# Compact and weakly compact multipliers on Fourier algebras of ultraspherical hypergroups

## TL;DR

This paper explores the properties of Fourier algebras on ultraspherical hypergroups, linking algebraic features like compact multipliers to the hypergroup's discreteness and analyzing Arens regularity of ideals.

## Contribution

It extends known results about Fourier algebras from groups to ultraspherical hypergroups, providing new characterizations of hypergroup discreteness and studying ideal regularity.

## Key findings

- Discreteness of hypergroups characterized by compact multipliers
- Several algebraic conditions equivalent to hypergroup discreteness
- Analysis of Arens regularity of closed ideals

## Abstract

A locally compact group $ G $ is discrete if and only if the Fourier algebra $ A(G) $ has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let $H$ be an ultraspherical hypergroup and let $A(H)$ denote the corresponding Fourier algebra. We will give several characterizations of discreteness of $ H $ in the terms of the algebraic properties of $A(H)$. We also study Arens regularity of closed ideals of $ A(H)$.