Cohomological characterization of $T$-Lau product algebras

TL;DR

This paper explores the algebraic and cohomological properties of T-Lau product Banach algebras, focusing on injectivity, projectivity, flatness, and Hochschild cohomology characterization.

Contribution

It provides a cohomological characterization of T-Lau product Banach algebras, extending understanding of their homological properties.

Findings

Characterization of Hochschild cohomology for T-Lau product algebras

Analysis of injectivity, projectivity, and flatness in these algebras

Structural insights into the algebraic properties of T-Lau products

Abstract

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 $A\times_{T}B$.