Common Fixed Points of Semihypergroup Representations

TL;DR

This paper explores fixed point properties of semihypergroup representations on convex sets and their relation to the amenability of almost periodic functions, extending to dual Banach spaces.

Contribution

It introduces new fixed point properties for semihypergroup representations and links them to the amenability of almost periodic functions, advancing the theory of semihypergroups.

Findings

Fixed point properties are equivalent to amenability of almost periodic functions.

Representation on the dual of a Banach space strengthens these equivalences.

The study extends the understanding of semihypergroup actions on topological spaces.

Abstract

In a series of previous papers, we initiated a systematic study of semihypergroups and had a thorough discussion on certain analytic and algebraic aspects associated to this class of objects. In particular, we introduced the notion of semihypergroup actions on a general topological space and discussed different continuity, equivalence and natural fixed point properties of the same in [6]. Now in this article, we consider different kinds of representations of a semihypergroup on compact convex subsets of a locally convex space and explore equivalence relations between certain fixed-point properties of such representations and amenability of the space of almost periodic functions. Finally, we investigate how far these equivalence relations can be strengthened when in particular, we consider representations on the dual of a Banach space.