The Radon-Nikodym theorem for von neumann algebras

Abstract

Let φ{symbol} be a faithful normal semi-finite weight on a von Neumann algebra M. For each normal semi-finite weight φ{symbol} on M, invariant under the modular automorphism group Σ of φ{symbol}, there is a unique self-adjoint positive operator h, affiliated with the sub-algebra of fixed-points for Σ, such that φ{symbol}=φ{symbol}(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. © 1973 Almqvist & Wiksell Informationsindustri AB.