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arXiv · 2212.06131

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

Maryam Aghakoochaki·Ali Rejali
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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.

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Maryam Aghakoochaki, Ali Rejali. 2022-12-12. $C^{*}$- properties of vector-valued Banach algebras. https://arxiv.org/abs/2212.06131

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