$C^{\ast }$- properties of vector-valued Banach algebras

Abstract

Let $X$ be a locally compact Hausdorff space, and $A$ be a commutative semisimple Banach algebra over the scalar field $\mathbb{C}$. The correlation between different types of BSE- Banach algebras $A$, and the Banach algebra $C_0(X,A)$ are assessed. It is found and approved that $C_0(X,A)$ is a $C^{\ast }$- algebra if and only if $A$ is so. Furthermore, $C_b(X,A)=C_0(X,A)$ if and only if $X$ is compact.