Some cohomological notions on ${\mathcal A}\times_T {\mathcal B}$

Abstract: Associated with two Banach algebras $\mathcal A$ and $\mathcal B$ and a norm decreasing homomorphism $T:{\mathcal B}\rightarrow{\mathcal A}$, 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, $w^*$-approximate Connes-amenability, essential amenability, essential $\phi$-amenability and essential character amenability. In particular, we improve the previous results for cyclic amenability of ${\mathcal A}\times_T {\mathcal B}$.