Abstract
In this article, we introduce and explore the notion of topological amenability in the broad setting of (locally compact) semihypergroups. We acquire several stationary, ergodic and Banach algebraic characterizations of the same in terms of convergence of certain probability measures, total variation of convolution with probability measures and translation of certain functionals, as well as the F-algebraic properties of the associated measure algebra. We further investigate the interplay between restriction of convolution product and convolution of restrictions of measures on a sub-semihypergroup. Finally, we discuss and characterize topological amenability of sub-semihypergroups in terms of certain invariance properties attained on the corresponding measure algebra of the parent semihypergroup. This in turn provides us with an affirmative answer to an open question posed by J. Wong in 1980.
Funding statement: The author would like to gratefully acknowledge the financial support provided by the Indian Institute of Technology Kanpur, India and Harish-Chandra Research Institute, India, where initial phases of the work were done.
Acknowledgements
The author is grateful to the reviewers for the valuable comments and suggestions which led to a better representation of the article.
References
[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. 10.1090/S0002-9939-1951-0045941-1Search in Google Scholar
[2] C. Bandyopadhyay, Analysis on semihypergroups: Function spaces, homomorphisms and ideals, Semigroup Forum 100 (2020), no. 3, 671–697. 10.1007/s00233-019-10065-6Search in Google Scholar
[3] C. Bandyopadhyay, Free product on semihypergroups, Semigroup Forum 102 (2021), no. 1, 28–47. 10.1007/s00233-020-10152-zSearch in Google Scholar
[4] C. Bandyopadhyay, Fixed points and continuity of semihypergroup actions, preprint (2022), https://arxiv.org/abs/2202.05956v2. Search in Google Scholar
[5] C. Bandyopadhyay and P. Mohanty, Equality in Hausdorff–Young for hypergroups, Banach J. Math. Anal. 16 (2022), no. 4, Paper No. 62. 10.1007/s43037-022-00211-8Search in Google Scholar
[6] J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1989. Search in Google Scholar
[7] W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter Stud. Math. 20, Walter de Gruyter, Berlin, 1995. 10.1515/9783110877595Search in Google Scholar
[8] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. 10.1215/ijm/1255380675Search in Google Scholar
[9] C. F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331–348. 10.1090/S0002-9947-1973-0320635-2Search in Google Scholar
[10] R. I. Jewett, Spaces with an abstract convolution of measures, Adv. Math. 18 (1975), no. 1, 1–101. 10.1016/0001-8708(75)90002-XSearch in Google Scholar
[11] B. E. Johnson, Cohomology in Banach Algebras, Mem. Amer. Math. Soc. 127, American Mathematical Society, Providence, 1972. 10.1090/memo/0127Search in Google Scholar
[12] A. T. M. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), no. 3, 161–175. 10.4064/fm-118-3-161-175Search in Google Scholar
[13] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. 10.1090/S0002-9947-1951-0042109-4Search in Google Scholar
[14] J.-P. Pier, Amenable Banach Algebras, Pitman Res. Notes Math. Ser. 172, Longman Scientific & Technical, Harlow, 1988. Search in Google Scholar
[15] K. A. Ross, Centers of hypergroups, Trans. Amer. Math. Soc. 243 (1978), 251–269. 10.1090/S0002-9947-1978-0493161-2Search in Google Scholar
[16] M. Skantharajah, Amenable hypergroups, Illinois J. Math. 36 (1992), no. 1, 15–46. 10.1215/ijm/1255987605Search in Google Scholar
[17] R. Spector, Aperçu de la théorie des hypergroupes, Analyse harmonique sur les groupes de Lie, Lecture Notes in Math. 497, Springer, Berlin (1975), 643–673. 10.1007/BFb0078026Search in Google Scholar
[18] J. C. S. Wong, Topologically stationary locally compact groups and amenability, Trans. Amer. Math. Soc. 144 (1969), 351–363. 10.1090/S0002-9947-1969-0249536-4Search in Google Scholar
[19] J. C. S. Wong, Invariant means on locally compact semigroups, Proc. Amer. Math. Soc. 31 (1972), 39–45. 10.1090/S0002-9939-1972-0289708-1Search in Google Scholar
[20] J. C. S. Wong, An ergodic property of locally compact amenable semigroups, Pacific J. Math. 48 (1973), 615–619. 10.2140/pjm.1973.48.615Search in Google Scholar
[21] J. C. S. Wong, A characterisation of locally compact amenable subsemigroups, Canad. Math. Bull. 23 (1980), no. 3, 305–312. 10.4153/CMB-1980-042-6Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
