On certain product of Banach modules

TL;DR

This paper introduces a unified framework for a product of Banach algebras called $A\bowtie B$, characterizing key properties like approximate identities, spectrum, and topological center, encompassing several known algebra constructions.

Contribution

It provides a unified approach to analyze properties of various Banach algebra products, including module extensions and Lau products, through the $A\bowtie B$ construction.

Findings

Characterizations of bounded approximate identities

Descriptions of spectrum and topological center

Unified approach for known Banach algebra products

Abstract

Let $A$ and $B$ be Banach algebras and let $B$ be an algebraic Banach $A-$bimodule. Then the $\ell^1-$direct sum $A\times B$ equipped with the multiplication $$(a_1,b_1)(a_2,b_2)=(a_1a_2,a_1\cdot b_2+b_1\cdot a_2+b_1b_2),~~ (a_1, a_2\in A, b_1, b_2\in B)$$ is a Banach algebra denoted by $A\bowtie B$. Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We obtain characterizations of bounded approximate identities, spectrum, and topological center of this product. This provides a unified approach for obtaining some known results of both module extensions and Lau product of Banach algebras.