Table Of ContentStudies in Advanced Mathematics
An Introduction to Operator Algebras
Studies in Advanced Mathematics
Series Editor
STEVEN G. KRANTZ
Washington University in St. Louis
Editorial Board
R. Michael Beals Gerald B. Folland
Rutgers University University of Washington
Dennis de Turck William Helton
University of Pennsylvania University of California at San Diego
Ronald DeVore Norberto Salinas
University of South Carolina University of Kansas
L. Craig Evans Michael E. Taylor
University of California at Berkeley University of North Carolina
Titles Included in the Series
Real Analysis and Foundations, Steven G. Krantz
CR Manifolds and the Tangential Cauchy-Riemann Complex, Albert Boggess
Elementary Introduction to the Theory of Pseudodifferential Operators,
Xavier Saint Raymond
Fast Fourier Transforms, James S. Walker
Measure Theory and Fine Properties of Functions, L. Craig Evans and
Ronald Gariepy
Partial Differential Equations and Complex Analysis, Steven G. Krantz
The Cauchy Transform, Potential Theory, and Conformal Mapping, Steven R. Bell
Several Complex Variables and the Geometry of Real Hypersurfaces,
John P. D’Angelo
Modem Differential Geometry of Curves and Surfaces, Alfred Gray
An Introduction to Operator Algebras, Kehe Zhu
Wavelets: Mathematics and Applications, John J. Benedetto and
Michael W. Frazier
KEHE ZHU
State University of New York at Albany
An Introduction to Operator Algebras
CRC Press
Taylor &. Francis Group
'v..— Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Library of Congress Cataloging-in-Publication Data
Zhu, Kehe 1961-
An introduction to operator algebras / Kehe Zhu.
p. cm. — (Studies in advanced mathematics)
Includes bibliographical references and index.
ISBN 0-8493-7875-3
1. Operator algebras. I. Title. II. Series.
QA326.Z48 1993
512'.55—dc20 93-7172
CIP
This book represents information obtained from authentic and highly regarded sources. Re
printed material is quoted with permission, and sources are indicated. A wide variety of
references are listed. Every reasonable effort has been made to give reliable data and informa
tion, but the authors and the publisher cannot assume responsibility for the validity of all
materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means,
electronic or mechanical, including photocopying, microfilming, and recording, or by any
information storage and retrieval system, without permission in writing from the publisher.
Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431.
©1993 by CRC Press, Inc.
International Standard Book Number 0-8493-7875-3
To Peijia and Peter
Contents
Preface ix
I Banach Algebras 1
1 Review on Functional Analysis 2
2 Banach Algebras and the Invertible Group 8
3 The Spectrum 15
4 Multiplicative Linear Functionals 21
5 The Gelfand Transform and Applications 27
6 Examples of Maximal Ideal Spaces 34
7 Non-Unital Banach Algebras 41
II (7*-Algebras 49
8 C*-Algebras 50
9 Commutative (7*-Algebras 55
10 The Spectral Theorem and Applications 60
11 Further Applications 66
12 Polar Decomposition 71
vii
CONTENTS
13 Positive Linear Functionals and States 77
14 The GNS Construction 84
15 Non-Unital C*-Algebras 89
III Von Neumann Algebras 97
16 Strong- and Weak-Operator Topologies 98
17 Existence of Projections 103
18 The Double Commutant Theorem 108
19 The Kaplansky Density Theorem 113
20 The Borel Functional Calculus 118
21 L°° as a von Neumann Algebra 124
22 Abelian von Neumann Algebras 129
23 The -Functional Calculus 134
24 Equivalence of Projections 140
25 A Partial Ordering 145
26 Type Decomposition 149
Bibliography 154
Index 155
Preface
This is a graduate text for a one semester course in operator algebras. I taught
such a course in the spring semester of 1992 at SUNY-Albany and this book
represents a slightly expanded version of the lecture notes that I prepared during
the semester. Since the theory of operator algebras has become such a vast
subject, a difficult task the instructor faces when teaching a one semester course
is deciding what topics to cover. Obviously the decision has to be somewhat
personal. I hope my way of choosing the topics would be of some help for
future instructors.
The book consists of twenty-six lectures. The two lectures on non-unital
Banach and C*-algebras can be omitted without affecting any other part of the
book. Almost all the exercises are elementary, so that they are appropriate for
homework assignments.
The latex style file for this book-series uses two different systems of number
ing in each chapter. Theorems, propositions, lemmas, corollaries, and remarks
are numbered as one group, while definitions and examples are numbered as
another.
I would like to thank all the students who attended my Spring 1992 class. They
are Robert Clark, Eric Grossman, Donghan Luo, Kevin Madigan, Jim Panariello,
John Racquet, Jose Reyes, and John Rice. I am grateful to Steven Krantz and
Norberto Salinas for their interest in the book project. Thanks are also due to
Wayne Yuhasz, the CRC Press executive editor in Physical Sciences, for his
enthusiasm and technical assistance in preparing the camera-ready manuscript.
Finally I would like to thank the National Science Foundation for its support
during the preparation of the manuscript.
