Cohomological characterization of $T$-Lau product algebras

Let $A$ and $B$ be Banach algebras and let $T$ be an algebra homomorphism from $B$ into $A$. The Cartesian product space $A\times B$ by $T$- Lau product and ${\mathrm{\ell }}^1$- norm becomes a Banach algebra $A{\times }_{T}B$. We investigate the notions such as injectivity, projectivity and flatness for the Banach algebra $A{\times }_{T}B$. We also characterize Hochschild cohomology for the Banach algebra $A{\times }_{T}B$.