Approximate biprojectivity of certain semigroup algebras

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.