Generalized Module Extension Banach Algebras: Derivations and Weak Amenability

Let $A$ and $X$ be Banach algebras and let $X$ be an algebraic Banach $A-$module. Then the ${\mathrm{\ell }}^1-$direct sum $A\times X$ equipped with the multiplication $$(a,x)(b,y)=(ab,ay+xb+xy)(a,b\in A,x,y\in X)$$ 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 $A\bowtie X\mathrm{.}$ We apply the results and the techniques of proofs for presenting some older results with simple direct proofs.