Standard systems for semifinite O*-algebras

Inoue, Atsushi

doi:10.1090/S0002-9939-97-03962-2

Standard systems for semifinite O$^{*}$-algebras
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by Atsushi Inoue

Proc. Amer. Math. Soc. 125 (1997), 3303-3312

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

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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 $(\mathcal {M}, \lambda , \lambda ’)$ of an O$^*$-algebra $\mathcal {M}$, a generalized vector $\lambda$ and the commutant $\lambda ’$ is unitarily equivalent to a standard system $\bigl ( \mathcal {N}, K’ \mu , (K’ \mu )’\bigr )$ constructed by a standard tracial generalized vector $\mu$ for an O$^*$-algebra $\mathcal {N}$ and a non-singular positive self-adjoint operator $K’$ affiliated with the commutant $\mathcal {N}’_{ \mathrm {w}}$ of $\mathcal {N}$.

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