The theory of semihypergroups is a natural extension to the theory of locally compact semigroups. In this article, we present different notions of amenability, namely amenability of function-spaces and topological amenability, in the broader setting of (locally compact) semihypergroups and survey some recent developments in this area of research regarding certain ergodic, stationary, hereditary, Banach algebraic and fixed-point characterizations of such notions on general (semitopological) semihypergroups.

Common fixed points of representations of different categories of topological and analytic objects have been a pivotal area of evolving interest in the studies of fixed-point theory and harmonic analysis for several reasons. In this text, we consider certain families of left/right coset, double coset, and orbit spaces arising from the category of locally compact groups. We solely investigate their actions on compact subsets of general locally convex spaces, as well as on certain Banach spaces. In particular, we use some recent developments in abstract harmonic analysis regarding the theory of Semihypergroups to provide an overview of several characterizations for the existence of common fixed points of such actions in terms of amenability of the underlying spaces.

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.

It was shown in [Colloq. Math. 131(2), 219--231 (2013)] that one can extend the domain of Fourier transform of a commutative hypergroup K to Lp(K) for 1 ≤ p≤ 2 , and the Hausdorff–Young inequality holds true for these cases. In this article, we examine the structure of non-zero functions in Lp(K) for which equality is attained in the Hausdorff–Young inequality, for 1 < p< 2 , and further provide a characterization for the basic uncertainty principle for commutative hypergroups with non-trivial center.

In a previous paper [1], we initiated a systematic study of semihypergroups and had a thorough discussion about some important analytic and algebraic objects associated to this class of objects. In this paper, we investigate free structures on the category of semihypergroups. We show that the natural free product structure along with the natural topology, although fails to give a free product for topological groups, works well on a vast non-trivial class of ‘pure’ semihypergroups containing most of the well-known examples including non-trivial coset and orbit spaces.

The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by Dunkl (Am Math Soc 179:331–348, 1973), Jewett (Adv Math 18(1):1–101, 1975) and Spector (Apercu de la theorie des hypergroups, (French) Analyse harmonique sur les groupes de Lie (Sém. Nancy–Strasbourg, 1973–75), Springer, New York, 1975) independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomorphisms and ideals, and thereby examine their behaviour (basic properties, structure of the kernel and relation of amenability to minimal ideals) in order to gain insight into the structure of a Semihypergroup itself. In the process, we further investigate where and why this theory deviates from the classical theory of semigroups.