Fundamentals of the Theory of Operator Algebras Vol.4: Special Topics-Advanced Theory, an Exercise Approach (Pure and Applied Mathematics (Academic Press), Volume 100)

603 Pages·1992·7.31 MB·English

by  Richard V. Kadison

FUNDAMENTALS OF THE THEORY
OF OPERATOR ALGEBRAS
SPECIATL OPICS
VOLUMEI V
Advanced Theory-An Exercise Approach
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FUNDAMENTALS OF THE THEORY
OF OPERATOR ALGEBRAS
SPECIATLO PICS
VOLUMIEV
A duanced The0 ry-A n Exercise Approach
Richard V. Kadison John R. Ringrose
Department of Mathematics School of Mathematics
University of Pennsylvania University of Newcastle
Philadelphia, Pennsylvania Newcastle upon Tyne, England
COPYRIGHT 0 1992, BY RICHARD V. KADISON
ALL PARTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELEC-
TRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORD-
ING, OR ANY INFORMATION STORAGE AND RETRIEVAL
SYSTEM, WITHOUT PERMISSION IN WRITING FROM
RICHARD V. KADISON.
THE STATEMENTS OF ALL EXERCISES
(0A CADEMIC PRESS 1986) APPEAR WITH THE
PERMISSION OF ACADEMIC PRESS.
BIRKHAIJSER BOSTON
675 Massachusetts Avenue, Cambridge, MA 02 139-3309
Library of Congress Cataloging-in-Publication Data
Kadison, Richard V., 1925-
Fundamentals of the theory of operator algebras.
(Pure and applied mathematics ; 100-100.2
(QA3.P8 vol. LO))
Vol. 4 has imprint : Boston : Birkhauser.
Vol. 3-4: lacks series statement.
Includes bibliographies and indexes.
Contents: v. 1. Elementary theory -- v. 2. Advanced
thcory [etc.] -- v. 4. Special topics : advanced
theory, an exercise approach.
1. Operator algebras. I. Ringrose, John R.
11. Title. 111. Series: Pure and applied mathematics
(Academic Press) ; 100-100, 2.
QA326.K26 1983 512l.55 82-13768
ISBN 0-8176-3498-3 (v. 4)
Printed on acid-free paper
Printed by Quinn-Woodbine, Woodbine, New Jersey
Printed in the U.S.A.
ISBN 0-8 176-3498-3
ISBN 3-7643-3498-3
9 8 7 6 5 4 3 2 1
CONTENTS
ix
Contents of Volume III
xi
Exercise Groupings
Chapter 6. Comparison Theory of Projections 274
- Exercises and Solutions
Chapter 7. Normal States and 312
Unitary Equivalence of
von Neumann Algebras
- Exercises and Solutions
Chapter 8. The Trace 368
- Exercises and Solutions
Chapter 9. Algebra and Commutant 45 1
- Exercises and Solutions
Chapter 10. Special Representations 546
of C*-Algebras
- Exercises and Solutions
Chapter 11. Tensor Products 680
-
Exercises and Solutions
Chapter 12. Approximation by Matrix Algebras 726
- Exercises and Solutions
Chapter 13. Crossed Products 783
-
Exercises and Solutions
Chapter 14. Direct Integrals and Decompositions 818
-
Exercises and Solutions
Bibliography 842
index 847
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PREFACE
These volumes are companions to the treatise; ‘‘Fundamentals
of the Theory of Operator Algebras,” which appeared as Volume 100
- I and I1 in the series, Pure and Applied Mathematics, published
by Academic Press in 1983 and 1986, respectively. As stated in the
preface to those volumes, “Their primary goal is to teach the sub-
ject and lead the reader to the point where the vast recent research
literature, both in the subject proper and in its many applications,
becomes accessible.” No attempt was made to be encyclopzedic; the
choice of material was made from among the fundamentals of what
may be called the “classical” theory of operator algebras.
By way of supplementing the topics selected for presentation
in “Fundamentals,” a substantial list of exercises comprises the last
section of each chapter. An equally important purpose of those exer-
cises is to develop “hand-on” skills in use of the techniques appearing
in the text. As a consequence, each exercise was carefully designed
to depend only on the material that precedes it, and separated into
segments each of which is realistically capable of solution by an at-
tenti ve, diligent, well-mot i vated reader.
The process by which the exercises were designed involved solv-
ing each of them completely and then subjecting the solutions to
detailed scrutiny. It became apparent, in the course of this oper-
ation, that the written solutions could be of considerable value, if
they were made generally available, as models with which a reader’s
solutions could be compared, as indicators of methods and styles for
producing further solutions on an individual basis, and as a speedy
route through one or another of the many special topics that sup-
plement those in the text proper of “Fundamentals” (for the reader
without the time or inclination to develop it as an exercise set). The
present texts contain those written solutions; the first of these texts
has the solutions to the exercises appearing in Volume I of “Fun-
damentals” and the second has the solutions to those appearing in
Volume 11. The statements of the exercises precede their solutions,
for the obvious convenience of the reader.
In most instances, where an exercise or group of exercises de-
viii P RE FAC E
velops a topic, the solutions have been given in what the authors
feel is optimal form. Solutions are, of course, geared to the state
of knowledge developed at the point in the book where the exer-
cise occurs. Very occasionally, this necessitates an approach that is
slightly less than optimal (for example, Exercise 2.8.10 occurs before
square roots of positive operators, which could be used to advantage
in its solution, are introduced). From time to time, an exercise reap-
pears, with the task of finding a solution involving newly acquired
information.
Of course, knowledge of and long experience with the literature
of the subject has had a major influence on both the content of the
exercises and the form of their solutions. In many cases, there is no
specific source for the exercise or its solution. In virtually no instance
was the solution of am exercise copied directly from the literature; the
solutions were constructed with the path and stage of development
of “Fundamentals” where the exercise occurs, very much in mind.
Often, where a set of exercises develops a special topic, the solutions
present a new and simpler route to the results appearing in the set.
References are placed after the solutions. As in “Fundamentals,”
no attempt is made to be thorough in referencing. The references
appearing are chosen with a few goals in mind: to supply the reader
with additional material, closely related to the exercise and its so-
lution, that may be of interest for further study, to provide a very
sketchy historical context that may be deepened by consulting the
papers cited and their bibliographies. Where a set of consecutive
exercises is largely inspired by a single article, for the most part, the
first of the set and/or the highpoints of the topic note the article.
A guide to the topics treated by sets of related exercises follows
this preface.
CONTENTS OF VOLUME I11
vii
Preface
ix
Contents of Volume iV
xi
Exercise Groupings
Chapter 1. Linear Spaces 1
- Exercises and Solutions
Chapter 2. Basics of Hilbert Space and 40
Linear Operators
- Exercises and Solutions
Chapter 3. Banach Algebras 84
-
Exercises and Solutions
Chapter 4. Elementary C*-Algebra Theory 139
- Exercises and Solutions
Chapter 5. Elementary von Neumann 207
Algebra Theory
- Exercises and Solutions
Bibliography 263
268
Indez