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Isomorphisms of the Figà-Talamanca-Herz algebras A p (G) for connected Lie groups G

  • Conference paper
Colloquium De Giorgi 2009

Part of the book series: Colloquia ((COLLOQUIASNS,volume 3))

Abstract

There are various Banach algebras of functions on a locally compact group G, made up of matrix coefficients of representations, such as the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G), which reflect the representation theory of the group. The question of whether these determine the group has been considered by many authors. Here we show that when 1 < p < ∞, the Figà-Talamanca-Herz algebras A p (G) determine the group G, at least if G is a connected Lie group.

The author wishes to thank the Centro di Ricerca Matematica Ennio De Giorgi and the Alexander von Humboldt Stiftung for their support.

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References

  1. M. Baronti, Algebre di Banach Abstract p di gruppi localmente compatti, Riv. Mat. Univ. Parma 11 (1985), 399–407.

    MathSciNet  MATH  Google Scholar 

  2. A. Čap, M. G. Cowling, F. De Mari, M. G. Eastwood and R. G. Mc Callum, The Heisenberg group, SL(3,R) and rigidity, pages 41–52, In: “Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory”, edited by Jian-Shu Li, Eng-Chye Tan, Nolan Wallach and Chen-Bo Zhu. World Scientific, Singapore, 2007.

    Google Scholar 

  3. P. M. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191–212.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. M. Cohen, On homomorphisms of group algebras, Amer. J. Math. 82 (1960), 213–226.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. G. Cowling and G. Fendler, On representations in Banach spaces, Math. Ann. 266 (1984), 307–315.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Derighetti, A property of Bp (G). Applications to convolution operators, J. Funct. Anal. 256 (2009), 928–939.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. H. Dooley, S. Gupta and F. Ricci, Asymmetry of convolution norms on Lie groups, J. Funct. Anal. 174 (2000), 399–416.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. P. Eymard, Algèbres Ap et convoluteurs de Lp , pages 55–72 (expos é 367) In: “Séminaire Bourbaki”, Vol. 1969/1970, Exposés 364-381, Springer-Verlag, Berlin, Heidelberg, New York, 1971.

    Chapter  Google Scholar 

  10. A. Figà-Talamanca, Translation invariant operators in Lp , Duke Math. J. 32 (1965), 495–501.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Guralnick, Invertibility preservers and algebraic groups, Linear Algebra Appl. 212/213 (1994), 249–257.

    Article  MathSciNet  Google Scholar 

  12. C. S. Herz, Remarques sur la note précédente de M. Varopoulos, C. R. Acad. Sci. Paris 260 (1965), 6001–6004.

    MathSciNet  MATH  Google Scholar 

  13. C. S. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23(3) (1973), 91–123.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Host, Le théorème des idempotents dans B(G), Bull. Soc. Math. France 114 (1986), 215–221.

    MathSciNet  MATH  Google Scholar 

  15. J. R. Hubbuck and R. M. Kane, The homotopy types of compact Lie groups, Israel J. Math. 51 (1985), 20–26.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Ilie and N. Spronk, Completely bounded homomorphisms of the Fourier algebras, J. Funct. Anal. 225 (2005), 480–499.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Jeffers, Lost theorems of geometry, Amer. Math. Monthly 107 (2000), 800–812.

    Article  MathSciNet  MATH  Google Scholar 

  18. N. Lohoué, Sur certaines propriétés remarquables des algbres Ap (G), C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A893–A896.

    Google Scholar 

  19. A. M. Mantero, Asymmetry of convolution operators on the Heisenberg group, Boll. Un. Mat. Ital. A (6) 4 (1985), 19–27.

    Google Scholar 

  20. D. Montgomery and L. Zippin, “Topological Transformation Groups”, Interscience Publishers, New York-London, 1955.

    MATH  Google Scholar 

  21. V. Runde, Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca-Herz algebras, J. Math. Anal. Appl. 329 (2007), 736–751.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Scheerer, Homotopieäquivalente kompakte Liesche gruppen, Topology 7 (1968), 227–232.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. E. Walter, A duality between locally compact groups and certain Banach algebras, J. Funct. Anal. 17 (1974), 131–160.

    Article  MATH  Google Scholar 

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Authors
  1. Michael G. Cowling
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Umberto Zannier

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© 2012 Scuola Normale Superiore Pisa

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Cowling, M.G. (2012). Isomorphisms of the Figà-Talamanca-Herz algebras A p (G) for connected Lie groups G . In: Zannier, U. (eds) Colloquium De Giorgi 2009. Colloquia, vol 3. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-387-1_1

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