Some cohomological notions on ${\mathcal A}\times_T {\mathcal B}$

TL;DR

This paper explores various forms of amenability in a specific Banach algebra constructed from two algebras and a homomorphism, extending previous results on cyclic amenability.

Contribution

It introduces and investigates multiple notions of amenability for the algebra ${ m extbf{A}} imes_T { m extbf{B}}$, improving existing results on cyclic amenability.

Findings

Extended the understanding of amenability properties in the algebra ${ m extbf{A}} imes_T { m extbf{B}}$

Provided new results on cyclic amenability of the algebra

Analyzed various approximate and essential amenability notions

Abstract

Associated with two Banach algebras $\mathcal A$ and $\mathcal B$ and a norm decreasing homomorphism $T:{\mathcal B}\rightarrow{\mathcal A}$, there is a certain Banach algebra product ${\mathcal A}\times_T {\mathcal B}$, which is a splitting extension of $\mathcal B$ by $\mathcal A$. We investigate some notions of amenability such as approximate weak amenability, approximate cyclic amenability, ideal amenability, approximate Connes-amenability, Pseudo-Connes amenability, $w^*$-approximate Connes-amenability, essential amenability, essential $\phi$-amenability and essential character amenability. In particular, we improve the previous results for cyclic amenability of ${\mathcal A}\times_T {\mathcal B}$.