The concept of BED-algebras, introduced by Inoue and Takahasi, explores the Gelfand image of commutative Banach algebras and draws inspiration from a theorem by Doss, which characterizes continuous functions on the dual group arising as the image of absolutely continuous measures under the Fourier-Stieltjes transform. Let \({\mathfrak {A}}\) and \({\mathfrak {B}}\) be commutative semisimple Banach algebras, and let \(\mathfrak {A}\tilde{\otimes }\mathfrak {B}\) denote a Banach algebra obtained by completing \(\mathfrak {A}\otimes \mathfrak {B}\) with respect to a submultiplicative cross-norm dominating the injective norm. In this paper, we examine the BED-properties of \(\mathfrak {A}\tilde{\otimes }\mathfrak {B}\) and establish necessary and sufficient conditions for \(\mathcal {C}_0(\Omega ,{\mathfrak {A}})\) to be a BED-algebra. Specifically, for a discrete space \(\Omega \) , we prove that \(\mathcal {C}_0(\Omega ,{\mathfrak {A}})\) is a BED-algebra if and only if \({\mathfrak {A}}\) itself is a BED-algebra. Furthermore, we extend this result to the algebra \(\ell ^p(X,{\mathfrak {A}})\) , showing that it is a BED-algebra if and only if \({\mathfrak {A}}\) is a BED-algebra.
