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

Let $G$ be a locally compact abelian group, and let $\omega :G\to [1,\mathrm{\infty })$ be a measurable weight, i.e., $\omega$ is measurable, and $\omega (s+t)\le \omega (s)\omega (t)$ 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 ${\mathrm{\Phi }}_{\mathcal{A}}\subset {\mathcal{A}}_{\ast }$. 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.