$C^*$ -algebra structure on certain Banach algebra products. Issue 3 (7th September 2020) - Abtahi, Fatemeh
Abstract: Let $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras and let $θ \in Δ (\mathcal B)$ . In this paper, we prove that $\mathcal A× _{θ }\mathcal B$ is a type I-BSE algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are so. As a main application of this result, we prove that $\mathcal A× _{θ }\mathcal B$ is isomorphic with a $C^*$ -algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are isomorphic with $C^* $ -algebras. Moreover, we derive related results for the case where $\mathcal A$ is unital.