(C^{}\)- properties of vector-valued Banach algebras

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}...

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Bibliographic Details
Published in:arXiv.org 2022-12
Main Authors: Aghakoochaki, Maryam, Rejali, Ali
Format: Article
Language:English
Subjects:
Algebra
Banach spaces
Scalars
Online Access:Get full text
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Summary: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.
ISSN:2331-8422