A Radon-Nikodym theorem for von Neumann algebras

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 $\varphi$ on a von Neumann algebra M and a strictly positive operator $\delta$, affiliated with 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 $\varphi ({\delta }^{1\mathrm{/}2}\mathrm{.}{\delta }^{1\mathrm{/}2})$. All the n.s.f. weights on M whose modular automorphisms commute with ${\sigma }^{\varphi }$ 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 ${\sigma }^{\varphi }$ are of this form, the invariance factor being a scalar.