On Amalgamated Banach algebras

Abstract

Let $A$ and $B$ be Banach algebras, $\theta: A\to B$ be a continuous Banach algebra homomorphism and $I$ be a closed ideal in $B$. Then the direct sum of $A$ and $I$ with respect to $\theta$, denoted $A\bowtie^{\theta}I$, with a special product becomes a Banach algebra which is called the amalgamated Banach algebra. In this paper, among other things, we compute the topological centre of $A\bowtie^{\theta}I$ in terms of that of $A$ and $I$. Using this, we provide a characterization of the Arens regularity of $A\bowtie^{\theta}I$. Then we determine the character space of $A\bowtie^{\theta}I$ in terms of that of $A$ and $I$. Moreover, we study the weak amenability of $A\bowtie^{\theta}I$.