The BSE property for some vector-valued Banach function algebras

In this paper, X is a locally compact Hausdorff space and \({\cal A}\) $$\mathcal{A}$$ is a Banach algebra. First, we study some basic features of C₀(X, \({\cal A}\) $$\mathcal{A}$$ ) related to BSE concept, which are gotten from \({\cal A}\) $$\mathcal{A}$$ . In particular, we prove that if C₀(X, \({\cal A}\) $$\mathcal{A}$$ ) has the BSE property then \({\cal A}\) $$\mathcal{A}$$ has so. We also establish the converse of this result, whenever X is discrete and \({\cal A}\) $$\mathcal{A}$$ has the BSE-norm property. Furthermore, we prove the same result for the BSE property of type I. Finally, we prove that C₀ (X, \({\cal A}\) $$\mathcal{A}$$ ) has the BSE-norm property if and only if \({\cal A}\) $$\mathcal{A}$$ has so.