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Standard systems for semifinite O$^*$-algebras

Author: Atsushi Inoue
Journal: Proc. Amer. Math. Soc. 125 (1997), 3303-3312
MSC (1991): Primary 47D40; Secondary 46K15, 46L10
MathSciNet review: 1403134
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Abstract: We shall continue the study of standard systems which make it possible to develop the Tomita-Takesaki theory in O$^*$-algebras. The main purpose of this paper is to give the necessary and sufficient conditions for which a standard system $({\cal M}, \lambda , \lambda ')$ of an O$^*$-algebra ${\cal M}$, a generalized vector $\lambda $ and the commutant $\lambda '$ is unitarily equivalent to a standard system $ \bigl ( {\cal N}, K' \mu , (K' \mu )'\bigr )$ constructed by a standard tracial generalized vector $\mu $ for an O$^*$-algebra ${\cal N}$ and a non-singular positive self-adjoint operator $K'$ affiliated with the commutant ${\cal N}'_{ \mathrm {w}} $ of ${\cal N}$.

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Additional Information

Atsushi Inoue
Affiliation: Department of Applied Mathematics, Fukuoka University, Fukuoka, 814-80, Japan

Keywords: O$^*$-algebra, standard generalized vector, Tomita-Takesaki theory
Received by editor(s): June 12, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society