Topological Invariant Means on dual Space of Multiplier algebra and weakly compact Multiplier on Herz-algebra

TL;DR

This paper explores the properties of multipliers in Herz-algebras, establishing the existence of a topological invariant mean and linking separability of dual spaces to the discreteness of the underlying group.

Contribution

It introduces the concept of a topological invariant mean on the dual space of multiplier algebras and characterizes when the group is discrete based on separability.

Findings

Existence of a topological invariant mean on $B_p^*(G)$

If $B_p^*(G)$ is separable, then $G$ is discrete

Analysis of compact and weakly compact multipliers of Herz-algebras

Abstract

In this paper we investigate the compact and weakly compact multipliers of the Herz-algebras $A_p(G)$. Let $B_p(G)$ be the space of pointwise multipliers of $A_p(G)$. We show that there is a topological invariant mean on $B^*_p (G)$. Furthermore, we show that if $B^*_p(G)$ is separable, then $G$ is a discrete group.