CHARACTERIZATION OF HOMOLOGICAL PROPERTIES OF $\theta$-LAU PRODUCT OF BANACH ALGEBRAS

Abstract

In this paper, we investigate biprojectivity and biflatness of ‎$‎\theta$-Lau product of Banach algebras $A\times_{\theta}B,$ where ‎$‎A$ and ‎$‎B$ are arbitrary Banach algebras and $\theta\in \sigma(B)$. Indeed, we show that $A\times_{\theta}B$ is biprojective if and only if ‎$‎A$ is contractible and ‎$‎B$ is biprojective. This generalizes some known results in \cite{khoddami}. Moreover, we characterize biflatness of $\theta$-Lau product of Banach algebras under some conditions. As an application, we give a negative answer to an open question arised in \cite{ebadjab}. Finally, we characterize pseudo-contractibility of $\theta$-Lau product of Banach algebras and give a positive answer to a question in \cite{ghaderi}.