Weighted discrete hypergroups

TL;DR

This paper explores the properties of weighted hypergroup algebras, focusing on Arens regularity and operator algebra isomorphisms, and examines specific classes of discrete weighted hypergroups with some surprising findings.

Contribution

It introduces the concept of weighted hypergroups and investigates their algebraic properties, extending known results from weighted group algebras to a broader hypergroup context.

Findings

Identifies conditions for Arens regularity in weighted hypergroup algebras

Establishes isomorphism criteria to operator algebras for these hypergroups

Provides examples with unexpected behaviors in regularity and isomorphism properties

Abstract

Weighted group algebras have been studied extensively in Abstract Harmonic Analysis where complete characterizations have been found for some important properties of weighted group algebras, namely amenability and Arens regularity. One of the generalizations of weighted group algebras is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study Arens regularity and isomorphism to operator algebras for them. We also examine our results on three classes of discrete weighted hypergroups constructed by conjugacy classes of FC groups, the dual space of compact groups, and hypergroup structure defined by orthogonal polynomials. We observe some unexpected examples regarding Arens regularity and operator isomorphisms of weighted hypergroup algebras.