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.

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. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024.

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