The Radon-Nikodym theorem for von neumann algebras

Gert K. Pedersen University of Copenhagen, Denmark Masamichi Takesaki University of Copenhagen, Denmark

TBD mathscidoc:1701.331433

Acta Mathematica, 130, (1), 53-87, 1972.5
Let ϕ be a faithful normal semi-finite weight on a von Neumann algebra$M$. For each normal semi-finite weight ϕ on$M$, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operator$h$, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ($h$·). Conversely, each such$h$determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result that$M$is semi-finite if and only if Σ consists of inner automorphisms.
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  title={The Radon-Nikodym theorem for von neumann algebras},
  author={Gert K. Pedersen, and Masamichi Takesaki},
  booktitle={Acta Mathematica},
Gert K. Pedersen, and Masamichi Takesaki. The Radon-Nikodym theorem for von neumann algebras. 1972. Vol. 130. In Acta Mathematica. pp.53-87.
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