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Jordan, Real and Lie Structures in Operator Algebras PDF

238 Pages·1997·8.217 MB·English
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Jordan, Real and Lie Structures in Operator Algebras Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 418 Jordan, Real and Lie Structures in Operator Algebras by Shavkat Ayupov Institute of Mathematics, Academy of Sciences, Tashkent, Uzbekistan Abdugafur Rakhimov University of World Economy and Diplomacy, Tashkent, Uzbekistan and Shukhrat U smanov Institute of Mathematics, Academy of Sciences, Tashkent, Uzbekistan Springer-Science+Business Media, B.Y. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-4891-2 ISBN 978-94-015-8605-4 (eBook) DOl 10.1007/978-94-015-8605-4 Printed on acid-free paper All Rights Reserved ©1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997. Softcover reprint of the hardcover 1st edition 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Dedicated to the memory of our teacher, Professor T .A.Sarymsakov CONTENTS INTRODUCTION PRELIMIN ARIES 7 Chapter 1: JORDAN OPERATOR ALGEBRAS 1;3 1.1. .IvV' -algebras and envdoping von Neumann algebras 14 1.2. Traces on ./ltV -alf,!;ebras 30 LL Types of JW-algebras and envelopiIlf,!; von Neumann algebra;,; :38 1.4. Classification of type I real aIld Jordan factors 44 l ..~ . Involutive antiautomorphisms of W* -algebras 48 1.6. I nvolutive antiautolllorphisms of injective W* -algebras 58 1. 7. Classification of injective real and Jordan factors type lit, ILx" and III,\, 0 < /\ ::; I 66 viii COMMENTS TO CHAPTER 1 70 Chapter 2: REAL STRUCTURE IN W*-ALGEBRAS 72 2.1. Real crossed products of real W*-algebras by an auto- n morphism 2.2. Discrete decomposition of real type Illo factors 92 2.3. Periodic antiautomorphisms and automorphisms of complex and real type III factors 110 2.4. Outer conjugacy classes of antiautomorphisms and auto- no morphisms of complex and real factors 2.5. Injectivity, amenability, semidiscreteness and hypf'rfini- teness in real W*-algebras 141 2.6. Diameters of state spaces of JW-algebras 151 COMMENTS TO CHAPTER 2 172 Chapter 3: LIE STRUCTURE IN OPERATOR ALGEBRAS 174 3.1. Theorem on isomophism of prime Lie rings. 17.1 3.2. Symmetric and skew-symmetric operators on real and quaternionian Hilbert spaces 182 ix :3.3. Commut.ators of skpw elements in real factors. 19S :3.4. Isomorphism of Lie operator algebras and conjugacy of involutivp antiautomorphisms. 206 COMMENTS TO CHAPTER 3 210 REFERENCES 212 INDEX 224 INTRODUCTION The theory of operator algebras acting on a Hilbert space was initiated in thirties by papers of Murray and von Neumann. In these papers they have studied the structure of algebras which later were called von Neu mann algebras or W* -algebras. They are weakly closed complex *-algebras of operators on a Hilbert space. At present the theory of von Neumann algebras is a deeply developed theory with various applications. In the framework of von Neumann algebras theory the study of fac tors (i.e. W* -algebras with trivial centres) is very important, since they are comparatively simple and investigation of general W* -algebras can be reduced to the case of factors. Therefore the theory of factors is one of the main tools in the structure theory of von Neumann algebras. In the middle of sixtieth Topping [To 1] and Stormer [S 2] have ini tiated the study of Jordan (non associative and real) analogues of von Neumann algebras - so called JW-algebras, i.e. real linear spaces of self adjoint opera.tors on a complex Hilbert space, which contain the identity operator 1. closed with respect to the Jordan (i.e. symmetrised) product S. Ayupov et al., Jordan, Real and Lie Structures in Operator Algebras © Springer Science+Business Media Dordrecht 1997 2 INTRODUCTION x 0 y = ~(Xy + yx) and closed in the weak operator topology. The structure of these algebras has happened to be close to the struc ture of von Neumann algebras and it was possible to apply ideas and meth ods similar to von Neumann algebras theory in the study of JW-algebras. Thus Topping [To 1] has classified JW-algebras into those of type I, Ih, IIoo, IlL later Stormer [S 2], [S 4] and Ayupoy [A 1], [A 3], [A 4], [A 9], [A 10]. [A 13] considered the problem on connections between the type of a JltV-algebra and the type of its enveloping von Neumann algebra. In [S 2] Stormer gave a com plete study of type I JW -algebras and has proved also that any reversible JW -algebra A (in particular of type II and III) is isomorphic to the direct sum Ac EEl AT, where the JW-algebra Ac is the self-adjoint part U(Ac)s of its enveloping von Neumann algebra U(Ac), whence the JW-algebra AT coincides with the self-adjoint part R(Ar)s n of the real enveloping algebra R( Ar) such that R( AT) iR( AT) = {O} (so called real von Neumann alge bra). In this connection the study of real von Neumann algebras was carried out parallel to the theory of JW algebras. A real von Neumann algebra is a real *-algebra R of bounded linear operators on a complex Hilbert space, containing the identity operator 1, which is closed in the weak topology and satisfies the condition op~rator n R iR = {O}. The smallest (complex) von Neumann algebra U(R) con taining R coincides with its complexification R+iR, i.e., U(R) = R+iR. Moreover R generates a natural involutive (i.e. of order 2) *-antiautomor phism aiR of U(R), namely aiR ( x + .) = * + . * 7y X 1y , where x + iy E U(R), x, y E R. = = It is clear that R {x E U(R): aiR(x) x*}. Conversely, given a von Neumann algebra U and any involutive *-antiautomorphism a on U, the set {;r E U: a(x) = x*} is real von Neumann algebra. It is not difficult to see that two real von Neumann algebras generating the same von Neumann algebra, are isomorphic if and only if the corre sponding illvolutive *-antiautomorphisms are conjugate [A 3], [A 6], [8 3], [84].

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