BED property for the tensor product of Banach algebras

Let $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras. Suppose that $\|\cdot\|_\gamma$ is an algebra cross-norm on $\mathcal A\otimes\mathcal B$ such that $\|\cdot\|_\gamma\geq\|\cdot\|_e$, and $\mathcal A\widehat{\otimes}_\gamma\mathcal B$ is a semisimple Banach algebra. In this paper, we verify the $\rm BED$ property for $\mathcal A\widehat{\otimes}_\gamma\mathcal B$. In fact, we show that if $\mathcal A\widehat{\otimes}_\gamma\mathcal B$ is of $\rm BED$, then both $\mathcal A$ and $\mathcal B$ are so, whenever either $\mathcal A$ or $\mathcal B$ is unital. We also show that if $\mathcal B$ (resp., $\mathcal A$) is unital and $\widehat{\mathcal A} \subseteq C^{0}_{\rm BSE}(\Delta(\mathcal A))$ (resp., $\widehat{\mathcal B} \subseteq C^{0}_{\rm BSE}(\Delta(\mathcal B))$), then $\widehat{\mathcal A\widehat{\otimes}_\gamma\mathcal B}\subseteq C^{0}_{\rm BSE}(\Delta(\mathcal A\widehat{\otimes}_\gamma\mathcal B))$. We also establish that if $\mathcal B$ (resp., $\mathcal A$) is finite dimensional, then $\mathcal A\widehat{\otimes}_\gamma\mathcal B$ is of $\rm BED$ if and only if $\mathcal A$ (resp., $\mathcal B$) is of $\rm BED$.