Some homological and cohomological notions on $T$-Lau product of Banach algebras

Abstract

Let $\mathcal A$ and $\mathcal B$ be Banach algebras and let $T:{\mathcal B}\rightarrow{\mathcal A}$ be a continuous homomorphism. Recently, we introduced a product ${\mathcal M}:={\mathcal A}\times_T{\mathcal B}$, which is a strongly splitting Banach algebra extension of $\mathcal B$ by $\mathcal A$. In the present paper, we characterize biprojectivity, approximate biprojectivity and biflatness of ${\mathcal A}\times_T{\mathcal B}$ in terms of ${\mathcal A}$ and ${\mathcal B}$. We also study some notions of amenability such as approximate amenability and pseudo-amenability of ${\mathcal A}\times_T{\mathcal B}$.