Johnson pseudo-contractibility and pseudo-amenability of $\theta$-Lau product of Banach algebras

Abstract

Given Banach algebras $A$ and $B$ with $\theta \in \Delta (B)$. We shall study the Johnson pseudo-contractibility and pseudo-amenability of $\theta$-Lau product $A{\times }_{\theta }B$. We show that if $A{\times }_{\theta }B$ is Johnson pseudo-contractible, then $A$ is Johnson pseudo-contractible and has a bounded approximate identity and $B$ is Johnson pseudo-contractible. In some particular cases complete characterization of Johnson pseudo-contractibility of $A{\times }_{\theta }B$ are given. Also, we show that pseudo-amenability of $A{\times }_{\theta }B$ implies approximate amenability of $A$ and pseudo-amenability of $B$.