Dynamical Entropy in Operator Algebras
During the last 30 years there have been several attempts at extending the notion of entropy to noncommutative dynamical systems. The authors present in the book the two most successful approaches to the extensions of measure entropy and topological entro
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A Series of Modern Surveys in Mathematics
Editorial Board M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollár, Princeton G. Laumon, Orsay H. W. Lenstra, Jr., Leiden J. Tits, Paris D. B. Zagier, Bonn/Paris G. M. Ziegler, Berlin Managing Editor R. Remmert, Münster
Volume 50
Sergey Neshveyev Erling Størmer
Dynamical Entropy in Operator Algebras
123
Sergey Neshveyev Erling Størmer Department of Mathematics University of Oslo P. B. 1053 Blindern 0316 Oslo, Norway e-mail: [email protected] [email protected]
Library of Congress Control Number: 2006928835
Mathematics Subject Classification (2000): 46L55, 28D20
ISSN 0071-1136 ISBN-10 3-540-34670-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34670-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors using a Springer LATEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper
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Preface
When the algebraic formalism of quantum statistical mechanics and quantum field theory gained momentum in the 1960’s, it started a very fruitful interplay between mathematical physics and operator algebras. The study of automorphisms and their invariant states became a blooming discipline, and a subject of noncommutative ergodic theory evolved. With the great success of entropy in classical (abelian) ergodic theory it was natural to extend that theory to operator algebras. In some cases, like that of quantum spin lattice systems, that is rather straightforward, since in those models the mean entropy definition in the classical case can be extended to C∗ -algebras by replacing partitions by local algebras. But in more general cases the C∗ -algebras generated by finite dimensional C∗ -algebras can easily be infinite dimensional, so the mean entropy cannot be used as a definition. In order to define dynamical entropy for automorphisms of C∗ -algebras one has to rewrite the classical definition in a form independent of the join of partitions and use that as the basis for a definition. This was done
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