On (approximate) homological notions of certain Banach algebras

TL;DR

This paper explores approximate homological properties of a new class of Banach algebras, providing examples and counterexamples related to biprojectivity and Johnson pseudo-contractibility.

Contribution

It introduces and analyzes new homological notions for Banach algebras, offering examples and non-examples of approximate biprojectivity and Johnson pseudo-contractibility.

Findings

Some Banach algebras are approximately biprojective.

Certain matrix Banach algebras are not Johnson pseudo-contractible.

New classes of Banach algebras with specific homological properties are identified.

Abstract

In this paper, we study the notion of $\phi$-biflatness, $\phi$-biprojectivity, approximate biprojectivity and Johnson pseudo-contractibility for a new class of Banach algebras. Using this class of Banach algebras we give some examples which are approximately biprojective. Also some Banach algebras are given among matrix algebras which are never Johnson pseudo-contractible.