Characterizations of amenable hypergroups

Document Type : Research Paper

Authors

1 Semnan University

2 Payame Noor University

10.22072/wala.2017.23365

Abstract

Let $K$ be a locally compact hypergroup with left Haar measure and let $L^1(K)$ be the complex Lebesgue space associated with it. Let $L^\infty(K)$ be the dual of $L^1(K)$. The purpose of this paper is to present some necessary and sufficient conditions for $L^\infty(K)^*$ to have a topologically left invariant mean. Some
characterizations of amenable hypergroups are given.

Keywords


[1] A. Azimifard, the alpha-amenability of hypergroups, Monatsh. Math., 155(1) (2008), 1-13.
[2] W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, vol. 20, de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1995.
[3] C.F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Am. Math. Soc., 179 (1973), 331-348.
[4] R.E. Edwards, Functional Analysis, New-York, Holt, Rinehart and Winston, 1965.
[5] E.E. Granirer, Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Am. Math. Soc., 40(2) (1973), 615-624.
[6] Z. Hu, M. Sangani Monfared and T. Traynor, On character amenable Banach algebras, Stud. Math., 193(1) (2009), 53-78.
[7] R.I. Jewett, Spaces with an abstract convolution of measures, Adv. Math., 18(1) (1975), 1-101.
[8] R.A. Kamyabi-Gol, Topological Center of Dual Banach Algebras Associated to Hypergroups, Ph.D. Thesis,
University of Alberta, 1997.
[9] R.A. Kamyabi-Gol, $P$-amenable locally compact hypergroups, Bull. Iran. Math. Soc., 32(2) (2006), 43-51.
[10] Lasser, Amenability and weak amenability of $l^1$-algebras of polynomial hypergroups, Stud. Math., 182(2) (2007), 183-196.
[11]  A.R. Medghalchi, The second dual of a hypergroup, Math. Z., 210 (1992), 615-624.
[12] M.S. Monfared, Character amenability of Banach algebras, Math. Proc. Camb. Philos. Soc., 144(3) (2008), 697-706.
[13] W. Rudin, Functional Analysis, McGraw Hill, New York, 1991.
[14] M. Skantharajah, Amenable hypergroups, Illinois J. Math., 36(1) (1992), 15-46.
[15] R. Spector, Mesures invariantes sur les hypergroupes (French, with English summary), Trans. Am. Math. Soc., 239 (1978), 147-165.
[16] N. Tahmasebi, Hypergroups and invariant complemented subspaces, J. Math. Anal. Appl., 414 (2014), 641-651.
[17] B. Wilson, Configurations and invariant nets for amenable hypergroups and related algebras, Trans. Am. Math. Soc., 366(10) (2014), 5087-5112.