Skip to main content
SpringerLink
Log in

The Radon-Nikodym theorem for von neumann algebras

  • Published:
Acta Mathematica

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 operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Combes, F., Poids sur uneC *-algèbre.J. Math. pures et appl., 47 (1968), 57–100.

    MATH  MathSciNet  Google Scholar 

  2. — Poids associè à une algèbre hilbertienne à gauche.Compositio Math., 23 (1971), 49–77.

    MATH  MathSciNet  Google Scholar 

  3. — Poids et espérances conditionnelle dans les algèbres de von Neumann.Bull. Soc. Math. France, 99 (1971), 73–112.

    MATH  MathSciNet  Google Scholar 

  4. Dixmier, J.,Les algébres d'opérateurs dans l'espace hilbertien. Gauthier-Villars, Paris, 2e édition, 1969.

    MATH  Google Scholar 

  5. —,Les C *-algèbres et leurs représentations. Gauthier-Villars, Paris, 1964.

    Google Scholar 

  6. — Formes linéaires sur un anneau d'opérateurs.Bull. Soc. Math. France, 81 (1953), 9–39.

    MATH  MathSciNet  Google Scholar 

  7. Dye, H. A., The Radon-Nikodym theorem for finite rings of operators.Trans. Amer. Math. Soc., 72 (1952), 243–280.

    Article  MATH  MathSciNet  Google Scholar 

  8. Haag, R., Hugenholtz, N. M. &Winnink, M., On the equilibrium states in quantum statistical mechanics.Comm. Math. Phys., 5 (1967), 215–236.

    Article  MATH  MathSciNet  Google Scholar 

  9. Halpern, H., Unitary implementation of automorphism groups on von Neumann algebras.Comm. Math. Phys., 25 (1972), 253–272.

    Article  MATH  MathSciNet  Google Scholar 

  10. Herman, R. &Takesaki, M., States and automorphism groups of operator algebras.Comm. Math. Phys., 19 (1970), 142–160.

    Article  MATH  MathSciNet  Google Scholar 

  11. Hille, E. & Phillips, R. S., Functional analysis and semi-groups.Amer. Math. Colloquium Publication, 31 (1957).

  12. Hugenholtz, N. M., On the factor type of equilibrium states in quantum statistical mechanics.Comm. Math. Phys., 6 (1967), 189–193.

    Article  MATH  MathSciNet  Google Scholar 

  13. Kadison, R. V., Transformations of states in operator theory and dynamics.Topology, 3 (1965), 177–198.

    Article  MATH  MathSciNet  Google Scholar 

  14. Kato, T.,Perturbation theory for linear operators. Springer-Verlag, 1966.

  15. 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.

    Article  MATH  Google Scholar 

  16. Pedersen, G. K., Measure theory forC *-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.

    MathSciNet  Google Scholar 

  17. Pedersen, G. K. Some operator monotone functions. To appear inProc. Amer. Math. Soc.

  18. Pedersen, G. K. & Takesaki, M., The operator equation THT=K. To appear inProc. Amer. Math. Soc.

  19. Perdrizet, E., Elements positif relativement à une algèbre hilbertienne à gauche.Compositio Math., 23 (1971), 25–47.

    MATH  MathSciNet  Google Scholar 

  20. Petersen, N. H., Invariant weights on semi-finite von Neumann algebras, to appear.

  21. Rudin, W.,Fourier analysis on groups. Interscience, New York, 1962.

    MATH  Google Scholar 

  22. Sakai, S., A Radon-Nikodym theorem inW *-algebras,Bull. Amer. Math. Soc., 71 (1965), 149–151.

    Article  MATH  MathSciNet  Google Scholar 

  23. Segal, I. E., A non-commutative extension of abstract integration.Ann. Math., 57 (1953), 401–457.

    Article  MATH  Google Scholar 

  24. Størmer, E., Types of von Neumann algebras associated with extremal invariant states.Comm. Math. Phys., 6 (1967), 194–204.

    Article  MATH  Google Scholar 

  25. Takesaki, M.,Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Mathematics no. 128, Springer-Verlag, 1970.

  26. — Disjointness of the KMS-states of different temperatures.Comm. Math. Phys., 17 (1970), 33–41.

    Article  MathSciNet  Google Scholar 

  27. Takesaki, M., Conditional expectations in von Neumann algebras. To appear inJ. Funct. Anal.

  28. Tomita, M.,Quasi-standard von Neumann algebras. Mimeographed notes, 1967.

  29. Tomita, M.,Standard forms of von Neumann algebras. The 5th Functional Analysis Symposium of the Math. Soc. of Japan, Sendai, 1967.

Download references

Author information

Authors and Affiliations

  1. University of Copenhagen, Denmark

    Gert K. Pedersen & Masamichi Takesaki

  2. University of California, Los Angeles, Calif., USA

    Gert K. Pedersen & Masamichi Takesaki

Authors
  1. Gert K. Pedersen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masamichi Takesaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by NSF Grant # 28976 X.

Partially supported by NSF Grant # GP-28737

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pedersen, G.K., Takesaki, M. The Radon-Nikodym theorem for von neumann algebras. Acta Math. 130, 53–87 (1973). https://doi.org/10.1007/BF02392262

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02392262

Keywords

Search

Navigation