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

TL;DR

This paper characterizes when certain vector valued Banach algebras, constructed from a commutative semisimple Banach algebra, are isomorphic to a C*-algebra, based on the properties of the algebra A.

Contribution

It establishes a necessary and sufficient condition for vector valued Banach algebras to be C*-algebras, extending the understanding of their algebraic structure.

Findings

Vector valued Banach algebras are C*-algebras if and only if A is a C*-algebra.

Isomorphism conditions are established for $C_0(X,A)$, $L^p(G,A)$, $l^p(X,A)$, and $l^{ abla}(X,A)$.

The results unify the structure theory of these algebras under the C*-algebra framework.

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 $C^*-$algebra if and only if $A$ is so.