BSE-property of tensor product Banach algebras

Abstract: Let $A$ and $B$ be commutative semisimple Banach algebras. In this paper, the $BSE$ and $BED$- property of tensor Banach algebra $A{\otimes}{\gamma} B$ with respect to the Banach algebras $A$ and $B$ are assesed. In particular, $BSE$ and $BED$- structure of vector-valued group algebra $L{1}(G, A)$ and $C^{*}$- algebra $C_{0}(X, A)$ characterized.