Jordan, Real and Lie Structures in Operator Algebras

By: Sh. Ayupov, Abdugafur Rakhimov, Shukhrat Usmanov

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Hardcover | 31 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.

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Non-Fiction Mathematics Algebra Applied Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Groups & Group Theory Geometry

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Published: 15th December 2010

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Introduction	p. 1
Prelimininaries	p. 7
Jordan Operator Algebras	p. 13
JW-algebras and enveloping von Neumann algebras	p. 14
Traces on JW-algebras	p. 30
Types of JW-algebras and enveloping von Neumann algebras	p. 38
Classification of type I real and Jordan factors	p. 44
Involutive antiautomorphisms of W*-algebras	p. 48
Involutive antiautomorphisms of injective W*-algebras	p. 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*-Algebras	p. 72
Real crossed products of real W*-algebras by an automorphism	p. 73
Discrete decomposition of real type III[subscript 0] factors	p. 92
Periodic antiautomorphisms and automorphisms of complex and real type II[subscript 1] factors	p. 110
Outer conjugacy classes of antiautomorphisms and automorphisms of complex and real factors	p. 130
Injectivity, amenability, semidiscreteness and hyperfiniteness in real W*-algebras	p. 141
Diameters of state spaces of JW-algebras	p. 151
Comments to Chapter 2	p. 172
Lie Structure in Operator Algebras	p. 174
Theorem on isomorphism of prime Lie rings	p. 175
Symmetric and skew-symmetric operators on real and quaternionian Hilbert spaces	p. 182
Commutators of skew elements in real factors	p. 195
Isomorphism of Lie operator algebras and conjugacy of involutive antiautomorphisms	p. 206
Comments to Chapter 3	p. 210
References	p. 212
Index	p. 224
Table of Contents provided by Blackwell. All Rights Reserved.

Jordan, Real and Lie Structures in Operator Algebras

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