Modular Automorphism Groups
As we have seen in the previous chapters, the modular operator gives rise to a one parameter automorphism group on the von Neumann algebra in question. First, we are going to identify this group. The first section is devoted to this task. This group is ch
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Subseries Editors: Joachim Cuntz Vaughan F. R. Jones
Springer-Verlag Berlin Heidelberg GmbH
M. Takesaki
Theory of Operator Algebras II
,
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 1 by M. Takesaki was published as VoI. 124 ofthe Encyclopaedia of Mathematical Sciences, ISBN 3-540-42248-X, Theory of Operator Algebras III by M. Takesaki was published as VoI. 127 of the Encyclopaedia of Mathematical Sciences, ISBN 3-540-42913-1 ISSN 0938-0396 ISBN 978-3-642-07689-3 DOI 10.1007/978-3-662-10451-4
ISBN 978-3-662-10451-4 (eBook)
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2003
Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 18t 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 eliempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Typeset in ID'EX by Johannes Kiister . typoma . www.typoma.com. based on the author's plain1l:;X files. Cover Design: E. Kirchner, Heidelberg, Germany Printed on acid-free paper SPIN: 11413134 46/3111 LK
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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, IT 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
