Amalgamated duplication of the Banach algebra $\mathfrak{A}$ along a $\mathfrak{A}$-bimodule $\mathcal{A}$

Abstract

Let $\mathcal{A}$ and $\mathfrak{A}$ be Banach algebras such that $\mathcal{A}$ is a Banach $\mathfrak{A}$-bimodule with compatible actions. We define the product $\mathcal{A}⋊\mathfrak{A}$, which is a strongly splitting Banach algebra extension of $\mathfrak{A}$ by $\mathcal{A}$. After characterization of the multiplier algebra, topological centre, (maximal) ideals and spectrum of $\mathcal{A}⋊\mathfrak{A}$, we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of $\mathcal{A}⋊\mathfrak{A}$ in relation to that of the algebras $\mathcal{A}$, $\mathfrak{A}$ and the action of $\mathfrak{A}$ on $\mathcal{A}$. We also compute the first cohomology group $H^1(\mathcal{A}⋊\mathfrak{A},(\mathcal{A}⋊\mathfrak{A}{)}^{(n)})$ for all $n\in \mathbb{N}\cup \{0\}$ as well as the first-order cyclic cohomology group $H_{\lambda }^1(\mathcal{A}⋊\mathfrak{A},(\mathcal{A}⋊\mathfrak{A}{)}^{(1)})$, where $(\mathcal{A}⋊\mathfrak{A}{)}^{(n)}$ is the n-th dual space of $\mathcal{A}⋊\mathfrak{A}$ when $n\in \mathbb{N}$ and $\mathcal{A}⋊\mathfrak{A}$ itself when $n=0$. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and $n$-weak amenability of $\mathcal{A}⋊\mathfrak{A}$.