A fixed point theorem and the existence of a Haar measure for hypergroups satisfying conditions related to amenability

TL;DR

This paper establishes a fixed point theorem for amenable hypergroups, linking invariant means to fixed points, and demonstrates the existence of a Haar measure for certain hypergroups, advancing harmonic analysis on hyperstructures.

Contribution

It introduces a fixed point property for amenable hypergroups and proves the existence of Haar measures under specific conditions, extending classical results to hypergroups.

Findings

Fixed point property for amenable hypergroups

Existence of Haar measure for certain hypergroups

Extension of classical harmonic analysis results

Abstract

In this paper we present a fixed point property for amenable hypergroups which is analogous to Rickert's fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, a certain class of hypergroups are shown to have a left Haar measure.