Generalized Module Extension Banach Algebras: Derivations and Weak Amenability

TL;DR

This paper introduces a new class of Banach algebras called generalized module extension Banach algebras, characterizes their derivations, and studies their weak amenability properties.

Contribution

It defines generalized module extension Banach algebras and provides a characterization of their derivations and weak amenability, simplifying proofs of existing results.

Findings

Characterization of n-dual valued derivations on the algebra

Conditions for n-weak amenability of the algebra

Unified framework encompassing module extension, Lau product, and direct sums

Abstract

Let $A$ and $X$ be Banach algebras and let $X$ be an algebraic Banach $A-$module. Then the $\ell^1-$direct sum $A\times X$ equipped with the multiplication $$(a,x)(b,y)=(ab, ay+xb+xy)\quad (a,b\in A, x,y\in X)$$ is a Banach algebra, denoted by $A\bowtie X$, which will be called "\textit{a generalized module extension Banach algebra}". Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We characterize the structure of $n-$dual valued ($n\in\mathbb N)$) derivations on $A\bowtie X$ from which we investigate the $n-$weak amenability for the algebra $A\bowtie X.$ We apply the results and the techniques of proofs for presenting some older results with simple direct proofs.