Skip to main content
SpringerLink
Log in

Normal ergodic actions and extensions

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We demonstrate that normal ergodic extensions of group actions are characterized as skew product actions given by cocycles into locally compact groups. As a consequence, Robert Zimmer’s characterization of normal ergodic group actions is generalized to the noninvariant case. We also obtain the uniqueness theorem which generalizes the von Neumann Halmos uniqueness theorem and Zimmer’s uniqueness theorem for normal actions with relative discrete spectrum.

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. E. G. Effros,The Borel space of von Neumann algebras on a separable Hilbert space, Pacific J. Math.15 (1964), 1153–1164.

    MathSciNet  Google Scholar 

  2. R. Fabec,Normal ergodic quasi-invariant actions, preprint.

  3. G. W. Mackey,The Theory of Group Representations, University of Chicago, Summer 1955.

  4. G. W. Mackey,Borel structures in groups and their duals, Trans. Amer. Math. Soc.85 (1957), 265–311.

    Article  MathSciNet  Google Scholar 

  5. G. W. Mackey,Point realizations of transformation groups, Illinois J. Math.6 (1962), 327–335.

    MATH  MathSciNet  Google Scholar 

  6. G. W. Mackey,Ergodic theory, group theory, and differential geometry, Proc. Nat. Acad. Sci. U.S.A.50 (1963), 1184–1191.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. W. Mackey,Ergodic transformations with a pure point spectrum, Illinois J. Math.8 (1964), 593–600.

    MATH  MathSciNet  Google Scholar 

  8. G. W. Mackey,Ergodic theory and virtual groups, Math. Ann.166 (1966), 187–207.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. C. Moore,Extensions and cohomology for locally compact groups, III, Trans. Amer. Math. Soc.221 (1976), 1–33.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Ramsay,Virtual groups and group actions, Advances in Math.6 (1971), 253–322.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Ramsay,Boolean duals of virtual groups, J. Functional Analysis15 (1974), 56–101.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Ramsay,Subobjects of virtual groups, preprint.

  13. C. Series,Ergodic actions of product groups Harvard Ph.D. Thesis, 1976.

  14. V. S. Varadarajan,Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc.109 (1963), 191–220.

    Article  MATH  MathSciNet  Google Scholar 

  15. V. S. Varadarajan,Geometry of Quantum Theory, Vol. II, Van Nostrand, Princeton, N.J., 1970.

    MATH  Google Scholar 

  16. J. J. Westman,Virtual group homomorphisms with dense range, Illinois J. Math.20 (1976), 41–47.

    MATH  MathSciNet  Google Scholar 

  17. R. J. Zimmer,Compact nilmanifold extensions of ergodic actions, Trans. Amer. Math. Soc.223 (1976), 397–406.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. J. Zimmer,Extensions of ergodic group actions, Illinois J. Math.20 (1976), 373–409.

    MATH  MathSciNet  Google Scholar 

  19. R. J. Zimmer,Ergodic actions with generalized discrete spectrum, Illinois J. Math.20 (1976), 555–588.

    MATH  MathSciNet  Google Scholar 

  20. R. J. Zimmer, Normal ergodic actions, J. Functional Analysis25 (1977), 286–305.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Lousiana State University, 70803, Baton Rouge, LA, USA

    R. C. Fabec

Authors
  1. R. C. Fabec
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fabec, R.C. Normal ergodic actions and extensions. Israel J. Math. 40, 175–186 (1981). https://doi.org/10.1007/BF02761908

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761908

Keywords

Search

Navigation