Approximate biprojectivity of certain semigroup algebras

TL;DR

This paper explores the concept of approximate biprojectivity in semigroup algebras, establishing conditions under which it coincides with biprojectivity and examining implications for related Banach algebras.

Contribution

It characterizes approximate biprojectivity for $ ext{l}^1$-semigroup algebras and links it to biprojectivity and pseudo-amenability in specific semigroup contexts.

Findings

$ ext{l}^1(S)$ is approximately biprojective iff biprojective for uniformly locally finite inverse semigroups

Approximate biprojectivity of $ ext{l}^1(S)^{**}$ implies pseudo-amenability for Clifford semigroups

Identifies classes of Banach algebras related to semigroup algebras that are not approximately biprojective

Abstract

In this paper, we investigate the notion of approximate biprojectivity for semigroup algebras and for some Banach algebras related to semigroup algebras. We show that $\ell^{1}(S)$ is approximately biprojective if and only if $\ell^{1}(S)$ is biprojective, provided that $S$ is a uniformly locally finite inverse semigroup. Also for a Clifford semigroup $S$, we show that approximate biprojectivity $\ell^{1}(S)^{**}$ gives pseudo amenability of $\ell^{1}(S)$. We give a class of Banach algebras related to semigroup algebras which is not approximately biprojective.