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

TL;DR

This paper investigates when vector-valued Banach algebras of the form $C_0(X, A)$ are $C^*$-algebras, establishing that this occurs if and only if the algebra $A$ itself is a $C^*$-algebra.

Contribution

It provides a characterization of $C^*$-properties in vector-valued Banach algebras based on the properties of the underlying algebra $A$ and the space $X$.

Findings

$C_0(X, A)$ is a $C^*$-algebra iff $A$ is a $C^*$-algebra.

$C_b(X, A) = C_0(X, A)$ iff $X$ is compact.

The correlation between BSE-Banach algebras and $C_0(X, A)$ is analyzed.

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 $C^{*}$- algebra if and only if $A$ is so. Furthermore, $C_{b}(X, A)= C_{0}(X, A)$ if and only if $X$ is compact.