C*-algebra structure on certain Banach algebra products

Abstract

Let A and B be commutative and semisimple Banach algebras and let > (B). In this paper,we prove thatA ? Bis a type I-BSE algebra if and only ifAe andBare so.As amain application of this result, we prove that A B is isomorphic with a C*-algebra if and only if Ae and B are isomorphic with C*-algebras. Moreover, we derive related results for the case where A is unital. © Canadian Mathematical Society 2020.