Tomita-Takesaki Theory in Algebras of Unbounded Operators
These notes are devoted to a systematic study of developing the Tomita-Takesaki theory for von Neumann algebras in unbounded operator algebras called O*-algebras and to its applications to quantum physics. The notions of standard generalized vectors and s
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1699
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Atsushi Inoue
Tomita-Takesaki Theory in Algebras of Unbounded Operators
Springer
Author Atsushi Inoue Department of Applied Mathematics Fukuoka University Fukuoka 814-0180, Japan e-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme Inoue, Atsushi: Tomitak-Takesaki theory in algebras of unbounded operators / Atsushi Inoue. - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo: Springer, 1998 (Lecture notesin mathematics; 1699) ISBN 3-540-65194-2
Mathematics Subject Classification (1991): 47D40, 47D25, 46L60, 46N50 ISSN 0075-8434 ISBN 3-540-65194-2 Springer-Verlag 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 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: Camera-ready TEX output by the author SPIN: 10650140 41/3143-543210 - Printed on acid-free paper
Preface
Main part of this note is a summary of my studies during several years of the Tomita-Takesaki theory in 0* -algebra. In 1995 I began this work for the preparation of the seminars during the summer of 1996 at the Mathematics Institute of Leipzig university. I wish to thank Professor K. D. Kiirsten and K. Schmiidgen for their warm hospitality, for their interest in this work and for their encouragement. Further, I wish to thank them for many helpful discussions and suggestions when they visited the Department of our university in 1996, 1997. I also acknowlege Professor J. P. Antoine (Louvain Catholique University), Van Daele (Leuven Katholieke University), W. Karwowski (Wroclaw University), G. Epifanio and C. Trapani (Palermo University) and A. Arai and M. Kishimoto (Hokkaido University) for giving me opportunities of the seminars and lectures about this work for their colleagues and graduate students of their Mathematical Departments and for many invaluable discussions and many helpful suggestions. I should like to thank Professor H. Kurose and Dr. Ogi fo
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