$C^{\ast }-$algebra structure on vector valued-Banach algebras

Abstract

Let $A$ be a commutative semisimple Banach algebra, $X$ be a locally compact Hausdorff topological space and $G$ be a locally compact topological group. In this paper, we investigate several properties of vector valued Banach algebras $C_0(X,A)$, $L^p(G,A)$, $l^p(X,A)$ and $l^{\infty }(X,A)$. We prove that these algebras are isomorphic with a $C^{\ast }-$algebra if and only if $A$ is so.