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Jordan, Real and Lie Structures in Operator Algebras : MATHEMATICS AND ITS APPLICATIONS (KLUWER ) - Shavkat Ayupov

Jordan, Real and Lie Structures in Operator Algebras



Published: 31st July 1997
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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 INTRODUCTION 2 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.

Introductionp. 1
Prelimininariesp. 7
Jordan Operator Algebrasp. 13
JW-algebras and enveloping von Neumann algebrasp. 14
Traces on JW-algebrasp. 30
Types of JW-algebras and enveloping von Neumann algebrasp. 38
Classification of type I real and Jordan factorsp. 44
Involutive antiautomorphisms of W*-algebrasp. 48
Involutive antiautomorphisms of injective W*-algebrasp. 58
Classification of injective real and Jordan factors type II[subscript 1], [actual symbol not reproducible]p. 66
Comments to Chapter 1
Real Structure in W*-Algebrasp. 72
Real crossed products of real W*-algebras by an automorphismp. 73
Discrete decomposition of real type III[subscript 0] factorsp. 92
Periodic antiautomorphisms and automorphisms of complex and real type II[subscript 1] factorsp. 110
Outer conjugacy classes of antiautomorphisms and automorphisms of complex and real factorsp. 130
Injectivity, amenability, semidiscreteness and hyperfiniteness in real W*-algebrasp. 141
Diameters of state spaces of JW-algebrasp. 151
Comments to Chapter 2p. 172
Lie Structure in Operator Algebrasp. 174
Theorem on isomorphism of prime Lie ringsp. 175
Symmetric and skew-symmetric operators on real and quaternionian Hilbert spacesp. 182
Commutators of skew elements in real factorsp. 195
Isomorphism of Lie operator algebras and conjugacy of involutive antiautomorphismsp. 206
Comments to Chapter 3p. 210
Referencesp. 212
Indexp. 224
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792346845
ISBN-10: 079234684X
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 230
Published: 31st July 1997
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 24.77 x 17.15  x 1.91
Weight (kg): 0.52