  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.
