Tomita's Theory of Modular Hilbert Algebras and its Applications
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128
M. Takesaki University of California, Los Angeles, CAlUSA and TOhoku University, Sendai, Japan
Tomita's Theory of Modular Hilbert Algebras and its Applications
"
Springer-Verlag Berlin· Heidelberg · NewYork 1970
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
128
M. Takesaki University of California, Los Angeles, CAlUSA and TOhoku University, Sendai, Japan
Tomita's Theory of Modular Hilbert Algebras and its Applications
"
Springer-Verlag Berlin· Heidelberg · NewYork 1970
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer-Verlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 79-117719 Printed in Germany. Title No. 3284
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Introduction
In
1967, Tomita clarified the algebraic relation between a
von Neumann al.gebra
M and its corcmutanb
M'
in two unpublished
papers [21] and [22], and then proved the commutation theorem for
(MI
tensor products of von Neumann algebras (i.e. In order to study the relation between
M and
M'
® M2)
normal state),
=
M{
2)·
®M
in a standard
representation, (for example, a cyclic representation of by a
I
M induced
he introduced two basic notions, called
a generalized and modular Hilbert al.gebr-as , respectively, both being algebra [4].
related to but different from Dixmier's See §2 for definitions.
It is not very difficult to show that every
von Neumann algebra is isomorphic to the left von Neuma,nn algebra of a generalized Hilbert algebra.
However, by means of generalized
Hilbert algebras we see how the involution of a von Neumann algebra is twisted in a Hilbert space structure. more clearly, suppose algebra
M.
\1>0
To explain his basic idea
is a faithful normal state of a von Neumann
Then the involution:
in the Hilbert space structure
x
M l-+
induced by
X
*
M is not an isometry however it is a
\1>0'
pre-closed operator, so that in the Hilbert space via,
\1>0
we can consider the adjoint operator
F
:It constructed
of the pre-closed
- 2 operator:
xs
Then
to
involution:
x *So where
O
x'So -e x ' *SO' x '
the involution: decomposition
is the cyclic vector corresponding
is nothing but the minimal closed extension of the
F
and the adjoint operator
6
So
XS S
o
S
M', of
-+ x *SO' X 1
= J6 2 ,
6
= FS,
F
of the commutant
M'
of
M,
is the minimal closed extension of
M. Therefore, we can get the polar of the involution
S.
The operator
is called the modular operator and plays a key role throughout
the theory.
As
an algebra of analytic vectors with respect to the
one parameter
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