$C^*$ -algebra structure on certain Banach algebra products. Issue 3 (7th September 2020)

• Record Type: Journal Article
• Authors: Abtahi, Fatemeh
• Is Part Of: Canadian mathematical bulletin =. Volume 64:Issue 3(2021)
• Page Start: 678
• Page End: 686
• Publication Date: 2020-09-07
• Subjects: 46J05 -- 46J25; BSE-algebra -- BSE-function -- C*-algebra -- commutative Banach algebra -- multiplier algebra -- θ-product; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439520000740
• Languages: English
• ISSNs: 0008-4395

Abstract

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.