Pseudo-amenable and pseudo-contractible generalized module extension Banach algebras

Let $A$ and $B$ be Banach algebras and $B$ be an algebraic Banach $A$-bimodule. Then we examine pseudo-amenability, pseudo-contractibility and property $\mathcal{F}$ of generalized module extension Banach algebra $A\bowtie B$. Indeed, we obtain essential and sufficient condition for pseudo-amenability, pseudo-constructibility and property $\mathcal{F}$ of $A\bowtie B$ in term of pseudo-amenability, pseudo-contractibility and property $\mathcal{F}$ of $A$ and $B$, respectively.