Harmonic Analysis on Internally Gelfand Pairs Associated to Groupoids

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Ibrahima Toure, Kinvi Kangni, Harmonic Analysis on Internally Gelfand Pairs Associated to Groupoids, Int. J. Anal. Appl., 17 (6) (2019), 928-939.

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Ibrahima Toure, Kinvi Kangni

Abstract

Let G be a topological locally compact, Hausdorff and second countable groupoid with a Haar system and K a proper subgroupoid of G with a Haar system too. (G, K) is an internally Gelfand pair if for any u in the unit space, the algebra of bi-K(u)-invariant functions on G(u) is commutative under convolution. In this work, we give some characterizations of these pairs and extend to this context some classical results of harmonic analysis.

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References

  1. M. Akouchi, A. Bakali, Une Generalisation des paires de Gel'fand, Bolletino U.M.I, 7 (1992), 795-822.
  2. M. Buneci, Groupoid Algebras For Transitive Groupoids, Math. Rep., 5(55) (2003), 9-26.
  3. R. Bos, Groupoids in Geometric Quantization, PhD. Thesis, Radboud University Nijmegen, 2007.
  4. J. Dieudonne, Gelfand Pairs And Spherical Functions, Int. J. Math. Math. Sci., 2(2) (1979), 153-162.
  5. V. Dijk, Introduction to Harmonic Analysis and Generalized Gelfand Pairs, De Gruyter Stud. Math., 36, Walter de Gruyter And Co, Berlin, 2009.
  6. J. Dixmier, Les C*-Algebres et Leurs Representations, Gauthier-Villars, Paris, 1964.
  7. J. Faraut, Infinite Dimensional spherical Analysis, COE Lecture Note, 10, Kyushu University, 2008.
  8. J. Faraut, Analyse Harmonique sur les Paires de Gelfand, Les cours du CIMPA, 1980.
  9. I. F. Gelfand, Spherical Functions On Symetrics Spaces, Dokl. Akad. Nauk., 70 (1960), 5-8.
  10. R. Godement, A Theory Of Spherical Functions I, Trans. Amer. Math. Soc., 73 (1952), 496-556.
  11. G. Goehle, Groupoid Crossed Products, PhD. Thesis, Dartmouth College, 2009.
  12. P. Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc., 242 (1978), 34-72.
  13. P. Hahn, Haar measure for Groupoids, Trans. Amer. Math. Soc., 242 (1978), 1-33.
  14. K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Note Ser., 213, Cambridge University Press, 2005.
  15. J. F. M {c}Clendon, Metric Families, Pac. J. Math., 57 (1975), 491-509.
  16. P. S. Muhly, J. N. Renault, D. P. Williams, Continuous-Trace Groupoid C*-algebras, III, Trans. Math. Soc., 348 (1996), 3621-3641.
  17. K. J. Oty, Fourier-Stieltjes Algebras of r-Discrete Groupoids, J. Oper. Theory, 4 (1999), 175-197.
  18. A. Ramsay, M. E. Walter, Fourier-Stieltjes Algebras of Locally Compact Groupoid, J. Funct. Anal., 148(2) (1997), 314-367.
  19. J. Renault, The Ideal Structure of Grupoid Crossed Product C*-Algebras, J. Oper. Theory, 25 (1991), 3-36.
  20. J. Renault, A Groupoid Approach to C*-Algebras, Lecture Notes in Math., 793, Springer-Verlag, 1980.
  21. I. Toure, K. Kangni, On Gelfand Pairs associated to transitive groupoids, Opuscula Math., 33(4) (2013), 751-762.
  22. I. Toure, K. Kangni, On Groupoid Algebras of Biinvariant Functions, Int. J. Math. Anal., 6 (43) (2012), 2101-2108.
  23. J. J. Westman, Harmonic Analysis on groupoids, Pacific J. Math., 27(3) (1968), 621-632.