Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001 pp. 477-489.

A Radon-Nikodym theorem for von Neumann algebras

Authors: Stefaan Vaes
Author institution: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium

Summary: In this paper we present a generalization of the Radon-Nikodym theorem proved by Pedersen and Takesaki in [7]. Given a normal, semifinite and faithful (n.s.f.) weight $\varphi$ on a von Neumann algebra $\mathcal M$ and a strictly positive operator $\delta$, affiliated with $\mathcal M$ and satisfying a certain relative invariance property with respect to the modular automorphism group $\sigma^\varphi$ of $\varphi$, with a strictly positive operator as the invariance factor, we construct the n.s.f. weight $\varphi(\delta^{\frac{1}{2}} \cdot \delta^{\frac{1}{2}})$. All the n.s.f. weights on $\mathcal M$ whose modular automorphisms commute with $\sigma^\varphi$ are of this form, the invariance factor being affiliated with the centre of $\mathcal M$. All the n.s.f. weights which are relatively invariant under $\sigma^\varphi$ are of this form, the invariance factor being a scalar.

Keywords: modular theory, Radon-Nikodym theorem, commuting weights

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