On the BSE and BED properties of the Beurling algebra $L^1(G,\omega)$

TL;DR

This paper investigates the properties of Beurling algebras on locally compact abelian groups, establishing conditions under which these algebras are both BSE and BED algebras, thus advancing understanding of their harmonic analysis structure.

Contribution

It proves that if the inverse of the weight function vanishes at infinity, then the associated Beurling algebra is both BSE and BED, providing new insights into their algebraic properties.

Findings

Beurling algebra is BSE if inverse weight vanishes at infinity.

Beurling algebra is BED under the same condition.

Conditions link weight decay to algebraic properties.

Abstract

Let $G$ be a locally compact abelian group, and let $\omega:G \to [1,\infty)$ be a weight, i.e., $\omega$ is measurable, $\omega$ is locally bounded and $\omega(s+t)\leq \omega(s)\omega(t)$ for all $s, t \in G$. If $\omega^{-1}$ is vanishing at infinity, then we show that the Beurling algebra $L^1(G,\omega)$ is both BSE- algebra and BED- algebra.