On the BSE- property of vector valued Beurling algebra $L^1(G,\omega, \mathcal{A})$

TL;DR

This paper investigates the BSE-property of vector-valued Beurling algebras on locally compact abelian groups, establishing a criterion linking the algebra's BSE-property to that of the underlying Banach algebra.

Contribution

It provides a necessary and sufficient condition for the Beurling algebra to be a BSE-algebra based on the properties of the Banach algebra involved.

Findings

The algebra $L^1(G, ext{ω}, ext{A})$ is BSE if and only if $ ext{A}$ is BSE.

The result applies when $ ext{ω}^{-1}$ vanishes at infinity.

The paper characterizes the BSE-property in the context of vector-valued Beurling algebras.

Abstract

Let $G$ be a locally compact abelian group, and let $\omega:G \to [1,\infty)$ be a measurable weight, i.e., $\omega$ is measurable, and $\omega(s+t)\leq \omega(s)\omega(t)$ for all $s, t \in G$. Let $\mathcal{A}$ be a semisimple commutative Banach algebra with a predual $\mathcal A_\ast$ such that the Gel'fand space $\Phi_{\mathcal A}\subset \mathcal{A}_\ast$. If $\omega^{-1}$ is vanishing at infinity, then we show that the Banach algebra $L^1(G,\omega,\mathcal{A})$ is a BSE- algebra if and only if $\mathcal A$ is a BSE- algebra.