Relative modular theory for a weight

Izumi, Hideaki

doi:10.1090/S0002-9939-99-04840-6

Relative modular theory for a weight
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by Hideaki Izumi PDF

Proc. Amer. Math. Soc. 127 (1999), 2703-2713 Request permission

Abstract:

We consider the balanced weight $\chi$ of a semi-finite weight $\varphi$ and a (not necessarily faithful) normal positive functional $\psi$ on a von Neumann algebra $\mathcal M$, and discuss how the modular operator $\Delta _\chi$ and the modular conjugation $J_\chi$ are described under the identification of the standard Hilbert space $\mathcal {H}_\chi$ with $\mathcal {H}_\varphi \oplus p\mathcal {H}_\varphi \oplus p’\mathcal {H}_\varphi \oplus pp’\mathcal {H}_\varphi$, where $p$ is the support projection of $\psi$ and $p’=J_\varphi p J_\varphi \in \mathcal {M}’$.

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