Some cohomological notions on $\mathcal{A}{\times }_T\mathcal{B}$

Associated with two Banach algebras $\mathcal{A}$ and $\mathcal{B}$ and a norm decreasing homomorphism $T:\mathcal{B}\to \mathcal{A}$, there is a certain Banach algebra product $\mathcal{A}{\times }_T\mathcal{B}$, which is a splitting extension of $\mathcal{B}$ by $\mathcal{A}$. We investigate some notions of amenability such as approximate weak amenability, approximate cyclic amenability, ideal amenability, approximate Connes-amenability, Pseudo-Connes amenability, $w^{\ast }$-approximate Connes-amenability, essential amenability, essential $\varphi$-amenability and essential character amenability. In particular, we improve the previous results for cyclic amenability of $\mathcal{A}{\times }_T\mathcal{B}$.