On approximately left phi-biprojective Banach algebras

TL;DR

This paper introduces new concepts of approximate left $$-biprojectivity in Banach algebras, exploring their properties and characterizations in relation to group and semigroup structures, with applications to Segal and measure algebras.

Contribution

It defines and studies approximately left $$-biprojective Banach algebras, establishing characterizations for Segal and measure algebras linked to group properties.

Findings

Segal algebra S(G) is approximately left $_1$-biprojective iff G is amenable

Measure algebra M(G) is approximately left character biprojective iff G is discrete and amenable

Hereditary properties of these notions are analyzed

Abstract

In this paper, for a Banach algebra A, we introduced the new notions of approximately left $\phi$-biprojective and approximately left character biprojective, where $\phi$ is a non-zero multiplicative linear functional on A. We show that for SIN group G, Segal algebra S(G) is approximately left $\phi_1$- biprojective if and only if G is amenable, where $\phi_1$ is the augmentation character on S(G). Also we showed that the measure algebra M(G) is approximately left character biprojective if and only if G is discrete and amenable. For a Clifford semigroup S, we show that `1(S) is approximately left character biprojective if and only if `1(S) is pseudo-amenable. We study the hereditary property of these new notions. Finally we give some examples among semigroup algebras and Triangular Banach algebras to show the differences of these notions and the classical ones.