Ergodic Transformation Groups and the Associated von Neumann Algebras
We now begin our systematic study of the von Neumann algebra associated with an ergodic transformation group of a standard measure space. We touched the subject lightly in Chapter V when we constructed a factor of type III. As ergodic transformation group
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Springer-Verlag Berlin Heidelberg GmbH
M. Takesaki
Theory of Operator Algebras III
Springer
Author Masamichi Takesaki University of California Department of Mathematics Los Angeles, CA 90095-1555 USA e-mail: [email protected]
Founding editor of the Encyclopaedia of Mathematical Sciences: R. V. Gamkrelidze
Mathematics Subject Classification
(2000 ):
22D25, 46LXX, 47CXX, 47DXX
Theory of Operator Algebras I by M. Takesaki was published as Vol. 124 of the Encyclopaedia of Mathematical Sciences, ISBN 978-3-642-07688-6 Theory of Operator Algebras II by M. Takesaki was published as Vol. 125 of the Encyclopaedia of Mathematical Sciences, ISBN 978-3-642-07688-6 ISSN 0938-0396 ISBN 978-3-642-07688-6 ISBN 978-3-662-10453-8 (eBook) DOI 10.1007/978-3-662-10453-8 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-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 1st edition 2003 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: Typeset in !5f]lX by johannes Kiister · typoma · www. typoma.com, based on the author's plainT]lX files. Cover Design: E. Kirchner, Heidelberg, Germany Spin:l0980613 46/3111 54 3 2 I Printed on acid-free paper
Preface to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry
The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. A factor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II and III. C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satis
