Noncommutative Dynamics and E-Semigroups
The term Noncommutative Dynamics can be interpreted in several ways. It is used in this book to refer to a set of phenomena associated with the dynamics of quantum systems of the simplest kind that involve rigorous mathematical structures associated
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Springer Science+Business Media New York
William Arveson
Noncommutative Dynamics and E-Semigroups
,
Springer
William Arveson University of Califomia at Berkeley Department of Mathematics Evans Hall Berkeley, CA 94720-000 I [email protected]
Mathcmatics Subject Classification (2000): 46L09, 46L55 Library ofCongress Cataloging-in-Publication Data Arveson, William. Noncommutative dynamics and E-semigroups I William Arveson. p. cm.-- (Springer monographs in mathematics) Includes bibliographical references and index. I. Noncommutative algebras. 2. Endomol1lhisms (Group thcory) 3. Semigroups. l. lilIe 11. Sen es QA251.4.A782oo3 512'.24--(jc21
Pnnted on acid-frce paper.
2002042734
ISBN 978-1-4419-1803-1 ISBN 978-0-387-21524-2 (eBook) . DOI 10.1007/978-0-387-21524-2 (02003 Springer Seience+Business Media New York Originally publishcd by Springer-Verlag New York, Ine in 2003 Softcover reprint of the hardcover 1st edition 2003 All rights re erved. This work may not be tran lated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpt in connection with review or cholarly analy i . U c inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or di si milar methodology now known or here·after developed i forbidden. The u ein this publication of trade names, trademarks, service marks, and similar terms, even ifthey are not identified as such, is not 10 be taken as an expre sion of opinion as 10 whether or notthey are subject to proprietary rights. 9 8 7 654 321
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Preface These days, the term Noncommutative Dynamics has several interpretations. It is used in this book to refer to a set of phenomena associated with the dynamical evolution of quantum systems of the simplest kind that involve rigorous mathematical structures associated with infinitely many degrees of freedom. The dynamics of such a system is represented by a one-parameter group of automorphisms of a noncommutative algebra of observables, and we focus primarily on the most concrete case in which that algebra consists of all bounded operators on a Hilbert space. If one introduces a natural causal structure into such a dynamical system, then a pair of one-parameter semigroups of endomorphisms emerges, and it is useful to think of this pair as representing the past and future with respect to the given causality. These are both Eo-semigroups, and to a great extent the problem of understanding such causal dynamical systems reduces to the problem of understanding Eo-semigroups. The nature of these connections is discussed at length in Chapter 1. The rest of the book elaborates on what the author sees as the important aspects of what has been learned about Eo-semigroups during the past fifteen years. Parts of the subject have evolved into a satisfactory theory with effective toolsj other parts remain quite mysterious. Like von Neumann algebras, Eo-semigroups divide
