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A Radon-Nikodym theorem for von Neumann algebras

Stefaan Vaes
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Abstract

In this paper we present a generalization of the Radon-Nikodym theorem proved by Pedersen and Takesaki. Given a normal, semifinite and faithful (n.s.f.) weight $ϕ$ on a von Neumann algebra M and a strictly positive operator $δ$, affiliated with M and satisfying a certain relative invariance property with respect to the modular automorphism group $σ^ϕ$ of $ϕ$, with a strictly positive operator as the invariance factor, we construct the n.s.f. weight $ϕ(δ^{1/2} . δ^{1/2})$. All the n.s.f. weights on M whose modular automorphisms commute with $σ^ϕ$ are of this form, the invariance factor being affiliated with the centre of M. All the n.s.f. weights which are relatively invariant under $σ^ϕ$ are of this form, the invariance factor being a scalar.

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Stefaan Vaes. 1998-11-20. A Radon-Nikodym theorem for von Neumann algebras. https://arxiv.org/abs/math/9811122

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