A Radon-Nikodym theorem for von Neumann algebras

Details A Radon-Nikodym theorem for von Neumann algebras A Radon-Nikodym theorem for von Neumann algebras

1998-11-20

arxiv.org/abs/math/9811122v1

Journal of Operator Theory 46 (3) (2001), 477--489.

Mathematics

Operator Algebras

14 pages, LaTeX

Scientific paper

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 $\phi$ 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 $\sigma^\phi$ of $\phi$, with a strictly positive operator as the invariance factor, we construct the n.s.f. weight $\phi(\delta^{1/2} . \delta^{1/2})$. All the n.s.f. weights on M whose modular automorphisms commute with $\sigma^\phi$ 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 $\sigma^\phi$ are of this form, the invariance factor being a scalar.

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