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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
DOI: https://doi.org/10.1090/S0002-9939-97-03962-2
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
Email: sm010888ssat.fukuoka-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-97-03962-2
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