Geodesic
$C^*$-algebra structure on certain Banach algebra products
Canadian mathematical bulletin, Tome 64 (2021) no. 3, pp. 678-686

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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.
DOI : 10.4153/S0008439520000740
Mots-clés : BSE-algebra, BSE-function, C*-algebra, commutative Banach algebra, multiplier algebra, θ-product 
Abtahi, Fatemeh. $C^*$-algebra structure on certain Banach algebra products. Canadian mathematical bulletin, Tome 64 (2021) no. 3, pp. 678-686. doi: 10.4153/S0008439520000740
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     title = {$C^*$-algebra structure on certain {Banach} algebra products},
     journal = {Canadian mathematical bulletin},
     pages = {678--686},
     year = {2021},
     volume = {64},
     number = {3},
     doi = {10.4153/S0008439520000740},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000740/}
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