A well-known theorem of Bochner, generalized by Eberlein, characterizes the image of Fourier-Stieltjes transform of the measure algebra of a locally compact abelian group. With the approach that arose from this theorem, the concept of BSE-Algebras was created by Takahasi and Hatori to study the Gelfand image of commutative Banach algebras. Let \({\mathfrak {A}}\) and \({\mathfrak {B}}\) be two semisimple commutative Banach algebras. We denote by \(\mathfrak {A}{\tilde{\otimes }}\mathfrak {B}\) a Banach algebra obtained by the completion of \(\mathfrak {A}\otimes \mathfrak {B}\) w.r.t. a submultiplicative cross-norm which dominates the injective norm. In this article, we provide necessary and sufficient conditions for \(\mathfrak {A}{\tilde{\otimes }}\mathfrak {B}\) to be a BSE-algebra. We give an extension of Bochner-Eberlein theorem. We also provide another proof for the well-known fact that the injective tensor product of \({\mathfrak {A}}\) and \({\mathfrak {B}}\) is isomorphic to a \(C^*\) -algebra if and only if \({\mathfrak {A}}\) and \({\mathfrak {B}}\) are so. For a discrete space \(\Omega\) , it is proved that the space \({\mathcal {C}}_0(\Omega ,{\mathfrak {A}})\) , the set of all continuous functions \(f:\Omega\rightarrow\mathfrak A\) that vanish at infinity, is a BSE-algebra if and only if \({\mathfrak {A}}\) is a BSE-algebra.
