Beurling-Figa-Talamanca-Herz algebras

TL;DR

This paper introduces and studies Beurling-Figa-Talamanca-Herz algebras for locally compact groups, establishing their properties and characterizing group amenability via bounded approximate identities.

Contribution

It defines Beurling-Figa-Talamanca-Herz algebras using inverse weights and characterizes group amenability through these algebras' bounded approximate identities.

Findings

For abelian groups, these algebras coincide with classical Beurling algebras.

The approach unifies and extends previous work for compact groups.

Group amenability is characterized by the existence of bounded approximate identities in these algebras.

Abstract

For a locally compact group $G$ and $p \in (1,\infty)$, we define and study the Beurling-Figa-Talamanca-Herz algebras $A_p(G,\omega)$. For $p=2$ and abelian $G$, these are precisely the Beurling algebras on the dual group $\hat{G}$. For $p =2$ and compact $G$, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that a locally compact group $G$ is amenable if and only if one - and, equivalently, every - Beurling-Figa-Talamanca-Herz algebra $A_p(G,\omega)$ has a bounded approximate identity.