Jordan, Real and Lie Structures in Operator Algebras

238 Pages·1997·8.217 MB·English

by  Shavkat Ayupov, Abdugafur Rakhimov, Shukhrat Usmanov (auth.)

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].