# Invariant $\varphi $-means on multiplier completion of Banach algebras with application to hypergroups

## TL;DR

This paper introduces invariant $$-means on the multiplier completion of Banach algebras, linking their properties to hypergroup Fourier algebras and providing criteria for hypergroup discreteness.

## Contribution

It defines and analyzes topologically invariant $$-means on multiplier completions of Banach algebras, connecting these concepts to hypergroup Fourier algebras and their properties.

## Key findings

- Invariant $$-means on ${
m A}_M^*$ and ${
m A}^*$ have the same cardinality under certain conditions.
- Characterizations of hypergroup discreteness are obtained via Fourier algebra properties.
- The study links Banach algebra invariance with hypergroup harmonic analysis.

## Abstract

Let ${\mathcal A}$ be a Banach algebra and let $\varphi $ be a non-zero character on ${\mathcal A}$. Suppose that ${\mathcal A}_M$ is the closure of the faithful Banach algebra ${\mathcal A}$ in the multiplier norm. In this paper, topologically left invariant $\varphi$-means on ${\mathcal A}_M^*$ are defined and studied. Under some conditions on ${\mathcal A}$, we will show that the set of topologically left invariant $\varphi$-means on ${\mathcal A}^*$ and on ${\mathcal A}_M^*$ have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. The main applications are concerned with the Fourier algebra of an ultraspherical hypergroup $H$. In particular, we obtain some characterizations of discreteness of $H$.