Acta Mathematica

The Radon-Nikodym theorem for von neumann algebras

Gert K. Pedersen and Masamichi Takesaki

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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.


Partially supported by NSF Grant # 28976 X.


Partially supported by NSF Grant # GP-28737


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

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

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

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1973 © Almqvist & Wiksell Informationsindustri AB


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

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  • Combes, F., Poids sur une C*-algèbre. J. Math. pures et appl., 47 (1968), 57–100.
  • — Poids associè à une algèbre hilbertienne à gauche. Compositio Math., 23 (1971), 49–77.
  • — Poids et espérances conditionnelle dans les algèbres de von Neumann. Bull. Soc. Math. France, 99 (1971), 73–112.
  • Dixmier, J., Les algébres d'opérateurs dans l'espace hilbertien. Gauthier-Villars, Paris, 2e édition, 1969.
  • —, Les C*-algèbres et leurs représentations. Gauthier-Villars, Paris, 1964.
  • — Formes linéaires sur un anneau d'opérateurs. Bull. Soc. Math. France, 81 (1953), 9–39.
  • Dye, H. A., The Radon-Nikodym theorem for finite rings of operators. Trans. Amer. Math. Soc., 72 (1952), 243–280.
  • Haag, R., Hugenholtz, N. M. & Winnink, M., On the equilibrium states in quantum statistical mechanics. Comm. Math. Phys., 5 (1967), 215–236.
  • Halpern, H., Unitary implementation of automorphism groups on von Neumann algebras. Comm. Math. Phys., 25 (1972), 253–272.
  • Herman, R. & Takesaki, M., States and automorphism groups of operator algebras. Comm. Math. Phys., 19 (1970), 142–160.
  • Hille, E. & Phillips, R. S., Functional analysis and semi-groups. Amer. Math. Colloquium Publication, 31 (1957).
  • Hugenholtz, N. M., On the factor type of equilibrium states in quantum statistical mechanics. Comm. Math. Phys., 6 (1967), 189–193.
  • Kadison, R. V., Transformations of states in operator theory and dynamics. Topology, 3 (1965), 177–198.
  • Kato, T., Perturbation theory for linear operators. Springer-Verlag, 1966.
  • Murray, F. J. & von Neumann, J., On rings of operators. Ann. Math., 37 (1936), 116–229; II, Trans. Amer. Math. Soc., 41 (1937), 208–248.
  • Pedersen, G. K., Measure theory for C*-algebras. Math. Scand., 19 (1966), 131–145; II, Math. Scand., 22 (1968), 63–74; III, Math. Scand., 25 (1969), 71–93; IV, Math. Scand., 25 (1969), 121–127.
  • Pedersen, G. K. Some operator monotone functions. To appear in Proc. Amer. Math. Soc.
  • Pedersen, G. K. & Takesaki, M., The operator equation THT=K. To appear in Proc. Amer. Math. Soc.
  • Perdrizet, E., Elements positif relativement à une algèbre hilbertienne à gauche. Compositio Math., 23 (1971), 25–47.
  • Petersen, N. H., Invariant weights on semi-finite von Neumann algebras, to appear.
  • Rudin, W., Fourier analysis on groups. Interscience, New York, 1962.
  • Sakai, S., A Radon-Nikodym theorem in W*-algebras, Bull. Amer. Math. Soc., 71 (1965), 149–151.
  • Segal, I. E., A non-commutative extension of abstract integration. Ann. Math., 57 (1953), 401–457.
  • Størmer, E., Types of von Neumann algebras associated with extremal invariant states. Comm. Math. Phys., 6 (1967), 194–204.
  • Takesaki, M., Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Mathematics no. 128, Springer-Verlag, 1970.
  • — Disjointness of the KMS-states of different temperatures. Comm. Math. Phys., 17 (1970), 33–41.
  • Takesaki, M., Conditional expectations in von Neumann algebras. To appear in J. Funct. Anal.
  • Tomita, M., Quasi-standard von Neumann algebras. Mimeographed notes, 1967.
  • Tomita, M., Standard forms of von Neumann algebras. The 5th Functional Analysis Symposium of the Math. Soc. of Japan, Sendai, 1967.