ( y ) }
for every b E F , so that f(x)zN
zf(y)). ~
N
Hence f E B. Since C(a(x) U a(y)) C B (Theorem 4.1.4.1), B = B. P r o p o s i t i o n 4.4.3.12
I
Take a E F+ and x E S h E . If xa = - a x ,
then xa=ax=O.
First suppose that x is selfadjoint. Then x +a -
a(-x)
§ =
ax-
(Theorem 4.1.4.1, Theorem 4.2.2.9 b),d)). Let p and q be the carriers of x + and x - , respectively (Theorem 4.4.1.8 j)). Since
354
4. C*-Algebras
pq--O
(Proposition 4.3.3.15), we have that x +ap = a x - p
= a x - q p = O,
(a,x+} = {a, p l x +) = (x+ap, 1 ) = O,
IIx+all 2 ~ (a,x+>llx+ll Ilall = o
(Proposition 4.3.1.4), x + a = O,
ax- =0,
x - a = ( a x - ) * = O,
x a = (x + - x - ) a = O,
ax
=
(za)*
= O.
Now let x be normal. By Theorem 4.1.4.1, (rex)a = - a ( r e x ) ,
(imx)a = - a ( i m x).
By the above considerations, (rex)a=a(rex)=0,
(imx)a=a(imx)=0,
so that x a = a x = O.
C o r o l l a r y 4.4.3.13
m
L e t a c F+ a n d p the carrier o f a ( C o r o l l a r y 4 . 4 . 2 . 1 7 ) .
The map
p(ReE)p is injective.
>F ,
x,
>xa + ax
4.4 W*-Algebras
355
Take x C p(Re E)p such that
xa + a x = 0 . By Corollary 4.4.3.12,
xa=O, so that
x=xp=O.
Corollary 4.4.3.14 and
Let a E F+,
I
p the carrier of a (Corollary ~.~.2.17),
L := {b e F+ i b < a}, l (xa -4- ax) E L} . Then KF and LE are compact and the map KF
~ LE,
x l
1 (xa + ax) ~ -~
is a homeomorphism. By Propositions 4.4.1.3 and 4.4.2.2., pE#+p is a compact set of E F . The map
EF
~FE,
x,
1
~(xa+ax)
is continuous (Corollary 4.4.2.10). Since F+ is a closed set of FE, L is also a closed set of FE and therefore K is closed in (pE#+p)F. Hence KF is compact. Assume the map
K
~L,
x,
~-~1 (xa + ax)
is not surjective. Then there is a
b e L\{-~l (xa + ax) I x E pE#+p} By Proposition 2.3.2.24, there is a y C Re E such that sup {~1(xa + ax), y} <
Then
356
4. C*-Algebras
1
sup (-~(xa + ax),pyp) = xEpE#+ p z.
=
sup (1 ~ePE~+P -~p(xa + ax)p, y) =
sup (1 (xa + ax) y) < (b, y ) = (pbp, y ) -
~,~+, -~
(b, pyp)
(Corollary 4.4.2.18, Proposition 4.3.3.2 a)). If q denotes the carrier of (pyp)+ (Theorem 4.4.1.8 j)), then q C pE#+p (Example 4.3.4.2, Corollary 4.4.1.13) and therefore 1
1
(b, (pyp)+) >_ (b, pyp) > (-~(qa + aq),pyp) = (a, -~(pypq + qpyp)) = (a, (pyp)+) (Proposition 4.3.3.15). This contradicts the relation b < a. Hence the map
K -----+ L,
x,
1 (xa + ax) > -~
is surjective. In particular, LE is compact. By Corollary 4.4.3.13, the map
KF
~ LE,
1 (xa + ax) ) -~
X,
is injective, and so it is a homeomorphism.
Remark.
I
x E pE#+p does not imply l (xa + ax) > 0 2
as one can see from the example in the remark to the next corollary. C o r o l l a r y 4.4.3.15 (Theorem of Radon-Nikodym) Take a C F+. Let p be the carrier of a and take b C R e F such that Ibl <_ a. Then there is a unique x C p(Re E)p such that
b=~1 (xa + ax) We have IxI _< p. We have b+, b- C F+, b+ < Ib ,
b- < Ibl
(Theorem 4.4.3.9). By Corollary 4.4.3.14, there are y, z E pE#+p such that
4.4 W*-Algebras
= -~(ya+ay),
b--
357
(za+az).
Put
x "=y-z. Then x C p(Re E # ) p and
~(xa + ax)
b+ - b- = b.
The uniqueness follows from Corollary 4.4.3.13. By Proposition 4.2.7.1 c ~ a,
Izl <_p. Remark.
m
In general,
-i1 (xa + ax) + ~- -~1 (x+a + ax+) Indeed, putting a
"~-
211 1
we obtain
,
X
--
2
[10] 0
x~=[10l 0
1
(x+a+ax+)=-2
0
-1
'
1([10][~ 11 [~ 1][101) 0 0
1
2
2
0 0
1
1 2
2 +
+
1 2
0
2
1 0
31
0 0
0
=
'
SO
1
-~(x+a + ax +) is not even positive. Corollary 4.4.3.16
( 8 ) L~t ~ =~ OK- ~).
The absolute convex hull of F#+ is a O-neighbourhood in F (in Re F ) .
358
b)
4. C*-Algebras
If x is an (involutive) linear form on F such that x(F+) C JR+, then x9
c)
For every x 9 E+,
Ilxll = sup (x, a}. ae F#+
d)
E'+# is the closure ofF#+ in E'E. a) follows from Theorem 4.4.3.9. b) By Theorem 4.4.3.9, x ( R e F ) C IlL so that x is involutive (Corollary
2.3.1.23 b =~ a). By a) and Corollary 1.7.1.8 b), x is continuous so that x 9 E . By Corollary 4.4.1.5, x 9 E + . c) Take b 9 Re F # . Then b+, b- 9 F+~ (Theorem 4.4.3.9), so that
- ( . . b-) <_ (.. b) _< (.. b+) I(x,b)l _< sup (z,a). aeF+e By Proposition 2.3.2.22 j), Ilxll =
sup bCRe F#
I(x,b)l = s u p ( x , a } . aC:_F#+
d) Let F+# be the closure of F+# in E ~ . It is contained in E '# (Proposition 1.2.6.6). Take x' 9 E'+#\F~+. We may assume
IIx'll-
1. By Proposition
2.3.2.25, there is an x 9 Re E such that sup (x, a} < (x, x'}. a~F+e Put y := Ilzl[1 + x e E + . By c),
I]Yi] = sup (y, a> _< ]]x]] + sup (x, a> <
< I1~11 + (x,~') < (11~111 + x , ~ ' ) = (y,~') < Ilyll (Corollary 2.3.4.7), which is a contradiction. Hence F+# - E~ # .
II
~.~ W*-Algebras
P r o p o s i t i o n 4.4.3.17
359
Put
5"F
~F,
a,
>xa
for every x 9 E .
a)
~ 9
L(F)
b)
The map
and
I1~11- Ilxll for
E
~y
~ e E.
>s
x~
>
is an algebra homomorphism preserving the norms. We identify E with
{~l x E E} and denote by 9 the topology induced on E by the topology of pointwise convergence of s
c) ~ is finer (coarser) then the topology of EF (of E ). d)
The map E #•
)E,
(x,y),
)xy
is continuous with respect to ff;.
e) Pr E is a closed set of E with respect to ?~. f) For every f e C([-1, 1]), the map Re E #
~E,
x,
~f(x)
is continuous with respect to ~ .
g) Let A be an upward (downward) directed set of E ,
x its supremum
(infimum), and ~ its upper (lower) section filter. Then ~ converges to x with respect to ~ .
h) Every linear map of E into a normed space is order continuous, whenever it is continuous with respect to ff;.
i) For every a C F , the map E--~
F,
is continuous with respect to ~ .
x~
) xa
(resp. ax)
360
4. C*-Algebras
a) 2 is linear and
II~all = Llzall < Ilxll Ilall for every a E F . Hence ~ E / : ( F ) and
Ilxll- sup I(x,a>l = sup I(1,xa>l aEF#
aEF#
On the other side, =
sup Ilxall-
aEF#
sup II~all <_ I1~11
aEF#
so that b) Take x,y E E and a , ~ E IK. Then for every a E F , c~x + ~ y a -- ((~x + ~ y ) a -- (~xa + ~ y a -- ~ a
x~a = ( ~ y ) a -
+ ~a
-- ( ~
+ ~y~a,
~(ya) -- ~(ya) = ~ y a ,
so that c~x+fly=c~+~,
x~'-y=~.
Hence the map of b) is an algebra homomorphism. By a), it preserves the norms. c) B y a ) , f o r a E F
and x E E ,
which proves the assertion. d) follows from b) and Proposition 1.2.6.7. e) By c) and Proposition 4.4.1.3, Re E is closed with respect to ~ and by d), {x E E # 1 x 2 = x } is also closed with respect to T. Hence P r E is closed with respect to T. f) By d), the assertion holds if f is a polynomial. Since f is the uniform limit of polynomials, the assertion holds, by b), also for f . g) follows from Corollary 4.4.3.10. h) follows from g). i) is obvious.
1
4.4 W*-Algebras
361
4.4.4 W * - H o m o m o r p h i s m s P r o p o s i t i o n 4.4.4.1
( 0 )
Take x', y' E E+' with x'(1) < y'(1).
If
pAsupx,,p, = x,
sup ,p ,,,p, = y, ( A Ap) pE
for every well-ordered set A of Pr E , then there is a p E Pr E \ { 0 } with x' (pxp) <_ y' (pxp) for every x
E+.
Pr E is order complete (Theorem 4.4.1.8 i)), so by our hypothesis (and by Zorn's Lemma), there is a maximal element q of the set {r E Pr E lx'(r) >_ y'(r)}. Put
p'--l-q. It is obvious that p # 0. Take r E P r E with r _< p. Then r+q--r+(1-p)
EPrE
(Corollary 4.2.7.7), so that
x'(~) _< y'(~). Take c > 0. By Proposition 4.4.2.2, p e p is a W*-algebra with p as unit. By Theorem 4.4.1.8 f), there is a finite family ((a~,p~))ec, in JR+ x Pr(pEp), such that
[pxp - E aepe eEl
liP'- ~'I1
We have Pe E Pr E and Pe _< p for every t E I (Corollary 4.2.7.6 e =~ a). Thus
(pxp, y' - x') - ( E a~pe, y' - x') eel
362
4. C*-Algebras
pxp - E
I1r
a,p,, y' - x'>
y'(pxp) - x'(pxp) = (pxp, y' - x') > < E
= ~
oLLp~,y' --
x'll < ~,
-- C =
"~(p~, y ' - ~') _ ~ >_ - ~ .
tel
Since r is arbitrary,
x' (pxp) < y' (pxp) .
Theorem 4.4.4.2
( 0 )
I
C i w n ~' c E'+, th~ SoUowing are ~q~ivat~nt:
a)
x'EF.
b)
x' is order continuous.
c)
Given any well-ordered set A of P r E ,
a =~ b follows from Theorem 4.4.1.8 c). b =~ c follows from Theorem 4.4.1.8 i). c =~ a. Consider A := {p C P r E I px' C F } . Let B be a nonempty, well-ordered set of A, ~ the upper section filter of B , and p the supremum of B (Theorem 4.4.1.8 i)). By c), lim(p - q, x') = 0 q,~
so that l i m ( p - q)x' - 0 q,t?
(Corollary 4.3.1.6 a =~ b),
px' - lim qx ~ C F . q,~
By Zorn's Lemma, A has a maximal element p.
4.4 W*-Algebras
363
Assume that p =/- 1. Then there is a y' E F+ with y'(1
-
p) > x'(1
-
p)
(Corollary 4.4.1.6 b)). By a =~ c, Proposition 4.4.2.2 and Proposition 4.4.4.1, there is a q E P r E \ { 0 } with q < 1 - p
and
x'(qxq) <_ Y'(qxq) for every x E E + . We deduce that
I<x, qx'>l 2 - I x ' ( x q ) l 2 <_ x'(qx*xq)x'(1) < y'(qx*xq)x'(1) = <x*x, qy'q)x'(1) for every x E E (Proposition 2.3.4.6 c)). Endow E # with the topology of uniform convergence on the weakly compact convex sets of F . Let ~U be the neighbourhood filter of 0 in E # . Since qy'q E F (Corollary 4.4.2.10), lim<x*x qy'q) = 0 (Corollary 4.4.2.12). Thus lim<x, qx') = O . xfU
Hence (qx')iE# is continuous at 0. By Proposition 1.3.7.9 a =v d, qx' E I m j F . Then ( p + q)x' E F , i.e. p + q E A (Corollary 4.2.7.7), and this contradicts the maximality of p. Hence p = 1 and x' E F . I
Corollary 4.4.4.3 ( 0 )
Let ]K =(~ ( ] K - IR). Given x' E E'
(x' E Re E')
the following are equivalent:
a) x ' E F . b)
x'EE ~ .
These conditions imply the following equivalent assertions: c)
x'lG E G ~ for every maximal Gelfand C*-subalgebra G of E .
d)
If A is a commutative downward directed set of Pr E with infimum 0 and ~ denotes its lower section filter, then x'(~) converges to O.
a =v b. Assume first x' selfadjoint. By Theorem 4.4.3.9, x '+, x ' - E F and by Theorem 4.4.4.2 a ~ b,
364
4. C*-Algebras
Xt = X l + - x
I - E E ~r.
If x' is arbitrary, then re x ' , im x' C F , so that by the above considerations, x' = rex' + i i m x ' E E ~ b ::v a follows from Theorem 4.4.4.2 b =:v a. b ~ c follows from Corollary 4.4.2.7. c =v d. Since A is commutative, there is a maximal Gelfand C*-subalgebra G of E containing A (Corollary 4.1.2.3). Then 0 is the infimum of A in G (Theorem 4.4.1.8 c),i)). Thus x'(~) converges to 0. d :=> c. There is a Radon measure # on a(G) such that
x'(x) = / ~d# for every x C G . Let K be a nowhere dense compact set of a(G) and ~A the set of clopen sets of a(G) containing K . s is downward directed and by Corollary 4.4.2.7,
AU=K. U EII
By d), # ( K ) =
0. Since K is arbitrary, the meager sets of a(G) are #-null
sets. By Example 1.7.2.14 a), # C C(a(G)) ~ , i.e. x'iG c G ' .
Remark.
I
The implication c =:~ a will be proved in Theorem 6.3.2.2 j4 :=~ j2.
C o r o l l a r y 4.4.4.4
(Dixmier, 1953) ( 0 )
Let I K - ~
( I K - I R ) a n d let
G be a W*-algebra. Then there is a unique closed vector subspace (a unique involutive G-submodule) H of G' such that the map G
>H',
x,
)(x,.)l H
is an isometry. H is an involutive G-submodule of G' and the above map is an isometry of involutive G-modules. H is called the predual of G and will be denoted by G. The existence of H is simply a restatement of the definition of a W*algebra. For ]K = r a r
the uniqueness follows immediately from Corollary 4.4.4.3
b. We prove the uniqueness when ]K = IR. Let H1,//2 be two involutive G -
submodules of G' enjoying the above property. By Theorem 4.3.6.4 (and using
4.4 W*-Algebras
~
365
o
the notation from there), H1,/-/2 are closed vector subspaces of (G)' and the map ' (/~k)',
z,
) (z, ")[/~k
is an isometry for k E {1,2}. By the uniqueness in the complex case (and ~--
o
Theorem 4.4.1.2), H1 = / / 2 . Take x' E H1. Then there is a (y', z') E H2 with
(x,, 0) = (y,, z , ) We get
9 '(x) = y'(~) + i / ( ~ ) for every x E G. Hence x' = y' E H2, H1 C H2, and so H1 =/-/2. By Corollary 4.4.2.10, H is an involutive G-submodule of G' and the map G
>H',
x,
)(x,-)lH
is an isometry of involutive E-modules.
I
Remark. a) We occasionally identify E canonically with (E)' via the above isometry. b) By the above corollary, the predual of a W*-algebra is unique. But the duals of different C*-algebras may "coincide". In fact, if T is a countable compact space, then C(T)' may be identified with t?l(T). Hence if S and T are countable compact spaces then there is an involutive isometry u:C(S)'
>C(T)'
such that u(C(S)+) = C(T)+.
D e f i n i t i o n 4.4.4.5
( 0 ) Let G be a W*-algebra and u" E--+ G an involu-
tive algebra homomorphism, u is called a W * - h o m o m o r p h i s m if u'(G) C [~. u is called an isometry or an i s o m o r p h i s m of W*-algebras if it is an isomorphism of C*-algebras (in which case it is automatically a W*-homomorphism). A (unitaO W*-subalgebra of E is a (unital)C*-subalgebra of E which is closed in E E . For A c E , the smallest (unital) W*-subalgebra of E containing A is called the (unitaO W*-subalgebra of E generated by A .
366
4. C*-Algebras
If E1,E2, E3 are W*-algebras and u :
E1 ---+ E2,
v:E2
-+ E3 are
W*-homomorphisms, then v o u is also a W*-homomorphism. If G is a C*-algebra and u : E -+ G a bijective involutive algebra homomorphism, then u is an isometry (Corollary 4.1.1.21), so t h a t G is a W * algebra and u is an isometry of W*-algebras.
Proposition
I
4.4.4.6
( 0 )
Let G be a W*-algebra and u " E =-+ G an
involutive algebra homomorphism. Then u is a W * - h o m o m o r p h i s m iff the map Ek
~ GG,
x,
> ux
is continuous. Take y' E G. We have u~y ~ - y' o u. Hence if u is a W*-homomorphism, then u'y' C E and therefore the map
Ee
>~,
~,
>(~z,y')=(~,~'y')
is continuous. Thus the map EE
>GG,
x~
~ux
is continuous. Conversely, if this map is continuous, then the map E~ ~
~ ,
~ ,
> <~, ~'~'> : < ~ , y'>
is continuous and therefore u'y' E E (Corollary 1.2.6.5). Thus u'(G) c E and u is a W*-homomorphism.
Proposition 4.4.4.7 ( 0 )
m Let E , G
be real W'-algebras and u" E - + G
an involutive algebra homomorphism. Put
Then u is a W * - h o m o m o r p h i s m iff ~t is a W*-homomorphism. I f H is a C*o
subalgebra of E , then H is a W*-subalgebra of E iff H is a W*-subalgebra o
orE. By Proposition 2.3.1.41, u is an involutive algebra homomorphism and o o
therefore continuous (Corollary 4.1.1.20). We identify (E)' with with
G'
~
o
E'
and (G)'
as in Corollary 4.3.6.2 b). Take (x', y') E G. By Proposition 4.3.6.6,
4-4 W*-Al9ebras
(~)'((x', v'))
367
(~'x', ~'y').
By Theorem 4.3.6.4 b), E (resp. G ) i s identified with E (resp. (~). Thus o
(u)'(G)
u'(G)
i.e. u is a W*-homomorphism iff u is a W*-homomorphism. The final assertion follows from Theorem 4.4.1.2.
Corollary 4.4.4.8
( 0 )
II
Let G be a W*-algebra, u " E ~
G a W*-
homomorphism, and v :G
) E,
y',
) u'y'.
a)
For every y e Im u, there is an x e -ul(y) with [Ixll=llyl[.
b)
If H is a W*-subalgebra of E , then u ( H ) is a W*-subalgebra of G .
c)
If we canonically identify E involutive,
u=v',
with (E,)' and G with (G)', then v is
Imv=~
Imu=(Kerv) ~
v is called the pretranspose of u.
d)
If A is a downward directed set of E with infimum O, then 0 is the infimum of u ( A ) .
a) Ker u is a closed ideal of EF (Proposition 4.4.4.6). Let q : E --+ E / K e r u be the quotient map and w : E / K e r u -+ I m u the factorization of u through E / K e r u. Since w is an isometry of C*-algebras (Theorem 4.2.6.6), we have
]lyll = ]lw-lvll 9 By Corollary 4.4.2.16, there is an x E E with q x -- w - l y ,
[]X[[--[[w--ly[[
We conclude that
x e ~(y),
b) By ~),
I1~11= Ilvll.
.
368
4. C*-Algebras
u(H) # = u ( H # ) .
Since H#. is compact (Alaoglu-Bourbaki Theorem) and since the map E
Ei~
~GG,
z,
~ux
is continuous (Proposition 4.4.4.6), it follows that u ( H ) ~ is compact. Hence u(H) is a closed set of G 5 (Corollary 1.3.7.5), i.e. it is a W*-subalgebra of G
(Theorem 4.2.6.6). c) By Proposition 2.3.2.22 d), u' is involutive, and so v is also involutive. The relation u = v ~ is obvious. By Theorem 4.2.6.6, I m u is closed and the algebraic isomorphism E / K e r u --+ Im u associated to u is an isometry of C*algebras. By Proposition 1.4.2.11, Im v is closed, so that by Proposition 1.3.5.8 and Proposition 1.4.2.9 b), Imu=(Kerv) ~
Imv=~
d) Let ~ be the lower section filter of A and take y~ C (~. Then u'y' C ]E, so that lim(ux, y') - lim(x u'y'} - 0 x,8
x,~
'
(Theorem 4.4.1.8 c)), and so u(~) converges to 0 in G&. By Theorem 4.4.1.8 c), 0 is the infimum of u ( A ) . C o r o l l a r y 4.4.4.9
m
( 0 ) Let G be a C*-subalgebra of E and u" G - + E
the inclusion map. Then the following are equivalent:
a)
G is a W*-subalgebra of E .
b)
G is a W*-algebra and u is a W*-homomorphism.
If these conditions are fulfilled, then G = u'(E) and the unit of G is an orthogonal projection in E .
First suppose that a) holds. By Theorem 4.4.2.20 (and Corollary 4.4.4.4), G is a W*-algebra and (~ = u ' ( E ) . Hence u is a W*-homomorphism. The unit of G is an orthogonal projection of E since it is idempontent and selfadjoint. b =a a follows from Proposition 4.4.4.8 b). Remark.
m
G can be a W*-algebra without being a W*-subalgebra of E .
Indeed, let T be the Stone- (~ech compactification of N and for every x C / ~ let ~ be its continuous extension on T . Then
4.4 W*-Algebras
369
G := {~1~ e e~} is a W*-algebra and a C*-subalgebra of g~(T). Since it is dense in g~176 it is not a W*-subalgebra of g~(T). C o r o l l a r y 4 . 4 . 4 . 1 0 "( 0 ) I f G is a W*-subalgebra of E and H is a C*subalgebra of G, then H is a W*-subalgebra of E iff it is a W*-subalgebra of G. Let u : G --+ E and v : H -+ G be the inclusion maps. By Corollary 4.4.4.9 a =~ b, G and H are W*-algebras and u is a W * - h o m o m o r p h i s m . Moreover, if H is a W*-subalgebra of G (resp. E ) , then v (resp. u o v ) i s a W * - h o m o m o r p h i s m . So, if H is a W * - s u b a l g e b r a of G , then u o v is a W * - h o m o m o r p h i s m and by Corollary 4.4.4.9 b =~ a, H is a W * - s u b a l g e b r a of E . Now suppose that H is a W*-subalgebra of E . By the last assertion of Corollary 4.4.4.9, G = u ' ( E ) . Thus
Hence v is a W * - h o m o m o r p h i s m and by Corollary 4.4.4.9 b =~ a, H is a W*-subalgebra of G .
Proposition 4.4.4.11 zcG.
m
( 0 ) Let G be a unital W*-subalgebra of E and
a)
The right and left carriers of x with respect to E (Theorem 4.4.1.8 j)) belong to G and are the right and left carriers of x with respect to G.
b)
The polar representation of x in E and G (Theorem 4.4.3.1) coincide.
c)
For every A C P r G , the supremum and the infimum of A in P r E (Theorem 4.~.1.8 i)) belong to G. a) By Theorem 4.4.1.8 d), G is an order faithful set of E and the assertion
follows from Corollary 4.3.3.12 (and T h e o r e m 4.4.1.8 c)). b) follows from a) and Theorem 4.4.3.1. c) By Proposition 4.2.7.14, it is sufficient to prove the assertion for the supremum. Moreover, we may assume A nonempty. Case 1
A finite
370
4. C*-Algebras
Put X
:----
1
Card A
Ep pEA
and denote by q the supremum of A in P r E . (X~)ke~ is an increasing sequence in G. By Proposition 4.2.7.21, q is the supremum of (x~)ke~ in E . Since G is an order faithful set of E (Theorem 4.4.1.8 d)), it follows q C G. Case 2
A infinite
The assertion follows from Case 1 since G and Pr E are order faithful sets of E (Theorem 4.4.1.8 d),i)). I C o r o l l a r y 4.4.4.12
( 0 )
a)
If G is an involutive subalgebra of E , then its closure in EF is the W*-subalgebra of E generated by G.
b)
Every maximal Gelfand C*-subalgebra of E is a W*-subalgebra of E . In particular, the W*-subalgebra of E generated by a self-normal element of E is a Gelfand W*-algebra and the maximal Gelfand C*-subalgebras of E and the maximal Gelfand W*-subalgebras of E coincide.
c)
For every p E Pr E , p e p is a hereditary W*-subalgebra of E and every hereditary W*-subalgebra of E is of this form. Moreover, pFp is an involutive pEp-submodule of F and the map pEp
~ (pFp)',
x ~--+ (x,.)lpFp
is an isometry of involutive pEp-modules, so that pFp may be identified .. with p e p .
d)
For every p C ECn Pr E , p e p is a closed ideal of EF and every closed ideal of Ep is of this form. Moreover, E / p E p is a W*-algebra, the quotient map q : E ~ E / p E p is a W*-homomorphism, and q l ( 1 -
p ) E ( 1 - p) is an isometry of W*-algebras, so that ( 1 - p ) F ( 1 - p) may be identified with E / p E p . e) If A is a subset of E such that
A c A* U S n E ,
4.4 W*-Algebras
371
then A c is a unital W*-subalgebra of E . If G is the W*-subalgebra of E generated by A , then G c A ~,
f)
A ~ - G ~.
If E is simple and I K = ( ~ ( I K = IR) then E c - r
( E ~ is isomorphic
to IR or ~ ).
g) If E is a Gelfand W*-algebra then F = {x' E E ' l x ' is order continuous}.
h)
An ideal of E is closed in EF iff it has a unit.
a) follows from Corollary 4.4.2.9. b) By Corollary 4.4.2.7, every maximal Gelfand C*-subalgebra of E is closed in E r . Hence the assertion follows from a). c) By Proposition 4.4.2.2, the map qo " EF
~ EF ,
x,
~p x p - x
is continuous. It follows from -1
pep
-
~ (0)
that p e p is closed in E F . By Example 4.3.4.2, pEp is a hereditary W*subalgebra of E . Every hereditary W*-subalgebra of E is of this form since its unit is an orthogonal projection in E . It is obvious that pFp is an involutive pEp-submodule of F . By Corollary 4.4.2.10, the map of c) is a homomorphism of involutive pEp-modules. For every x E p E p , Ilxll -
s u p I(x, a)l = aEF#
=
s u p I(pxp, a)l -aEF#
s u p I(x, pap)l = sup I(x,a)l-II(x,')lpFpII, aEF# aE(pFp)#
which shows (by Hahn-Banach Theorem) that this map is an isometry. d) It is obvious that p e p is an ideal of E . By c), it is closed in EF. Let now G be a closed ideal of EF. By Proposition 4.3.4.5 e), G is a hereditary W*-subalgebra of E . By c), there is a p E Pr E , such that
372
4. C * - A l g e b r a s
G = pEp.
Take x c E . Then px, xp c G = pEp,
so that p x -- p x p -- x p .
Hence p C E c . Moreover, x-
(1- p)x(1 -p)
e G
and therefore q x = q(1 - p ) x ( 1 - p) .
Thus q[(1 - p)E(1 - p) is surjective. From q(1 - p)x(1 - p ) = 0 we get that (1 - p ) z ( 1 - p) ~ G ,
(1-p)x(1-p) Hence q l ( 1 - p ) E ( 1 - p )
= (1-p)x(1-p)p=O.
is injective. By Theorem 4.2.6.6, q l ( 1 - p ) E ( 1 - p )
an isometry of C*-algebras. Hence E / G
is
is a W*-algebra and q l ( 1 - p ) E ( 1 - p )
is an isometry of W*-algebras. By c), (1 - p ) F ( 1
-p)
may be identified with
E/G. e) By Corollary 4.1.4.2 b), A ~ is a unital C*-subalgebra of E . By Corollary 4.4.2.5, A ~ is closed in E F and so A ~ is a unital W*-subalgebra of E . By the above considerations, A ~c is a W*-subalgebra of E . Since A C A ~ , we deduce successively that Ac
GcA
~ ,
A ~ = A ccc c G ~ c A ~,
AC ~ GC..
~.~ W*-Algebras
373
f) By e), E c is a W*-subalgebra of E . By d), E c A P r E = {0, 1} so that E c = r 1 ( R e E c = IR 1) (Theorem 4.4.1.8 g)). By Theorem 4.2.8.24, E c is isomorphic to lR or to C in the real case. g) follows from Corollary 4.3.4.3 c r
d (and Proposition 4.2.7.17).
h) Let G be an ideal of E having a unit p. Then G = E p , so that G is closed in E F . The converse implication was proved in d). Remark.
II
The reverse implication in f) does not hold (Corollaries 6.1.7.14 b),
5.3.2.14, and 3.1.1.13). Proposition
4.4.4.13
Let E, G be complex W*-algebras and u : E -+ G an
involutive algebra homomorphism. Then the following are equivalent:
a)
u is a W*-homomorphism.
b) ~'(G+) E+.
~) ~'(c+) a =~ b =~ c
is trivial.
c =~ a. By Theorem 4.4.3.9, Re(~
G+.
Hence by c), u'(ReG)
u ( G + ) - u(G+)
It follows that
II
i.e. u is a W*-homomorphism. C o r o l l a r y 4.4.4.14
Let E, G be complex W*-algebras and u : E -+ G a
homomorphism of involutive algebras. Then the following are equivalent:
a)
u is a W*-homomorphism.
b)
(resp. c)) If A is a downward directed (and commutative) set of E+ with infimum 0, then 0 is the infimum of u ( A ) .
d)
If A is a well-ordered set of P r E supremum of u ( A ) .
with supremum p, then up is the
374
C*-Algebras
a =~ b follows from Proposition 4.4.4.8 d). b :=> c =~ d is trivial. d =~ a. Take y' E G+. Let A be a well-ordered set of Pr E with supremum p and let ~ be its upper section filter. By d) and Theorem 4.4.1.8 c), i), y'ou(~) converges to y'(up). By Theorem 4.4.4.2 c =~ a, u'y' e E , i.e. u'(G+) c E . By Proposition 4.4.4.13 c =~ a, u is a W * - h o m o m o r p h i s m . Corollary 4.4.4.15
Assume ]K = ~ .
m
Let G be a C*-subalgebra of E . Then
the following are equivalent: a)
G is a W*-subalgebra of E .
b)
G is a W*-algebra and an order faithful set of E . a =~ b follows from Theorem 4.4.1.8 d). b ==~ a follows from Corollary 4.4.4.14 b =~ a and Corollary 4.4.4.9 b =~
a.
m
Proposition 4 . 4 . 4 . 1 6 homomorphism, r : G --+ G / K e r v
Let G
v : G ~
E
be a W*-algebra,
u 9E
-+ G
a W*-
the pretranspose of u,
q : E ~
E/Keru,
the quotient maps, i : I m u --+ G , j
: Imv ~
E the in-
clusion maps, and ~ : E / K e r n
--~ I m u , ~ : G / K e r v ~
Imv
the associated
algebraic isomorphism of u and v , respectively. We canonically identify E with ([~)' and G with (G)' and define wl" ((~/Kerv)' and let
> (Kerv) ~
a',
> r'a'
E / ( I m v) ~ -+ (Im v)' be the factorization of j' through E / ( I m v) ~ .
a)
v, r, j , and ~ are involutive.
b)
K e r u = (Imv) ~ I m u
C)
W 1
d)
E/Kern
(Kerv) ~ K e r v = ~
Imv
and w2 are involutive isometrics. and I m u are W*-algebras, q aund i are W*-homomorphisms,
is an isometry of W*-algebras, ~ is an involutive isometry, and W1
W2
a) By Proposition 4.4.4.8 c), v is involutive. Hence Ker v is an involutive set of G and Im v is an involutive set of F . By Proposition 2.3.2.22 o), ~ is involutive; j and r are obviously involutive.
~.~ W*-Algebras
375
b) By Theorem 4.2.6.6, Im u is closed and so by Proposition 1.4.2.11, Im v is closed as well. The assertion now follows from Corollary 1.4.2.10 a). c) follows from a) and Proposition 2.3.2.22 g). d) By b), Ker u and Im u are closed in E r and GG, respectively. By Corollary 4.4.4.12 d), E / K e r u is a W*-algebra and q is a W*-homomorphism. By Proposition 4.4.4.8 b) and Corollary 4.4.4.9 a =~ b, Im u is a W*-algebra and i is a W*-homomorphism. By Theorem 4.2.6.6, ~ is an isometry of C*algebras. If by c), we identify ((~/Kerv)' with (Kerv) ~ - Im u via wl and E/(Imv)~
with (Imv)' via w2, then Imv becomes the predual of
E/Keru
E / K e r u, (~/Ker v becomes the predual of Im u, and ~ becomes the pretranspose of ~. By the above considerations, ~ is an isometry of W*-algebras. By Corollary 1.4.2.10 d), ~-wl
oV' o w 2 ,
so that
By c), ~' is an involutive isometry. Hence ~ is an involutive isometry as well (a) and Corollary 1.4.2.5). P r o p o s i t i o n 4.4.4.17 W*-homomorphism. and f
9
Let E , G
be W * - a l g e b r a s and u " E -+ G a unital
Take x E Sn E
and let it be the spectral measure of x
a bounded Borel f u n c t i o n on or(x).
a) u f (x) : f (ux) . b)
u o it is the spectral measure of u x .
By Proposition 4.4.4.8 d), u o it is a spectral measure. If (A~)~EI is a finite family of Borel sets of a(x) and (C~)~eI is a family in IK, then
By continuity,
uf (x) = u ( f f dp) - f f d(u o p) (Theorem 4.3.2.19 b)). Hence u o it is the spectral measure of u x and u f (x) : f (ux) .
II
376
4. C*-Algebras
T h e o r e m 4.4.4.18
( 0 ) We identify E with F' via the map E
~ F',
x,
>(x,.)lF.
Let G be a C*-subalgebra of E such that every continuous linear form on G is continuous on GF. Put H : - (~ map. Put I := F / ~
~ and let j : G -+ H be the inclusion
and let q : F -+ I be the quotient map. Let u be the
involutive isometry I'
>H ,
x' ,
~ q' x'
(Proposition 2.3.2.22 h)). If G # is dense in H#F , then:
a) H is the closure of G in EF and the W*-subalgebra of E generated by G .
b)
The map v:I
)G',
a:
~ (jla) o u - l o j
is an involutive isometry and nov' oje=
j .
c) v q a - (jFa)IG for every a 9 F . d)
(a, x) 9 F • G ::v x(vqa) = v q ( x a ) , (vqa)x = v q ( a x ) .
e) (a,x") 9 F • G" =:~ (vqa, x") = (a, u v ' x " ) . f) (a, x") C F • G" :=~ x"(vqa) = vq((uv'x")a), (vqa)x" = vq(a(uv'x")) .
g) a i v ~ ~", y" e a" ~ I ~v'(x" ~ r
( h)
- ~'(~" -~ r
= (~v'x")(~v'r
,
x" ~ y" -- x" -~ y".
The Arens multiplications of G" (Definition 2.2.7.13) coincide, G" endowed with this multiplication is a W*-algebra and u o v' : G" ~
H
is a W*-isometry. If we identify G" with H via u o v', then j is the evaluation. If G is commutative, then G" and H are also commutative.
a) By Proposition 1.3.5.4, H is the closure of G in E F . By Corollary 4.4.4.12 a), H is the W*-subalgebra of E generated by G. b) Since jI is involutive (Proposition 2.3.2.22 e)), v is also involutive and the assertion follows from Proposition 1.3.6.21 d).
377
4.4 W*-Algebras
c) By Proposition 1.3.6.21 a), (x, vqa) -- (x, (jiqa)
o u -1
o
j) = (x, a) = (x, (jFa)IG)
for every x E G. Thus vqa = (jFa)IG.
d) By c), Proposition 2.2.7.5, and Corollary 4.4.2.10, (y,x(vqa)) = (yx, v q a ) = (yx, jFa) = (y, xjFa) = ( y , j F ( x a ) ) =
(y, vq(xa)),
(y, ( v q a ) x ) = (xy, v q a ) = (xy, jFa) = (y, (jFa)X) = (y, jF(aX)) - (y, vq(ax)) for every y E G and thus x(vqa) = vq(xa),
(vqa)x = vq(ax).
e) We have (vqa, x") -- (a,q'v'x") = (a, uv'x") .
f) By b), d), and e), (X, x"(vqa)) -- ((vqa)x, x") = (vq(ax), x") -- (ax, uv'x") = = ( ( u v ' x " ) a , x ) = ((uv'x")a, j x ) =
((uv'x")a, u v ' j G x ) =
= (vq((uv'x")a),jax) -- (x, vq((uv'x")a)), (x, (vqa)x") = (x(vqa), x") - (vq(xa), x " ) = (xa, uv'x") = (a(uv'x"), x ) = (a(uv'x"), j x ) =
(a(uv'x"), u v ' j a x ) -
= (vq(a(uv'x")),jex) - (x, vq(a(uv'x")))
for every x C G. Thus x"(vqa) = vq((uv'x")a),
g) By e) and f),
(vqa)x" = vq(a(uv'x")).
378
4. C*-Algebras
(a, uv'(x" H y")) = (vqa, x" ~ y") = (y"(vqa),x") = (vq((uv'y")a),x") = = ((uv'y")a, u v ' x " ) = (a, ( u v ' x " ) ( u v ' y " ) } , (a, uv'(x" --t y")) = (vqa, x" -t y") = ((vqa)x", y") = (vq(a(uv'x")), y") -= (a(uv'x"), u v ' y " ) = (a, (uv'x")(uv'y")) for every a E F . Hence
~v'(x" ~ y") = (uv'~")(uv'y") = u~'(~" e y"), x" --I y" = x" t- y ' . h) By b) and Proposition 2.3.2.22 d), n o v ' is an involutive isometry. By g), the Arens multiplications of G" coincide, G" endowed with this multiplication is a C*-algebra and nov' : G" -4 H is an isometry of C*-algebras. By Theorem 2.2.7.15 e), G' is a G " - m o d u l e and by Proposition 2.2.7.5, I m j c , is a G ' submodule of G t" . Hence G" is a W*-algebra and it is easy to see that u o v p is a W*-isometry. By b), if we identify G" and H via u o v', then j becomes the evaluation. The last assertion follows from Theorem 2.2.7.15 g). Y~
I
4.4.4.19
Let ( a n ) n ~ be a sequence in F+ such that ((a~,p))n~IN converges whenever p E Pr E . Then there is an a C F+ such that lim (an, x) = (a, x) n---+ ( x )
whenever x E E . By Corollary 4.4.1.14, there is an order continuous a C E' such that lim (an, x) = (a, x) n----~ (3(3
whenever x E E . Then a C E~ (Corollary 4.4.1.7) and by Theorem 4.4.4.2 b=>a,
aCF+.
P r o p o s i t i o n 4.4.4.20
I Let (an)neIN be a sequence in F such that for each
p c Pr E ((an, P})ne~ converges. If E is a Gelfand W*-algebra, then there is an a C F such that lirn (an, x) - (a, x} whenever x C E .
4.4 l/i~*-Algebras
379
By Corollary 4.4.1.14, there is an order continuous a 9 E' such that lim (an, x> -- (a, x)
n - - + (:X:)
I
whenever x 9 E . By Corollary 4.4.4.12 g), a 9 F . Remark.
We do not know if the above proposition holds for W*-algebras which
are not Gelfand W*-algebras. P r o p o s i t i o n 4.4.4.21
Let (Et)tE T be a family of W*-algebras and E the
C*-direct product of this family (Example ~. 1.1.6). Put
Y:~Ila~ll< ~ }
H := { a E H and for a 9 H
define
"5.E
)IK,
x,
~E<xt, at>. tET
a)
E is a W*-algebra and ~E -
b)
For every s c T , the maps
{~la
ps " E qs " E~
9 H}.
) Es , > E,
x,
Xs l
> xs
> (SstXs)tET,
are W * - h o m o m o r p h i s m s and q~(Es) is a W * - subalgebra of E .
a) It is easy to see that F - - {~la ~ H} is an involutive E-submodule of E'. By Proposition 1.2.2.13, the map
E
>F ' ,
x,
> <x,->IF
is an isometry. Hence E is a W*-algebra and E - F . b) By a), the maps Ei~ (Es)Es
> (E~)k ,
x,
>psX
> EE ,
x,
> qsx
are continuous. Hence the maps ps and qs are W*-homomorphisms (Proposition
4.4.4.6). By
Corollary 4.4.4.8 b), qs(Es) is a W*-subalgebra of E .
I
380
4. C*-Algebras
I
~ 0 ) Let T be a nonempty compact space T . Then the following are equivalent: Theorem
4.4.4.22
a)
C(T) is a W*-algebra.
b)
T is a Stone space and C(T)c(T), is Hausdorff.
c)
T is hyperstonian.
If these conditions are fulfilled, then: d)
C(T) ~ is the predual of C(T).
e) If x is a bounded Borel function on T and ~ ' C(T)'
~ IK,
#,
~ / xd# ,
then ~ E C(T)" = (C(T)'~) ''' and {x # j"~} is meager, where j denote8 the evaluation of C(T) '~ . a :=> c follows from Corollary 4.4.1.10. b r
c follows from Example 1.7.2.14 d).
c~a&d.
Let ~ be an ultrafilter on C(T) #
Take # E g ( T ) ' + . T h e n
~:
converges in L~(P)#L,O,) to a function x , . By Proposition 1.7.2.13 a ~ b (and Example 1.7.2.14 a)), we may assume x , E C(T) # . Let #, u E C(T)'+. Then # + u E C(T)~_ and X#
-'- X t t + u = X u
on (Supp) Cl (Supp v). Hence we may define S'-
U
Supp#
)IK,
t,
#EC(T)~_ where t E S u p p #
for some # E C(T)~, and x is continuous. Since T is
homeomorphic to the Stone-(~ech compactification of S , we may consider x E
C(T) # . Then ~ converges to x in C(T)#c(T)~ and C (T)c(T). # is compact. By Proposition 1.3.6.27, C(T) is a W*-algebra with C(T) '~ as predual. e) By Proposition 1.7.2.13 a ~ b, there is a y E C(T), such that {x # y} is meager. Assume that {x # j'~} is not meager. Then y # j ' ~ . By Example 1.7.2.14 d), there is a p E C(T) '~ , such that
4.4 W*-Algebras
381
By Example 1.7.2.14 a), J
J
= (j#, Y> = <#, Y> =/= <#,j'~> = <j#,~>.
I
This is a contradiction. Proposition
( 8 ) Let (x,y) C E . If there is a filter ~ on E
4.4.4.23
such that
lim<(z, 0) (a, b)> - <(x, y) (a, b)> z~
~
.. o
for any (a,b) c E+ then y - O
and
lim - <x a> z~
~
for any a C Re E:. .. o
For every (a, b) C E+, <x, a > - (y, b> + i(<x, b> + ) = = <(x, y) (a, b)> = l~<(z, 0), (a, b)> - lim + i lim. '
,
z,i}
z,i~
'
In particular, for every a E E+,
<x, a> + i
= l i m < z , a> z,~
i.e.
(y,a> - 0 ,
<x, a> = lim z~
(Proposition 4.3.6.1 bl =:~ b2). It follows
-0. By Theorem 4.4.3.9 (and Corollary 4.4.2.10), y - 0 and lim(z a > - (x a / z~
for every a C E .
'
I
382
4. C*-Algebras
Proposition 4.4.4.24
Let E be a W*-algebra and
~E := {a 9 [~ l x 9 E =~ xa = a x } . Take a 9 ~E and let (x, lal) be its polar representation.
a) x e E ~,
[ale~E.
b)
(x, [a[[E c) is the polar representation of a l E ~.
c)
IlalE~ll = Ilalla)
Case 1
Let u C U n E
IK = ~ and put
b := u*lalu 9 E+ (Proposition 2.3.6.4). Then ua = au = xlalu = x u b ,
a = u*xub.
Since (xu)(xu)* = xuu*x* = xx* E Pr E ,
it follows (u*xu)* (u*xu) = u*x*xu = (xu)* (xu) C
(Corollary 4.1.2.22 b =v a). Moreover,
(Theorem 4.4.3.5 a)). Let y C E such that yb=O.
Then 0 = yu* x* xub = yu* x * a u ,
0 -- yu*x*a,
E
4.4 W*-Algebras
383
so that 0 = yu*x*xx* = yu*x*
(Theorem 4.4.3.5 b), Corollary 4.1.2.22 a =:~ c), yU*X*XU -
O.
Thus u*x*xu is the carrier of b and (xu, b) is the polar representation of u a . It follows t h a t (u*xu, b) is the polar representation of a. By the uniqueness of the polar representation, ~*x~ = ~,
~*lal~ = b = lal,
so that xu = ux,
lalu = ulal.
This implies x 9 E c and lal 9 ~E (Corollary 4.1.3.7). Case 2
IK = IR
By complexification, we reduce this case to Case 1. We have o
o " C(E)
(a, 0 ) 9 ~E
((x, 0), (Ia], 0))
and it is easy to see that
is the polar representation of
(a, 0)
(Corollary 4.4.3.7). By Case 1, o
o
~
(x, 0 ) e ( E ) ~ =
o
E~ ,
( a l , 0) e ~ ( E ) =
,
and so xGE
~,
laiECE.
b) E c is a W*-subalgebra of E (Corollary 4.1.4.2 b), Corollary 4.4.2.5). Denote by ~a : E c
~, E
the inclusion map and by ~:E
> E~
384
4. C*-Algebras
its pretranspose (Corollary 4.4.4.8 c)). Then alE ~ = ~a 9
~",
lall E~
-
~lal
e
E ~ ,
so that (by a)), (x, lallE ~) is the polar representation of al E~ . c) By b) (and Corollary 2.3.4.7),
IlalECll : II lallECll : <1, lal) : II lal II : Ilall 9
m
385
Name Index
Name Index Alaoglu, L. 1.2.8.1 Arens, R.F. 1.5.2.10,2.2.7.13 Arzel/~, C. 1.1.2.16 Ascoli, G. 1.1.2.14,1.1.2.16 Atkinson, F.V. 3.1.3.7, 3.1.3.11, 3.1.3.12, 3.1.3.21 Autonne, L. 2.3.1.3 Banach, S. 1.1.1.2, 1.2.8.2, 1.3.1.2, 1.3.2, 1.3.3.1, 1.3.4.1, 1.3.4.10, 1.4.1.2, 1.4.2.3, 1.4.2.19 Beurling, A. 2.2.5.4 Bourbaki, N. 1.2.8.1 Branges, L. de 1.3.5.14 Carleman, T. 3.1.3.1 Cauchy, A. 1.3.10.6 Dedekind, R. 1.7.2.1 Dieudonn~, J. 1.2.8.2,3.1.3.9 Dixmier, J. 4.4.4.4 Drewnowski, L. 4.3.2.13 Dworetzky, A. 1.1.6.14 Dye, H.A. 4.1.3.7 Eberlein, W.F. 1.3.7.15 Effros, E.G. 4.2.4.15 Enflo, P. 3.1.1.7 Ford, J.W.M. Fourier, J.-B.-J. Fr~chet, M. Fredholm, E. Frobenius, G.F. Fuglede, B. Fukamiya, M. Gelfand, I.M. Gohberg, I. Goldstine, H.H. Goodearl, K.R. Gowers, W.T.
386
Grothendieck, A. 1.6.1.1, 3.1.6.25, 4.2.8.13 Hahn, H. 1.1.1.2, 1.2.1.3, 1.3.3.1, 1.3.6.1, 1.3.6.3, 1.3.8.1, 1.4.1.4 Hamilton, W.R. 2.1.4.17 Helly, E. 1.1.1.2, 1.3.3.13 Hilbert, D. 2.1.3.1 Hirschfeld, R.A. 2.2.5.6 Jacobson, N. 2.1.3.10 James, R.C. 1.3.8.1 Kadison, R.K. 4.3.3.20 Kaplanski, I. 4.1.2.1, 4.2.6.5, 4.4.2.24 Kelley, J.L. 4.2.1.1 Kojima, ?? 1.2.3.11 Kolmogoroff, A. 1.1.1.2 Kottman, C.A. E 1.3.5 Krein, M.G. 1.3.1.10, 1.3.7.3 Laguerre, E.N. 2.2.3.5 Laurent, P.A. 1.3.10.8 Le Page, C. 2.2.3.8, 2.2.4.3, 2.2.5.6 Lindenstrauss, J. 1.2.5.13 Liouville, J. 1.3.10.6 Lomonsov, V.I. 3.1.5.10 Mackey, G.W. 1.3.7.2 Mazur, S. 2.2.5.5 Mihlin, S.G. 3.1.3.12 Milman, D.P. 1.3.1.10 Minkowski, H. 1.1.1.2, 1.1.3.4 Murray, F.J. 1.2.5.8 Nagumo, M. 2.2.1.1 Naimark, M.A. 4.1.1.1, 4.1.2.5, 4.1.3.1, 4.2.6.6 Neumann, C. 2.2.3.5 Neumann, J. von 1.1.1.2, 3.1.3.1 Nikodym, O. 4.3.2.14 Noether, F. 3.1.3.1 Palmer, T.W. 4.1.1.1 Pedersen, G.K. 4.2.4.16 Peter, F. 2.2.1.15 Pettis, P.J. 1.3.8.4, 1.3.8.5 Phillips, R.S. 1.2.5.14, E 1.3.3, 2.1.4.9
Name Index
Pierce, B. Putnam, I.F. Rickart, C.E. Riesz, F.
2.1.1.1, 2.1.3.6 4.1.4.1 4.1.1.20, 4.1.2.12 1.1.1.2, 1.1.4.4, 1.2.1.1, 1.2.2.5, 2.2.5.1 3.1.1.1, 3.1.3.8, 3.1.3.17, 3.1.5.1 Rogers, C.A. 1.1.6.14 Rosenblum, M. 4.1.4.1 Russo, B. 4.1.3.7 Sakai, S. 4.4, 4.4.1.1, 4.4.3.5 Schauder, J.P. 3.1.1.22 Schmidt, E 1.1.1.2 Schur, I. 1.2.3.11, 1.2.3.12, 1.3.6.11 Schwartz, L. 2.4.6.5, 2.4.6.8 Segal, I.E. 4.2.6.2, 4.2.6.5, 4.2.8.2 Shirali, S. 2.4.2.4 Sierpifiski, W. 1.1.2.17 Silow, G. 2.2.4.27 Smulian, V. 1.3.7.3, 1.3.7.15 Steinhaus, H.D. 1.4.1.2 Stone, M.H. 1.3.4.10, 1.3.5.16, 2.3.3.12, 4.1.2.5 St~rmer, E. 4.2.6.3 Toeplitz, O. 1.2.3.4, 2.3.1.3 Vaught, R.L. 4.2.1.1 Vitushkin, A.G. 2.4.3.7 Volterra, V. 2.2.4.22 Weierstrass, K. 1.3.5.16 Weyl, H. 2.2.1.15 Wielandt, H. 2.2.5.8 Wiener, N. 2.4.5.7 Yood, B. 3.1.3.11, 3.1.3.12, 4.1.1.13 Zelazko, W. 2.2.5.6
387
388
Subject Index NT means Notation and Terminology
(.A, B, (:)-multiplication
1.5.1.1
absolute value of a number
1.1.1.1
absolute value of a measure
NT
absolutely convex 1.2.7.1 absolutely convex closed hull 1.2.7.6 absolutely convex hull 1.2.7.4 absolutely summable family 1.1.6.9 additive group NT adherence, point of NT adherent point NT adjoint 2.3.1.1 adjoint differential operator 3.2.2.3 adjoint kernel 3.1.6.5 adjoint sesquilinear form 2.3.3.1 adjoint sesquilinear map 2.3.3.1 algebra 2.1.1.1 algebra, Calkin 3.1.1.13 algebra, complex 2.1.1.1 algebra, degenerate 2.1.1.1 algebra, division 2.1.2.1 algebra, Gelfand 2.4.1.1 algebra, Gelfand unital 2.4.1.1 algebra, involutive 2.3.1.3 algebra, involutive Gelfand 2.4.2.1 algebra, involutive unital Gelfand algebra, normed 2.2.1.1 algebra, real 2.1.1.1 algebra, semi-simple 2.1.3.18 algebra, strongly symmetric 2.3.1.26 algebra, symmetric 2.3.1.26 algebra, unital 2.1.1.3 algebra, unital Gelfand 2.4.1.1 algebra homomorphism 2.1.1.6
Subject Index
389
algebra homomorphism, unital algebra isomorphism
2.1.1.6
2.1.1.6
algebra isomorphism, unital algebraic dimension
2.1.1.6
1.1.2.18
algebraic dual 1.1.1.1 algebraic isomorphism, associated
1.2.4.6
algebras, isomorphism of involutive analytic function
1.3.10.1
approximate unit
2.2.1.15
2.3.1.3
approximate unit of a C*-algebra, canonical Arens multiplication, left
1.5.2.10, 2.2.7.13
Arens mutliplication, right
1.5.2.10, 2.2.7.13
associated algebraic isomorphism associated quadratic form associated quadratic map
2.3.3.1 2.3.3.1
associated unital C*-algebra atom
1.2.4.6
4.1.1.13
4.3.2.20
atomic 4.3.2.20 atomless 4.3.2.20 Baire function 1.7.2.12 Baire set
1.7.2.12
ball, unit 1.1.1.2 Banach algebra 2.2.1.1 Banach algebra, involutive
2.3.2.1
Banach algebra, quasiunital 2.2.1.15 Banach algebra, unital 2.2.1.1 Banach categories, functor of 1.5.2.1 Banach categories, functor of unital 1.5.2.1 Banach category
1.5.1.1
Banach category, unital Banach space
1.5.1.1
1.1.1.2
Banach space, complex Banach space, involutive
1.1.1.2 2.3.2.1
Banach space, ordered 1.7.1.4 Banach space, real 1.1.1.2 Banach subalgebra generated by 2.2.1.9 Banach system 1.5.1.1 Banach system, bidual of a 1.5.1.9
390
Banach system, dual of a
1.5.1.9
Banach systems, isometric band
1.5.2.1
1.7.2.1
bicommutant
2.1.1.16
bidual of a Banach system
1.5.1.9
bidual of a normed space bijective
1.3.6.1
NT
bilinear map
1.2.9.1
binomial theorem bitranspose
2.2.3.12
1.3.6.15
bound, lower
1.7.2.1
bound, upper
1.7.2.1
bounded map
1.1.1.2
bounded operator
1.2.1.3
bounded operator, lower bounded sequence bounded set
1.2.1.18
1.1.1.2
1.1.1.2
boundedness, principle of uniform
1.4.1.2
boundedness theorem, Nikodym's
4.3.2.14
C*-algebra
4.1.1.1
C*-algebra, canonical approximate unit of a C*-algebra, canonical order of a C*-algebra, complex
C*-algebra, complex unital C*-algebra, Gelfand
4.1.1.1
4.1.1.1
C*-algebra, purely real C*-algebra, real
4.1.1.8
4.1.1.1
C*-algebra, real unital
4.1.1.1
C*-algebra, simple
4.3.5.1
C*-algebra, unital
4.1.1.1
C*-algebra associated, unital Calkin algebra
4.2.1.2
4.1.1.1
4.1.1.13
3.1.1.13
Calkin category
3.1.1.12
canonical approximate unit of a C*-algebra canonical involution of E p
2.3.1.1
canonical metric of a normed space canonical norm of
s
F)
canonical order of a C*-algebra
1.1.1.2
1.2.1.9 4.2.1.2
4.2.8.2
Subject Index
391
canonical projection of the tridual of E cardinal number
NT
cardinality, topological carrier
1.3.6.19
NT
4.3.3.1
carrier, left carrier, right
4.3.3.1 4.3.3.1
carrier of a function
NT
carrier of a Radon measure category, Banach
1.5.1.1
C*-direct product C*-direct sum character
NT
4.1.1.6 4.1.1.6
2.4.1.1
characteristic function of a set C*-hull
1.1.2.1
4.1.1.22
class NT closed graph theorem
1.4.2.19
closed involutive subalgebra generated by
2.3.2.14, 2.3.2.15
closed involutive unital subalgebra generated by
2.3.2.14, 2.3.2.15
closed subalgebra generated by 2.2.1.9 closed unital subalgebra generated by 2.2.1.9 closed vector subspace generated by 1.1.5.5 codimension 1.2.4.1 codomain NT cokernel of a linear map commutant commutative
1.2.4.5
2.1.1.16 2.1.1.1
commutative monoid compact, relatively compact operator
E 2.1.1 1.1.2.9 3.1.1.1
compatible, simultaneously
1.5.1.1
compatible (left and right) multiplications complement of a subspace complemented subspace complete, C-order complete, order complete norm
1.2.5.3 1.2.5.3
4.3.2.3 1.7.2.1 1.1.1.2
complete normed space complete ordered set
1.1.1.2 1.7.2.1
1.5.1.1
392
2.2.1.13
completion of a normed algebra
1.3.9.1
completion of a normed space complex algebra
2.1.1.1
complex Banach space
1.1.1.2
complex C*-algebra 4.1.1.1 complex C*-algebra, unital 4.1.1.1 complex normed algebra
2.2.1.1
complex normed space
1.1.1.2
complex unital C*-algebra complex W*-algebra
4.1.1.1
4.4.1.1
complexification of algebras
2.1.5.7
complexification of Banach algebras complexification of involutive algebras
2.2.1.19 2.3.1.40
complexification of involutive vector spaces complexification of vector spaces 2.1.5.1 composition of functors 1.5.2.1 composition of maps
NT
cone 1.3.7.4 cone, sharp 1.3.7.4 conjugacy class
2.2.2.7
conjugate exponent of 1.2.2.1 conjugate exponents 1.2.2.1 conjugate exponents, weakly 1.2.2.1 2.3.1.3 1.3.7.10
conjugate involution conjugate linear map
1.1.1.1
conjugate number
1.7.2.3 continuous, order 1.1.6.22 convergence, radius of convex
1.2.7.1
convex, absolutely
1.2.7.1
convex closed hull
1.2.7.6
convex closed hull, absolutely convex hull
1.2.7.4
convex hull, absolutely 1.2.7.4 convolution 2.2.2.7, 2.2.2.10 ~-order complete
4.3.2.3
~-order a-complete 4.3.2.3 C*-subalgebra 4.1.1.1
1.2.7.6
2.3.1.38
Subject Index
393
C*-subalgebra, unital 4.1.1.1 C*-subalgebra generated by 4.1.1.1 C*-subalgebra generated by, hereditary 4.3.4.1 C*-subalgebra generated by, unital 4.1.1.1 decomposition, spectral 4.3.2.19 degenerate algebra 2.1.1.1 derivative 1.1.6.24 differentiable 1.1.6.24 differential operator, adjoint 3.2.2.3 differential operator, selfadjoint 3.2.2.3 dimension, algebraic 1.1.2.18 Dirac measure 1.2.7.14 direct sum 1.2.5.3 directed, downward 1.1.6.1 directed, upward 1.1.6.1 disjoint family of sets 1.2.3.9 distance of a point from a set 1.1.4.1 division algebra 2.1.2.1 domain NT downward directed 1.1.6.1 dual, algebraic 1.1.1.1 dual of a Banach system dual of a normed space dual space 1.3.1.11 E-algebra 2.2.7.1 E-algebra, involutive
1.5.1.9 1.2.1.3
2.3.6.1
E-algebra, involutive unital E-algebra, unital 2.2.7.1 E-algebras, homomorphism of
2.3.6.1 2.2.7.1
E-algebras, homomorphism of involutive 2.3.6.1 E-algebras, homomorphism of involutive unital 2.3.6.1 E-algebras, homomorphism of unital 2.2.7.1 eigenspace 3.1.4.1 eigenvalue 3.1.4.1 eigenvector 3.1.4.1 E-module 2.2.7.1 E-module, involutive 2.3.6.1 E-module, involutive unital 2.3.6.1
394
E-module, unital
2.2.7.1
E-modules, homomorphism of
2.2.7.1
E-modules, homomorphism of involutive equicontinuous
1.1.2.14
equivalence class
NT
equivalence class of a point equivalence relation
NT
NT
equivalent norms
1.1.1.2
essential spectrum E-submodule
3.1.3.24 2.2.7.1
Euclidean norm evaluation
1.1.5.2 1.2.1.8
evaluation functor
1.5.2.1
evaluation operator of a normed space E-valued spectral measure exact set
4.3.2.16
1.7.2.12
exponential function
2.2.3.5
exponents, conjugate
1.2.2.1
exponents, weakly conjugate extreme point
1.2.2.1
1.2.7.9
face of a convex set
1.2.7.9
factorization of a linear map faithful, order family
1.2.4.6
4.2.2.18
NT
family, absolutely summable family, sum of a
1.1.6.9
1.1.6.2
family, summable
1.1.6.2
family of sets, disjoint
1.2.3.9
filter, lower section
1.1.6.1
filter, upper section
1.1.6.1
filter of cofinite subsets
NT
finite-dimensional F-invariant
1.1.2.18
3.1.4.4
Fourier integral Fourier transform Fredholm alternative Fredholm operator
2.4.6.2 2.4.6.2 3.1.6.23 3.1.3.1
Fredholm operator, index of a
3.1.3.1
2.3.6.1
Subject Index
free ultrafilter NT function NT function, Baire 1.7.2.12 function, step NT functional calculus 4.1.3 functor 1.5.2.1 functor, identity 1.5.2.1 functor, inclusion 1.5.2.16 functor, isometric 1.5.2.1 functor, quotient 1.5.2.17 functor, transpose of a 1.5.2.3 functor of (unital) Banach categories 1.5.2.1 functor of (unital) A-categories 1.5.2.1 functor of (left, right) A-modules 1.5.2.1 functors, composition of 1.5.2.1 Gelfand, Theorem of 2.2.5.4 Gelfand algebra 2.4.1.1 Gelfand algebra, involutive 2.4.2.1 Gelfand algebra, involutive unital 2.4.2.1 Gelfand algebra, spectrum of a 2.4.1.1 Gelfand algebra, unital 2.4.1.1 Gelfand C*-algebra 4.1.1.1 Gelfand-Mazur, Theorem of 2.2.5.5 Gelfand transform 2.4.1.2 graph NT, 1.4.2.18 Green function 3.2.1.2 group, additive NT Hahn-Banach Theorem 1.3.3.1 hereditary 4.3.4.1 hereditary C*-subalgebra generated by 4.3.4.1 Hermitian sesquilinear map 2.3.3.3 HSlder inequality 1.2.2.5 homomorphism of C*-algebras 4.1.1.20 homomorphism of E-algebras 2.2.7.1 homomorphism of E-modules 2.2.7.1 homomorphism of involutive E-algebras 2.3.6.1 homomorphism of involutive E-modules 2.3.6.1 homomorphism of involutive unital E-algebras 2.3.6.1
395
homomorphism of unital E-algebras hyperstonian space ideal
2.2.7.1
1.7.2.12
2.1.1.1
ideal, left
2.1.1.1
ideal, maximal proper
2.1.1.1
ideal, maximal proper left
2.1.1.1
ideal, maximal proper right ideal, proper
2.1.1.1
2.1.1.1
ideal, proper left
2.1.1.1
ideal, proper right
2.1.1.1
ideal, regular maximal proper
2.1.3.17
ideal, regular maximal proper left
2.1.3.17
ideal, regular maximal proper right ideal, right
ideal generated by idempotent
2.1.1.2
2.1.3.6
identity functor identity map
1.5.2.1 NT
identity operator iff
2.1.3.17
2.1.1.1
1.2.1.3
NT
image of a linear map imaginary part
1.2.4.5
1.1.1.1, 2.3.1.22
inclusion functor
1.5.2.16
inclusion map
NT
index of a Fredholm operator index of U induced norm infimum
1.1.1.2 1.7.2.1
infinite-dimensional infinite matrix mjective
1.1.2.18 1.2.3.1
NT
tuner multiplication interior point
3.1.3.1
3.1.3.21
1.5.1.1
NT
invariant vector subspace
3.1.4.4
reverse of a bijective map
NT
inverse of a morphism
1.5.1.6
inverse of an element in a unital algebra inverse operators, principle of
1.4.2.4
2.1.2.4
Subject Index
invertible
397
1.5.1.5, 2.1.2.1
invertible, left
1.5.1.5
invertible, right involution
1.5.1.5 2.3.1.1
involution, conjugate
2.3.1.3
involution of E p, canonical involutive algebra
2.3.1.1
2.3.1.3
involutive algebra, complexification of an involutive algebra, strongly symmetric involutive algebra, symmetric
2.3.1.26
2.3.1.26
involutive algebras, isomorphism of involutive Banach algebra
2.3.1.40
2.3.1.3
2.3.2.1
involutive Banach space
2.3.2.1
involutive Banach unital algebra associated to involutive E-algebra involutive E-module
2.3.6.1 2.3.6.1
involutive Gelfand algebra
2.4.2.1
involutive map 2.3.1.1 involutive normed algebra
2.3.2.1
involutive normed space
2.3.2.1
involutive normed unital algebra associated to involutive set 2.3.1.1 involutive space 2.3.1.1 involutive subalgebra generated by
2.3.1.18
involutive unital algebra associated to
2.3.1.9
involutive unital E-algebra
2.3.6.1
involutive unital E-module
2.3.6.1
involutive unital Gelfand algebra
2.4.2.1
involutive unital subalgebra generated by involutive vector space
2.3.2.9
2.3.1.18
2.3.1.3
involutive vector spaces, isomorphism of
2.3.1.3
involutive vector subspace generated by
2.3.1.18
isometric Banach systems isometric functor
1.5.2.1
1.5.2.1
isometric normed algebras isometric normed spaces
2.2.1.1 1.2.1.12
isometric normed unital algebras 2.2.1.1 isometry of normed algebras 2.2.1.1
398
isometry of W*-algebras
4.4.4.5
Isometry of normed spaces
1.2.1.12
lsometry of normed unital algebras Isomorphic algebras
2.2.1.1
2.1.1.6
isomorphic normed algebras Isomorphic normed spaces
2.2.1.1 1.2.1.12
Isomorphic normed unital algebras Isomorphic unital algebras Isomorphism, algebra
2.2.1.1
2.1.1.6 2.1.1.6
Isomorphism associated to a linear map, algebraic Isomorphism of involutive algebras
2.3.1.3
isomorphism of involutive vector spaces isomorphism of normed algebras
2.3.1.3
2.2.1.1
isomorphism of normed spaces 1.2.1.12 isomorphism of normed unital algebras 2.2.1.1 kernel of a linear map 1.2.4.5 Kronecker's symbol lattice 1.7.2.1 lattice, vector
1.2.2.6
1.7.2.1
Laurent series 1.3.10.8, 1.3.10.9 left Arens multiplication 1.5.2.10, 2.2.7.13 left carrier left ideal
4.3.3.1 2.1.1.1
left ideal, maximal proper left ideal, proper
2.1.1.1
2.1.1.1
left ideal, regular maximal proper left ideal generated by 2.1.1.2 left invertible
1.5.1.5
left multiplication left shift
2.1.3.17
1.5.1.1
1.2.2.9, E 1.2.11
left (unital) A-module linear form 1.1.1.1 linear form, positive linear map, conjugate lower bound 1.7.2.1 lower bounded operator
1.5.1.10 1.7.1.9 1.3.7.10 1.2.1.18
lower section filter 1.1.6.1 L2-distributions, in the sense of
3.2.2.3
1.2.4.6
Subject Index
map
399
NT
map, bilinear
1.2.9.1
map, bounded
1.1.1.2
map, conjugate linear map, identity
1.3.7.10
NT
map, inclusion
NT
map, inverse of a bijective map, involutive
NT
2.3.1.1.
map, nuclear
1.6.1.1
map, quotient
1.2.4.1
maps, composition of matrix, infinite
NT
1.2.3.1
maximal proper ideal
2.1.1.1
maximal proper ideal, regular maximal proper left ideal
2.1.3.17 2.1.1.1
maximal proper left ideal, regular maximal proper right ideal
2.1.3.17
2.1.1.1
maximal proper right ideal, regular measure, Dirac
measure, E-valued spectral measure, Radon
4.3.2.16
NT
measure of x, spectral
4.3.2.19
measure space, a-finite
3.1.6.14
metric of a normed space, canonical module
2.2.7.1
module, involutive
2.3.6.1
module, involutive unital module, unital
2.3.6.1
2.2.7.1
modules, homomorphism of
2.2.7.1
modules, homomorphism of involutive modulo
NT
modulus monoid
4.2.5.1, 4.4.3.5 E 2.1.1
monoid, commutative morphism
E 2.1.1
1.5.1.1
morphism, inverse of a multipliable sequence multiplication
2.1.3.17
1.2.7.14
2.1.1.1
1.5.1.6 2.2.4.33
1.1.1.2
400
multiplication, (A, B, C) -
1.5.1.1
multiplication, compatible (left and right) multiplication, inner
1.5.1.1
multiplication, left
1.5.1.1 1.5.2.10, 2.2.7.13
multiplication, left (right) Arens multiplication, right
1.5.1.1
multiplication operator multiplicity
2.2.2.22
3.1.4.1
negative 1.7.1.1 negative part 4.2.2.9, 4.2.8.13 Nikodym's boundedness theorem nilpotent
4.3.2.14
2.1.1.1
norm 1.1.1.2 norm, complete norm, Euclidean
1.1.1.2 1.1.5.2
norm, induced
1.1.1.2
norm, quotient norm, supremum
1.2.4.2 1.1.2.2, 1.1.5.2
norm of an operator 1.2.1.3 F ) , canonical 1.2.1.9 norm of s norm topology normal
1.1.1.2
2.3.1.3
normed algebra 2.2.1.1 normed algebra, completion of a normed algebra, complex
normed algebra, real
2.2.1.13
2.2.1.1
normed algebra, involutive normed algebra, quasiunital
2.3.2.1 2.2.1.15
2.2.1.1
normed algebras, isometric
2.2.1.1
normed algebras, isometry of
2.2.1.1
normed algebras, isomorphic
2.2.1.1
normed algebras, isomorphism of normed space
1.5.1.1
2.2.1.1
1.1.1.2
normed space, bidual of a
1.3.6.1
normed space, complete 1.1.1.2 normed space, completion of a 1.3.9.1 normed space, complex normed space, involutive
1.1.1.2 2.3.2.1
Subject Index
401
normed space, ordered
1.7.1.4
normed space, real
1.1.1.2
normed spaces, isometric
1.2.1.12
normed spaces, isometry of
1.2.1.12
normed spaces, isomorphic
1.2.1.12
normed spaces, isomorphism of normed unital algebra
1.2.1.12
2.2.1.1 2.2.1.1
normed unital algebras, isometric normed unital algebras, isometry of
2.2.1.1
normed unital algebras, isomorphic
2.2.1.1
normed unital algebras, isomorphism of norms, equivalent nuclear map
1.6.1.1
number, cardinal
NT
number, ordinal
NT
object of a Banach system onto
2.2.1.1
1.1.1.2
1.5.1.1
NT
open mapping principle operator
1.4.2.3
1.2.1.3
operator, adjoint differential
3.2.2.3
operator, bounded
1.2.1.3
operator, compact
3.1.1.1
operator, Fredholm
3.1.3.1
operator, identity
1.2.1.3
operator, index of a Fredholm operator, lower bounded
3.1.3.1 1.2.1.18
operator, multiplication
2.2.2.22
operator, order of an
3.1.3.18
operator, selfadjoint differential operators, principle of inverse order complete
1.7.2.1
order continuous order faithful
3.2.2.3 1.4.2.4
1.7.2.3 4.2.2.18
order of a pole
1.3.10.9
order relation of a C*-algebra, canonical order summable order a-complete order a-continuous
1.7.2.10 1.7.2.1 1.7.2.3
4.2.1.2
402
order a-faithful
4.2.2.18
ordered Banach space
1.7.1.4
ordered normed space
1.7.1.4
ordered set, complete
1.7.2.1
ordered set, totally NT ordered set, a-complete ordered vector space ordinal number
NT
orthogonal projection parallelogram law
4.1.2.18 2.3.3.2
partition of a set p-norm
1.7.2.1
1.7.1.1
NT
1.1.2.5, 1.1.5.2
point, adherent
NT
point, extreme 1.2.7.9 point, interior NT point of adherence NT point spectrum 3.1.4.1 polar 1.3.5.1 polar representation 4.2.6.9, 4.4.3.1, 4.4.3.5 polarization identity pole (of order)
2.3.3.2
1.3.10.9
positive 1.7.1.1, 2.3.3.3, 2.3.4.1 positive linear form 1.7.1.9, 2.3.4.1 positive part
4.2.2.9, 4.2.8.13
power series
1.1.6.22
precompact
1.1.2.9
predual of a Banach space predual of a W*-algebra prepolar
1.3.1.11 4.4.1.1, 4.4.4.4
1.3.5.1
pretranspose of an operator
1.3.4.9, 4.4.4.8
principal part 1.3.10.8, 1.3.10.9 principle of inverse operators 1.4.2.4 principle of open mapping
1.4.2.3
principle of uniform boundedness product 2.1.1.1 product, C*-direct
4.1.1.6
product of a family of sets NT product of a sequence 2.2.4.33
1.4.1.2
Subject Index
projection
403
1.2.5.7
projection, orthogonal
4.1.2.18 1.3.6.19
projection of the tridual of E , canonical proper ideal
2.1.1.1
proper ideal, maximal
2.1.1.1
proper ideal, regular maximal proper left ideal
2.1.3.17
2.1.1.1
proper left ideal, maximal
2.1.1.1
proper left ideal, reguar maximal proper right ideal
2.1.3.17
2.1.1.1
proper right ideal, maximal
2.1.1.1
proper right ideal, regular maximal pure state
2.3.5.1
pure state space
2.3.5.1
purely real C*-algebra
4.1.1.8
quadratic form, associated
2.3.3.1
quadratic map, associated
2.3.3.1
quasinilpotent
2.2.4.20
quasiunital
2.2.1.15
quaternion
2.1.4.17
quotient functor
1.5.2.17
quotient map
NT, 1.2.4.1
quotient norm
1.2.4.2
quotient space
1.2.4.2
quotient A-category
1.5.2.17
quotient A-module
1.5.2.17
Raabe's ratio test radical
2.2.3.11
2.1.3.18
radius of convergence Radon measure
1.1.6.22
NT
Radon-Nikodym Theorem range of values real algebra
4.4.3.15
NT 2.1.1.1
real Banach space real C*-algebra
1.1.1.2 4.1.1.1
real C*-algebra, purely
4.1.1.8
real C*-algebra, unital
4.1.1.1
real normed space
1.1.1.2
2.1.3.17
404
real part 1.1.1.1, 2.3.1.3 real W*-algebra 4.4.1.1 reflexive 1.3.8.1 2.1.3.17 regular maximal proper ideal 2.1.3.17 regular maximal proper left ideal 2.1.3.17 regular maximal proper right ideal relatively compact 1.1.2.9 residue 1.3.10.8, 1.3.10.9 resolvent 2.1.3.1 resolvent equation 2.1.3.9 Riesz, theorem of 2.2.5.1 1.5.2.10, 2.2.7.13 right Arens multiplication right carrier 4.3.3.1 right ideal 2.1.1.1 2.1.1.1 right ideal, maximal proper right ideal, proper 2.1.1.1 2.1.3.17 right ideal, regular maximal proper right ideal generated by 2.1.1.2 right invertible 1.5.1.5 right multiplication 1.5.1.1 right shift 1.2.2.9, E 1.2.11 right (unital) A-module, 1.5.1.10 scalar 1.1.1.1 Schwartz space of rapidly decreasing C~-functions Schwarz inequality 2.3.3.9 section filter, lower 1.1.6.1 section filter, upper 1.1.6.1 selfadjoint 2.3.1.1 selfadjoint differential operator 3.2.2.3 self-normal 2.3.1.3 seminorm 1.1.1.2 semi-simple algebra 2.1.3.18 sequence NT series, Laurent 1.3.10.8, 1.3.10.9 series, power 1.1.6.22 sesquilinear form 2.3.3.1 sesquilinear form, adjoint 2.3.3.1 sesquilinear map 2.3.3.1
2.4.6.5
Subject Index
405
sesquilinear map, adjoint
2.3.3.1
sesquilinear map, Hermitian set, Baire
2.3.3.3
1.7.2.12
set, bounded
1.1.1.2
set, complete ordered set, exact
1.7.2.1
1.7.2.12
set, partition of a
NT
set, totally ordered set, p-null
NT
NT
set, a-complete ordered sharp cone shift, left
1.7.2.1
1.3.7.4 1.2.2.9
shift, right
1.2.2.9
simple C*-algebra
4.3.5.1
simultaneously compatible space, Banach
1.5.1.1
1.1.1.2
space, bidual of a normed
1.3.6.1
space, complete normed
1.1.1.2
space, completion of a normed
1.3.9.1
space, complex Banach
1.1.1.2
space, complex normed
1.1.1.2
space, dual
1.3.1.11
space, hyperstonian
1.7.2.12
space, involutive
2.3.1.1
space, involutive Banach
2.3.2.1
space, involutive normed
2.3.2.1
space, involutive vector space, normed
2.3.1.3
1.1.1.2
space, ordered Banach
1.7.1.4
space, ordered normed
1.7.1.4
space, ordered vector space, pure state
1.7.1.1 2.3.5.1
space, quotient
1.2.4.2
space, real Banach
1.1.1.2
space, real normed
1.1.1.2
space, state space, Stone
2.3.5.1 1.7.2.12
space, subspace of a normed
1.1.1.2
406
space, vector
1.1.1.1
space, a-Stone
1.7.2.12
spaces, isometric normed
1.2.1.12
spaces, isometry of normed
1.2.1.12
spaces, isomorphic normed
1.2.1.12
spaces, isomorphism of involutive vector spaces, isomorphism of normed spectral decomposition
4.3.2.19
spectral measure, E-valued spectral measure of x spectral r~dius
4.3.2.16 4.3.2.19
2.1.3.1
spectrum, essential
3.1.3.24
spectrum, point
3.1.4.1
spectrum of an element
2.1.3.1
spectrum of a Gelfand algebra state
2.4.1.1
2.3.5.1
state, pure
2.3.5.1
state space
2.3.5.1
state space, pure
2.3.5.1
step function
NT
Stone space
1.7.2.12
strongly symmetric involutive algebra subalgebra
2.3.1.26
2.1.1.1
subalgebra, unital
2.1.1.3
subalgebra generated by
2.1.1.4
subalgebra generated by, involutive subspace, complemented subspace of a normed space sum, C*-direct
1.1.1.2
4.1.1.6
sum, direct
1.2.5.3
sum of a family
1.1.6.2
summable, absolutely
1.1.6.9
summable, order
1.7.2.10
summable family
1.1.6.2
support of a function
NT
support of a Radon measure 1.7.2.1
2.3.1.18
1.2.5.3
subspace generated by, closed vector
supremum
2.3.1.3
1.2.1.12
NT
1.1.5.5
Subject Index
407
supremum norm surjective
1.1.2.2, 1.1.5.2
NT
symbol, Kronecker's
1.2.2.6
symmetric involutive algebra
2.3.1.26
symmetric involutive algebra, strongly Theorem of Alaoglu Bourbaki Theorem of Banach
1.2.8.1
1.3.1.2
Theorem of Banach-Steinhaus Theorem of closed graph Theorem of Gelfand
1.4.1.2 1.4.2.19
2.2.5.4
Theorem of Gelfand-Mazur
2.2.5.5
Theorem of Hahn-Banach Theorem of Laurent
1.3.3.1
1.3.10.8
Theorem of Liouville
1.3.10.6
Theorem of Krein-Milman
1.3.1.10
Theorem of Krein-Smulian
1.3.7.3
Theorem of Minkowski
1.1.3.4
Theorem of Murray
1.2.5.8
Theorem of Radon-Nikodym Theorem of Riesz
4.4.3.15
2.2.5.1 1.3.5.16
Theorem of Weierstrass-Stone topological cardinality
NT
topological zero-divisor
2.2.4.24
topology, norm
1.1.1.2
topology, weak
1.3.6.9
totally ordered set transpose kernel
NT 3.1.6.5
transpose of a functor
1.5.2.3
transpose of an operator
1.3.4.1 1.5.2.2
transpose unital category of s transposition functor of s triangle inequality
1.5.2.2
1.1.1.2
tridual of a Banach system tridual of a normed space u-invariant ultrafilter, free
1.5.1.9 1.3.6.1
3.1.4.4 NT
uniform boundedness, principle of unit
1.5.1.1, 1.5.1.4, 2.1.1.1
1.4.1.2
408
unit, approximate unit ball
2.2.1.15
1.1.1.2
umt of an inner multiplication unital algebra
1.5.1.1
2.1.1.3
unital algebra, normed 2.2.1.1 umtal algebra associated to 2.1.1.8 umtal algebra associated to, involutive unital algebra homomorphism unital algebra isomorphism
2.3.1.9
2.1.1.6 2.1.1.6
unital algebras, isometric normed
2.2.1.1
unital algebras, isometry of normed unital algebras, isomorphic
2.2.1.1
2.1.1.6
2.2.1.1 2.2.1.1 unital algebras, isomorphism of normed unital Banach algebra 2.2.1.1 2.2.1.4 umtal Banach algebra associated to 2.2.1.1 umtal Banach algebras, isomorphism of
unital algebras, isomorphic normed
unital Banach category 1.5.1.1 unltal Banach subalgebra generated by umtal C*-algebra
2.2.1.9
4.1.1.1
unital C*-algebra, complex 4.1.1.1 unital C*-algebra, real 4.1.1.1 umtal C*-algebra, associated to a C*-algebra umtal C*-subalgebra 4.1.1.1 unital C*-subalgebra generated by unltal E-algebra
2.3.6.1
2.2.7.1
unital E-module, involutive unital Gelfand algebra
4.1.1.1
2.2.7.1
unital E-algebra, involutive unital E-module
4.1.1.13
2.3.6.1
2.4.1.1
unital Gelfand algebra, involutive
2.4.2.1
unital involutive algebra associated to
2.3.1.9
unital involutive Banach algebra associated to
2.3.2.9
unital involutive normed algebra associated to
2.3.2.9
unital left A-module 1.5.1.10 unital normed algebra associated to unital normed algebras, isomorphism of unital right A-module
1.5.1.10
2.2.1.4 2.2.1.1
Subject Index
unital subalgebra 2.1.1.3 unital subalgebra generated by 2.1.1.4 2.3.1.18 unital subalgebra generated by, involutive unital W*-subalgebra 4.4.4.5 unital W*-subalgebra generated by 4.4.4.5 unital A-category 1.5.1.14 unital A-module 1.5.1.12 unital (A, A)-module 1.5.1.12 unitary 2.3.1.3 upper bound 1.7.2.1 upper section filter 1.1.6.1 upward directed 1.1.6.1 vector lattice 1.7.2.1 vector space 1.1.1.1 vector space, involutive 2.3.1.3 2.3.1.3 vector spaces, isomorphisms of involutive Volterra integral equation 2.2.4.22 W*-algebra 4.4.1.1 W*-algebra, complex 4.4.1.1 W*-algebra, predual of a 4.4.1.1, 4.4.4.4 W*-algebra, real 4.4.1.1 W*-algebras, isometry of 4.4.4.5 W*-homomorphism 4.4.4.5 W*-subalgebra 4.4.4.5 W*-subalgebra, unital 4.4.4.5 W*-subalgebra generated by 4.4.4.5 W*-subalgebra generated by, unital 4.4.4.5 weak topology 1.3.6.9 weakly conjugate exponents 1.2.2.1 zero-divisor 2.1.1.1 zero-divisor, topological 2.2.4.24 A-categories, functor of (unital) 1.5.2.1 A-category 1.5.1.14 A-category, quotient 1.5.2.17 A-category, unital 1.5.1.14 A-module 1.5.1.12 A-module, left (right) 1.5.1.10 A-module, quotient 1.5.2.17
409
410
A-module, unital 1.5.1.12 1.5.1.10 A-module, unital left (right) A-modules, functor of left (right) 1.5.2.1 A-subcategory 1.5.2.16 A-submodule 1.5.2.16 (A, A)-module 1.5.1.12 (A, A)-module, unital 1.5.1.12 p-null set NT a-complete, C-order 4.3.2.3 a-complete order 1.7.2.1 a-complete ordered set 1.7.2.1 a-continuous, order 1.7.2.3 a-faithful, order 4.2.2.18 a-finite measure space 3.1.6.14 a-Stone space 1.7.2.12
Symbol Index
411
Symbol Index N T means N o t a t i o n and Terminology lal
4.4.3.5
a*
2.3.1.30
A*
2.3.1.1
A
NT
~1
~
NT
~
1.3.5.1
A c , A c~ , A ~
2.1.1.16
.A', .A", .A"'
1.5.1.9
AA, 1.3.6.9 aa' , a' a 2.2.7.8 ab 2.1.4.23 ax 2.2.7.23 a'x" 2.2.7.11 a"x' 1.5.2.8 A/B 1.5.2.17 A + B
1.2.4.1
A\B AAB AxB
NT NT NT
A + z
1.2.4.1
B
1.1.2.4 NT
c
1.1.2.3, 2.1.4.3
c(T) Co
co(T) C(T) C(T,E) Co(T)
1.1.2.3, 2.1.4.3 1.1.2.3, 2.1.4.3 1.1.2.3, 2.1.4.3 1.1.2.4, 2.1.4.4 1.1.2.8 1.2.2.10, 2.1.4.4
Card
NT
Coker
1.2.4.5
dA
1.1.4.1
Det
NT
Dim
1.1.2.18
412
T~(k, #, v)
3.1.6.15
T~o(k, ~)
3.1.6.1
E'
1.2.1.3
E"
1.3.6.1
E'"
1.3.6.1
/~
2.1.5.1, 2.1.5.7, 2.3.1.38, 2.3.1.40
eA
1.1.2.1
et eT
1.1.2.1 1.1.2.1
eT
1.1.2.1
ex
2.2.3.5
Ea(u) Eb(u)
3.1.3.18 3.1.3.18
Em,n
2.1.4.23, 2.3.1.30
En,n
2.1.4.24, 2.3.1.31 1.1.2.1
ET E (T)
1.1.2.1
E~
1.7.2.3
E~
1.7.2.3 4.4.4.4
E+
1.7.1.1, 4.2.1.1
E#
1.1.1.2
E+#
1.7.1.4
E_~, E~
2.2.7.15
E(x)
2.3.2.15
E(x, 1)
2.3.2.15
E -5+ F
1.5.1.1
E -~ F
1.5.1.1
A
E/F
1.2.4.1, 2.1.1.13, 2.3.1.42 4.3.1.7
f'
1.1.6.24
f
2.3.3.1
.T'A .T'(E)
1.2.6.1 3.1.3.1
.T'(E, F)
3.1.3.1
~I
1.1.6.1
flS
NT
Symbol Index
f(a, .) f(.,b) f(A) f(x)
413
NT NT NT NT, 4.1.3.1, 4.1.3.2, 4.3.2.5
f-1
NT
--1
f (B)
NT
-1
f (y)
NT
f :X--+ Y NT f : X --+ Y , x ~ T(x) F[s,t] NT F[t] NT F @G
NT
1.2.5.3
{f=g} {f # g} {f > a} gof
NT NT NT NT, 1.5.2.1
IH
1.7.2.3 2.1.4.17, 2.3.1.46, 4.1.1.31
im Im
1.1.1.1, 2.3.1.22 1.2.4.5
-~A
Ind u
3.1.3.1
Ind U
3.1.3.21 1.3.6.3, 1.5.2.1
jE jEF
1.5.2.1 1.1.1.1
IK
IK1 2.1.1.3 IK[-], IK[.,-] 1.1.1.1 K:(E)
3.1.1.1
K:(E, F)
kx
3.1.1.1
kI
3.1.6.1, 3.1.6.15 3.1.6.5
k*
3.1.6.5
k~x N
k
3.1.6.1, 3.1.6.15 1.2.3.1
u
k Ker s
1.2.3.1 1.2.4.5 1.2.1.3, 1.5.1.1, 2.1.4.6
414
s
1.5.1.1
s
1.2.1.3 1.6.1.1, 1.6.1.3
•1
/:]~
1.6.1.13
ep
1.1.2.5 1.1.2.5 1.1.2.3
eP(T) go t~~
1.1.2.3
~
1.1.2.2, 2.1.4.3 1.1.2.2, 2.1.4.3
~~176 (T)
gP'q(S, T) go'q(S,T)
1.2.3.2
1.2.3.2 2.2.3.9, 4.2.4.4
log A/ib IN
1.1.2.26 NT
INn Nx,
1.1.3.3 2.3.4.1
No Pr
2.3.1.3 4.1.2.18
~3
1.1.2.1
~[~f
1.1.2.1 NT NT
(~ IR
IR
NT
re
1.1.1.1, 2.3.1.3
Re
2.3.1.1
Re E # r(x),
2.3.2.1
rE(x)
2.1.3.1
Sn 2.3.1.3 $(IR n) 2.4.6.5 Suppf Supp#
NT NT
T
2.4.4.1
u'
1.3.4.1
u" u*
1.3.6.15 2.3.1.1, 3.2.2.3
u
2.1.5.11, 2.3.1.41
o
Un
2.3.1.3
S y m b o l Index
415
Uo(t)
1.1.1.2
uT(t)
1.1.1.2
X~ ~
NT
xn
2.1.1.1
x':'
4.2.4.1, 4.2.4.4
x -n
2.1.2.5
x~
2.1.1.3
x -1
1.5.1.6, 2.1.2.4
x*
2.3.1.1 E
~, ~ v
x
2.4.1.1 2.3.5.1
Ixl
4.2.5.1
x+,x -
4.2.2.9
x ' + , x '-
4.2.8.13
(Xt)eE I NT xa 2.2.7.23 x'a"
1.5.2.8
x"a'
2.2.7.11
xx' , x' x
1.5.2.5
(x, x'}, (x', x} x 9y
1.2.1.3
2.2.2.7
x" -A y", x" ~ y" x | y
1.5.2.10, 2.2.7.13
3.1.1.25
{x I P(x)}
NT
{x e X I P ( x ) } (.,J}y 2Z
NT
1.3.3.3 NT
z+ A
1.2.4.1 1.1.1.1
lal
1.1.1.1
aA
1.2.4.1
(:)
2.2.3.10
]~, 9[, ]~, ~], [~, 9[, [~, ~] A
NT
5st
1.2.2.6
~t
1.2.7.14
5(s,t)
I,I
1.2.2.6 NT
NT
416
#
u
2.2.2.10 2.1.4.1, 2.3.1.4
HEt tEE
H X~
H zn, H x. nc A
2.2.4.33
nEIN
a ( E ) , or0(E) 2.4.1.1 a(T) 2.4.4.1 a(x), erE(X) 2.1.3.1 ae(u) 3.1.3.24
a~(u) y] x(t)
3.1.4.1 1.1.2.1
tET q E Xn n=p
1.1.6.2
y]~x~
E(-, z'~)y~
1.3.3.3
tCl oo
E
ognxn
n=O <
y]~-x~
1.1.6.22
1.7.2.10
~(E), ~o(E) 1 1E 1 O:--X
2.3.5.1
1.2.1.3, 2.1.1.3 1.2.1.3, 1.5.1.5 2.2.3.5
+ x] ~ + • \ <., .>
1.2.4.1 NT NT 1.2.1.3
{'1"} NT { . - .}, { - ~ - } , { - > .} = (modp) NT
I1" [[1
1.6.1.1
NT
Symbol Index
I1 I]0 ]1-I1~
1.1.2.3 1.1.2.2
V, 3, 3! o @
NT
NT, 1.5.2.1 1.2.5.3
[., .], ]., .[, [., .[, ]., .] f x d#
417
4.3.2.17
NT
