On Amalgamated Banach algebras

TL;DR

This paper studies the structure and properties of amalgamated Banach algebras formed from two Banach algebras, focusing on their topological centers, Arens regularity, character spaces, and weak amenability.

Contribution

It provides explicit descriptions of the topological center, character space, and conditions for Arens regularity and weak amenability of amalgamated Banach algebras.

Findings

Computed the topological center of amalgamated Banach algebras.

Characterized Arens regularity in terms of component algebras.

Determined the character space of the amalgamated algebra.

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 $A\bowtie^{\theta}I$.