Kazhdan’s Property and Weak Containment for Hypergroups

  • Alireza Bagheri Salec
  • Seyyed Mohammad Tabatabaie
  • Benyamin Karami
Original Paper


We introduce the notion of strong and weak containment for representations of hypergroups, and investigate their relationships to functions of positive type. We also define and study the notion of Kazhdan’s property for hypergroups.


Locally compact hypergroup Kazhdan’s property Weak containment Strong containment 

Mathematics Subject Classification

Primary 43A62 



We would like to thank the referee for careful reading and very useful comments and suggestions.


  1. 1.
    Bekka, B., Laharpe, P., Valette, A.: Kazhdan’s property (T), new mathematical monographs, 11. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  2. 2.
    Bloom, W.R., Heyer, H.: Harmonic analysis of probability measures on hypergroups. De Gruyter, Berlin (1995)CrossRefMATHGoogle Scholar
  3. 3.
    Conway, J.B.: A course in functional analysis, 2nd edn. Springer, New York (1990)MATHGoogle Scholar
  4. 4.
    Dixmier, J.: Les \(C^*\)-algebres et Leurs representations. Gauthier-Villars, Paris (1969)MATHGoogle Scholar
  5. 5.
    Dunkl, C.F.: The measure algebra of a locally compact hypergroup. Trans. Am. Math. Soc. 179, 331–348 (1973)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dunkl, C.F., Ramirez, D.E.: A family of countably compact \(P_*\)-hypergroups. Trans. Am. Math. Soc. 202, 339–356 (1975)MATHGoogle Scholar
  7. 7.
    Fell, J.M.G.: The dual space of C-algebras. Trans. Am. Math. Soc. 94, 365–403 (1960)MATHGoogle Scholar
  8. 8.
    Fell, J.M.G.: Weak containment and induced representations. Can. J. Math. 14, 237–268 (1962)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Folland, G.B.: A course in abstract harmonic analysis. CRC Press, Inc., (1995)Google Scholar
  10. 10.
    Godement, R.: Les fonctions de type de positif et la thorie des groupes. Trans. Am. Math. Soc. 63, 1–84 (1948)MATHGoogle Scholar
  11. 11.
    Hermann, P.: Induced representations of hypergroups. Math. Z. 211, 687–699 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18, 1–101 (1975)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kazhdan, D.A.: Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. J. Prilozen 1, 71–74 (1967)MathSciNetMATHGoogle Scholar
  14. 14.
    Pavel, L.: On hypergroups with Kazhdan’s property (T). Math. Reports-Studii Cerc. Math. 2(52), 345–350 (2000)MathSciNetMATHGoogle Scholar
  15. 15.
    Pavel, L.: Ergodic sequences of probability measures on commutative hypergroups. Int. J. Math. Math. Sci. 7, 335–343 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Spector, R.: Apercu de la theorie des hypergroups. In: Analyse Harmonique sur les Groups de Lie, 643–673, Lec. Notes Math. Ser., 497, Springer (1975)Google Scholar
  17. 17.
    Tabatabaie, S.M.: The problem of density on \(L^2(G)\). Acta Math. Hungar. 150, 339–345 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Tabatabaie, S.M., Haghighifar, F.: The associated measure on locally compact cocommutative KPC-hypergroups. Bull. Iran. Math. Soc. 43(1), 1–15 (2017)MathSciNetMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • Alireza Bagheri Salec
    • 1
  • Seyyed Mohammad Tabatabaie
    • 1
  • Benyamin Karami
    • 1
  1. 1.Department of MathematicsUniversity of QomQomIran

Personalised recommendations