Left $\phi$-biprojectivity of some Banach algebras

TL;DR

This paper introduces a new homological concept called left φ-biprojectivity for Banach algebras, exploring its implications for group and semigroup algebras, and establishing conditions under which these algebras are φ-biprojective.

Contribution

It defines left φ-biprojectivity for Banach algebras and characterizes this property for Segal and Fourier algebras in relation to properties of the underlying groups.

Findings

Segal algebra S(G) is left φ-contractible iff G is compact

Fourier algebra A(G) is left φ-biprojective iff G is discrete

Examples show differences from classical biprojectivity

Abstract

In this paper, we introduce a homological notion of left $\phi$-biprojectivity for Banach algebras, where $\phi$ is a non-zero multiplicative linear functional. We show that for a locally compact group $G$, the Segal algebra $S(G)$ is left $\phi$-contractible if and only if $G$ is compact. Also, we prove that the Fourier algebra $A(G)$ is left $\phi$-biprojective if and only if $G$ is discrete. Finally, we study left $\phi$-biprojectivity of certain semigroup algebras. We give some examples which show the differences between our new notion and the classical ones.