Acta Mathematica

The Radon-Nikodym theorem for von neumann algebras

Gert K. Pedersen and Masamichi Takesaki

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Abstract

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, 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 thatM is semi-finite if and only if Σ consists of inner automorphisms.

Note

Partially supported by NSF Grant # 28976 X.

Note

Partially supported by NSF Grant # GP-28737

Note

This revised version was published online in November 2006 with corrections to the Cover Date.

Article information

Source
Acta Math., Volume 130 (1973), 53-87.

Dates
Received: 2 May 1972
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485889766

Digital Object Identifier
doi:10.1007/BF02392262

Mathematical Reviews number (MathSciNet)
MR412827

Zentralblatt MATH identifier
0262.46063

Rights
1973 © Almqvist & Wiksell Informationsindustri AB

Citation

Pedersen, Gert K.; Takesaki, Masamichi. The Radon-Nikodym theorem for von neumann algebras. Acta Math. 130 (1973), 53--87. doi:10.1007/BF02392262. https://projecteuclid.org/euclid.acta/1485889766


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