Some vector-valued Banach function algebras, satisfying the Bochner—Eberlein—Doss property

The notion of $\rm BED$ algebras was introduced by Japanese mathematicians. Afterwards, some authors investigated the BED property of some commutative and semisimple Banach algebras. In this paper, assuming that $X$ is a Hausdorff locally compact space and $\mathcal{A}$ is a commutative and semisimple Banach algebra, we verify the BED property for $C_0(X,\mathcal{A})$. We first prove that $\mathcal{A}$ is a BED algebra, whenever $C_0(X,\mathcal{A})$ is so. Then, we verify the converse of this statement, and prove that $\widehat{\mathcal{A}}\subseteq C^0_{\operatorname{BSE}}(\Phi_{\mathcal{A}})$ implies that $\widehat{C_{0}(X,\mathcal{A})}\subseteq C^0_{\operatorname{BSE}}(\Phi_{C_{0}(X,\mathcal{A})})$. Finally, similar to the BSE case, we prove that if $X$ is discrete, then $C_0(X,\mathcal{A})$ is a BED algebra if and only if $\mathcal{A}$ is so.