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A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability

Published online by Cambridge University Press:  20 November 2018

Benjamin Willson
Benjamin Willson*
Affiliation:
Department of Mathematics, Hanyang University, zzz Wangsimni-ro, Seongdong-gu, Seoul, Korea. e-mail: [email protected]
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Abstract

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In this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

Keywords

43A6243A0543A07invariant measureHaar measurehypergroupamenabilityfunction translations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 415 - 422
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Alaghmandan, M., Amenability notions of hypergroups and some applications to locally compact groups. arxiv:1 402.2263Google Scholar
[2] Bloom, W. R. and Heyer, H., Harmonie analysis of probability measures on hypergroups. de Gruyter Studies in Mathematics, 20, Walter de Gruyter & Co., Berlin, 1995.Google Scholar
[3] Dunkl, C. F., The measure algebra of a locally compact hypergroup. Trans. Amer. Math. Soc. 179 (1973), 331–348. http://dx.doi.Org/10.1090/S0002–9947-1973-0320635-2 Google Scholar
[4] Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. I. Structure of topological groups, integration theory, group representations. Grundlehren der Mathematischen Wissenschaften, 115, Springer-Verlag, Berlin-New York, 1979.Google Scholar
[5] Izzo, A. J., A functional analysis proof of the existence ofHaar measure on locally compact abelian groups. Proc. Amer. Math. Soc. 115 (1992), no. 2, 581–583. http://dx.doi.Org/10.1090/S0002-9939-1992-1097346-9 Google Scholar
[6] Jewett, R. I., Spaces with an abstract convolution of measures. Advances in Math. 18 (1975), no. 1, 1–101. http://dx.doi.Org/10.1016/0001-8708(75)90002-X Google Scholar
[7] Lasser, R. and Skantharajah, M., Reiter's condition for amenable hypergroups, Monatsh Math 63 (2010), no. 3, 327–338. http://dx.doi.Org/10.1007/s00605-010-0206-z Google Scholar
[8] Paterson, A. L. T., Amenability. Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988.Google Scholar
[9] Rosenblatt, J. M., Invariant measures and growth conditions. Trans. Amer. Math. Soc. 193 (1974), 33–53. http://dx.doi.Org/10.1090/S0002-9947-1974-0342955-9 Google Scholar
[10] Singh, A. I., Completely positive hypergroup actions. Mem. Amer. Math. Soc. 124 (1996), no 593.Google Scholar
[11] Skantharajah, M., Amenable hypergroups. Illinois J. Math. 36 (1992), no. 1,15–46.Google Scholar
[12] Spector, R., Mesures invariantes sur les hypergroupes. Trans. Amer. Math. Soc. 239 (1978), 147–165. http://dx.doi.Org/10.1090/S0002–9947-1978-0463806-1 Google Scholar
[13] Tahmasebi, N., Hypergroups and invariant complemented subspaces. J. Math. Anal. Appl. 414 (2014), no. 2, 641–655. http://dx.doi.Org/10.1016/j.jmaa.2014.01.026 Google Scholar
[14] Willson, B., Configurations and invariant nets for amenable hypergroups and related algebras. Trans. Amer. Math. Soc. 366 (2014), no. 10, 5087–5112. http://dx.doi.org/10.1090/S0002–9947-2014-05731-0 Google Scholar