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$C^*$-algebra structure on certain Banach algebra products

Part of: Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  07 September 2020

Fatemeh Abtahi
Fatemeh Abtahi*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran, 81746-73441
*
e-mail: [email protected][email protected]
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Abstract

Let $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras and let $\theta \in \Delta (\mathcal B)$ . In this paper, we prove that $\mathcal A\times _{\theta }\mathcal B$ is a type I-BSE algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are so. As a main application of this result, we prove that $\mathcal A\times _{\theta }\mathcal B$ is isomorphic with a $C^*$ -algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are isomorphic with $C^* $ -algebras. Moreover, we derive related results for the case where $\mathcal A$ is unital.

Keywords

BSE-algebraBSE-functionC*-algebracommutative Banach algebramultiplier algebraθ-product

MSC classification

Primary: 46J05: General theory of commutative topological algebras
Secondary: 46J25: Representations of commutative topological algebras
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 678 - 686
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Copyright
© Canadian Mathematical Society 2020

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