Journal of Mathematical Sciences

, Volume 140, Issue 3, pp 398–425 | Cite as

Unitary representations and modular actions

  • A. S. Kechris


We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense “orthogonal” to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity and strong mixing. Bibliography:s 25 titles.


Invariant Measure Unitary Representation Cayley Graph Regular Representation Countable Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. Bekka, P. de la Harpe, and A. Valette, “Kazhdan’s property (T),” Preprint (2002).Google Scholar
  2. 2.
    B. Bekka and M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, London Math. Soc. Lect. Notes Series, 269, Cambridge Univ. Press, Cambridge (2000).MATHGoogle Scholar
  3. 3.
    V. Bergelson and J. Rosenblatt, “Mixing actions of groups. III,” J. Math., 32, No. 1, 65–80 (1988).MATHMathSciNetGoogle Scholar
  4. 4.
    M. Burger and P. de la Harpe, “Constructing irreducible representations of discrete groups,” Proc. Ind. Acad. Sci. (Math. Sci.), 107, No. 3, 223–235 (1997).MATHGoogle Scholar
  5. 5.
    P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette, Groups With the Haagerup Property, Progress Math., 197, Birkhäuser Verlag, Basel (2001).MATHGoogle Scholar
  6. 6.
    M. Cowling, U. Haagerup, and R. Howe, “Almost L 2 matrix coefficients,” J. Reine Angew. Math., 387, 97–110 (1988).MATHMathSciNetGoogle Scholar
  7. 7.
    J. Dixmier, C*-Algebras, North Holland, Amsterdam-New York-Oxford (1977).MATHGoogle Scholar
  8. 8.
    G. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, Florida (1995).MATHGoogle Scholar
  9. 9.
    H. Furstenberg and B. Weiss, “The finite multipliers of infinite ergodic transformations. The structure of attractors in dynamical systems,” Lect. Notes Math., 668, 127–132 (1978).MathSciNetCrossRefGoogle Scholar
  10. 10.
    U. Haagerup, “An example of a nonnuclear C*-algebra which has the metric approximation property,” Invent. Math., 50, No. 3, 279–293 (1978/79).CrossRefMathSciNetGoogle Scholar
  11. 11.
    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Springer-Verlag, Berlin-New York (1979).MATHGoogle Scholar
  12. 12.
    G. Hjorth, “A converse to Dye’s theorem,” Trans. Amer. Math. Soc., 357, No. 8, 3083–3103 (2002).CrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Hjorth and A. S. Kechris, Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations, Mem. Amer. Math. Soc., 177, No. 833 (2005).Google Scholar
  14. 14.
    S. Jackson, A. S. Kechris, and A. Louveau, “Countable Borel equivalence relation,” J. Math. Logic, 2, No. 1, 1–80 (2002).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    V. Jones and K. Schmidt, “Asymptotically invariant sequences and approximate finiteness,” Amer. J. Math, 109, 91–114 (1987).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    B. Kitchens and K. Schmidt, “Automorphisms of compact groups,” Ergodic Theory Dynam. Systems, 9, 691–735 (1989).MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Knapp, Representation Theory of Semisimple Groups, Princeton Univ. Press, Princeton, New Jersey (1986).MATHGoogle Scholar
  18. 18.
    A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser Verlag, Basel (1994).MATHGoogle Scholar
  19. 19.
    R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin-New York (1977).MATHGoogle Scholar
  20. 20.
    K. Schmidt, “Asymptotic properties of unitary representations and mixing,” Proc. London Math. Soc., 48, No. 3, 445–460 (1925).CrossRefGoogle Scholar
  21. 21.
    K. Schmidt, Dynamical Systems of Algebraic Origin, Progress Math., 128, Birkhäuser Verlag, Basel (1995).MATHGoogle Scholar
  22. 22.
    K. Schmidt and P. Walters, “Mildly mixing actions of locally compact groups,” Proc. London Math. Soc., 45, 506–518 (1982).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    S. Thomas, “Superrigidity and countable Borel equivalence relations.” Ann. Pure Appl. Logic, 120, 237–262 (2003).MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    S. Wagon, The Banach-Tarski Paradox, Cambridge Univ. Press, Cambrigde (1985).MATHGoogle Scholar
  25. 25.
    R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser Verlag, Basel (1984).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • A. S. Kechris
    • 1
  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations