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

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.