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 $\ell^{ 1}$- 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 $A\times_{ T}B$.
Categories: math.FA