Topological Amenability of Semihypergroups

TL;DR

This paper introduces the concept of topological amenability for semihypergroups, providing multiple characterizations and exploring its properties and implications, including an answer to a question from 1980.

Contribution

It develops a comprehensive framework for topological amenability in semihypergroups, including new characterizations and the relation between sub-semihypergroups and measure invariance.

Findings

Multiple characterizations of topological amenability.

Relation between convolution restrictions and sub-semihypergroups.

Resolution of an open question by J. Wong (1980).

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.