Elementary Theory of C*-algebras and von Neumann Algebras
The first section of this chapter contains basic material on C*-algebras. The second section presents definitions and elementary results concerning representations of C*-algebras. The third section is on the special von Neumann algebra ß(H). Here some of
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Erik M. Alfsen Frederic W. Shultz
State Spaces of Operator Algebras Basic Theory, Orientations, and C*-products
Springer Science+Business Media, LLC
Frederic W. Shultz Department of Mathematics Wellesley College Wellesley, MA 02481 U.SA.
Erik M. Alfsen Mathematical Institute University of Oslo N-0136 Oslo Norway
Library of Congress Cataloging-in-Publication Data Alfsen, Erik M. (Erik Magnus), 1930State spaces of operator algebras : basic theory, orientations and C*-products I E. Alfsen, F. Shultz. p. cm - (Mathematics : theory & applications) lncludes bibliographical references and index. ISBN 978-1-4612-6634-1 ISBN 978-1-4612-0147-2 (eBook) DOI 10.1007/978-1-4612-0147-2 1. Operator algebras. 2. State-space methods. 1. Shultz, Frederic W., 1945- II. Mathematics (Boston, Mass.) QA326.A43 2001 512'.55-dc21
2001025873 CIP
AMS Subject Classifications: Primary: 46L30, 46L05, 46L10,46L52, 46L60, 46L70, 46B04, 46B20, 46B40,46N50, 47L50, 47L90, 52A05; Secondary: 46C05, 46E25, 46110, 47 A60, 47CI0, 47C15, 47L30 Printed on acid-free paper. © 2001 Springer Science+Business Media New York Originally published by Birkhăuser Boston in 2001 Softcover reprint ofthe hardcover Ist edition 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-6634-1 Typeset by the authors in 1EX
987654321
SPIN 10535316
Preface
The topic of this book is the theory of state spaces of operator algebras and their geometry. The states are of interest because they determine representations of the algebra, and its algebraic structure is in an intriguing and fascinating fashion encoded in the geometry of the state space. From the beginning the theory of operator algebras was motivated by applications to physics, but recently it has found unexpected new applications to various fields of pure mathematics, like foliations and knot theory, and (in the Jordan algebra case) also to Banach manifolds and infinite dimensional holomorphy. This makes it a relevant field of study for readers with diverse backgrounds and interests. Therefore this book is not intended solely for specialists in operator algebras, but also for graduate students and mathematicians in other fields who want to learn the subject. We assume that the reader starts out with only the basic knowledge taught in standard graduate courses in real and complex variables, measure theory and functional ana
