Weighted group algebras

TL;DR

This paper investigates the algebraic and topological properties of weighted group algebras on locally compact Abelian groups, establishing conditions for regularity and Tauberian properties based on the weight function.

Contribution

It characterizes when weighted group algebras are regular and Tauberian, linking these properties to the nature of the weight function.

Findings

Weighted group algebra $L^{1}(G, w)$ is regular iff $w$ is nonquasianalytic.

$L^{1}(G, w)$ is Tauberian for any Borel measurable weight.

Provides criteria connecting weight functions to algebraic properties.

Abstract

Let $G$ be a locally compact Abelian group, and $w: G\to (0, \infty)$ be a Borel measurable weighted function. In this paper, the algebraic and topological properties of group algebra are studied and assessed. We show that the weighted group algebra $L^{1}(G, w)$ is regular if and only if $w$ is a nonquasianalytic weight function. Also $L^{1}(G, w)$ is Tauberian, for any Borel measurable weight function $w$ on the group $G$.