Skip to main content

Non-unitarizable uniformly bounded group representations

  • Chapter
  • 750 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1618)

Summary

In this chapter, we introduce the space B (G) of coefficients of unitary representations on a discrete group G and a related space T p (G), (1 p < ) of complex valued functions on G. We show that if G = IF N , the free group on N 2 generators, there are non-unitarizable uniformly bounded representations on G. We give several related characterizations of amenable groups. Then we extend the method to the case of semi-groups. This allows us to produce (this time for G = IN) examples of power bounded operators which are not polynomially bounded.

Keywords

  • Unitary Representation
  • Toeplitz Oper
  • Discrete Group
  • Amenable Group
  • Hankel Operator

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Greenleaf F.: Invariant means on topological groups. Van Nostrand, New York 1969.

    MATH  Google Scholar 

  2. Paterson A.: Amenability. A.M.S. Math. Surveys 29 (1988).

    Google Scholar 

  3. Pier J.P.: Amenable locally compact groups. Wiley, Interscience, New York 1984.

    Google Scholar 

  4. Akemann C. and Ostrand P.: Computing norms in group C*-algebras. Amer. J. Math. 98 (1976) 1015–1047.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Eymard P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92 (1964) 181–236.

    MathSciNet  MATH  Google Scholar 

  6. Haagerup U.: MoA(G) functions which are not coefficients of uniformly bounded representations. Handwritten manuscript 1985.

    Google Scholar 

  7. Pisier G.: Multipliers and lacunary sets in non-amenable groups. Amer. J. Math. 117 (1995) 337–376.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Boiejko M.: Littlewood functions, Hankel multipliers and power bounded operators on a Hilbert space. Colloquium Math. 51 (1987) 35–42.

    Google Scholar 

  9. Wysoczanski J. Radial Herz-Schur multipliers on free products of discrete groups. Journal Funct. Anal. 129 (1995) 268–292.

    CrossRef  Google Scholar 

  10. Ann. Inst. Fourier 41 (1991) 797–822.

    CrossRef  Google Scholar 

  11. Boiejko M.: Positive definite bounded matrices and a characterization of amenable groups. Proc. A.M.S. 95 (1985) 357–360.

    Google Scholar 

  12. Nebbia C. Multipliers and asymptotic behaviour of the Fourier algebra of non amenable groups. Proc. Amer. Math. Soc. 84 (1982) 549–554.

    Google Scholar 

  13. Fendler G.: A uniformly bounded representation associated to a free set in a discrete group. Colloq. Math. 59 (1990) 223–229.

    MathSciNet  MATH  Google Scholar 

  14. Wysoczanski J.: Characterization of amenable groups and the Littlewood functions on free groups. Colloquium Math. 55 (1988) 261–265.

    MathSciNet  MATH  Google Scholar 

  15. Leinert M.: Faltungsoperatoren auf gewissen diskreten Gruppen. Studia Math. 52, (1974) 149–158.

    MathSciNet  MATH  Google Scholar 

  16. Haagerup U.: An example of a non-nuclear C’-algebra which has the metric approximation property. Invent. Mat. 50 (1979) 279–293.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Haagerup U. and Pisier G.: Linear operators between C’-algebras. Duke Math. J. 71 (1993) 889–925.

    MathSciNet  MATH  Google Scholar 

  18. Figa-Talamanca A. and Picardello M.: Harmonic Analysis on Free groups. Marcel Dekker, New-York, 1983.

    MATH  Google Scholar 

  19. Figa-Talamanca A. and Nebbia C.: Harmonic analysis and representation theory for groups acting on homogeneous trees. Cambridge Univ. Press, LMS Lecture notes series 162, Cambridge, 1990.

    Google Scholar 

  20. Mantero A.M. and Zappa A. The Poisson transform on free groups and uniformly bounded representations. J. Funct. Anal. 51 (1983) 372–399.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Mantero A.M. and Zappa A.: Uniformly bounded representations and L P convolution theorems on a free group. Harmonic Analysis (Proc. Cortona 1982) Springer Lecture Notes 992 (1983) 333–343.

    MathSciNet  Google Scholar 

  22. Kunze R.A. and Stein E.: Uniformly bounded representations and Harmonic Analysis of the 2 x 2 real unimodular group. Amer. J. Math. 82 (1960) 1–62.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Ehrenpreis L. and Mautner F.: Uniformly bounded representations of groups. Proc. Nat. Acad. Sc. 41 (1955) 231–233.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Pytlik T. and Szwarc R. An analytic family of uniformly bounded representations of free groups. Acta Math. 157 (1986) 287–309.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. Fournier J.: An interpolation problem for coefficients of H∞ functions. Proc. Amer. Math. Soc. 42 (1974) 402–408.

    MathSciNet  MATH  Google Scholar 

  26. Havin V. P. and Nikolski N. K. (editors): Linear and complex analysis problem book 3. Part 1. Lecture Notes in Math. 1573, Springer Verlag, Heidelberg, 1994.

    Google Scholar 

  27. Lust-Piquard F.: Opérateurs de Hankel 1-sommants de (IN) dans £°°(IN) et multiplicateurs de H 1 (T). Comptes Rendus Acad. Sci. Paris. 299 (1984) 915–918.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Pisier, G. (1996). Non-unitarizable uniformly bounded group representations. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21537-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-21537-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60322-1

  • Online ISBN: 978-3-662-21537-1

  • eBook Packages: Springer Book Archive