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

TL;DR

This paper addresses when Beurling algebras $L^1(G,\omega)$ on a locally compact abelian group $G$ possess the $BSE$ and $BED$ properties. It proves that if $\omega$ is a weight with $\omega^{-1}$ vanishing at infinity, then $L^1(G,\omega)$ is both a $BSE$-algebra and a $BED$-algebra, and it shows that these properties are preserved under isomorphisms induced by continuous weight changes. The main technical contributions are a uniform continuity bridge from $\widehat{G}$ to $L^1(G,\omega)^*$, a density argument identifying $M(G,\omega)$ with $C_0(G,\omega^{-1})^*$, and a two-case BED proof that separates discrete and non-discrete $G$ and eliminates the singular part of the representing measure. Together, these results extend the classical $L^1(G)$ BSE/BED properties to the weighted Beurling setting, with implications for multiplier algebras and spectral theory in weighted group algebras.

Abstract

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

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof