Approximate biprojectivity and biflatness of some algebras over certain semigroups

Abstract

‎We investigate (bounded) approximate biprojectivity of $l^1(S)$ for uniformly locally finite inverse semigroups‎. ‎As a consequence‎, ‎we show that when $S=mathcal{M}(G‎, ‎I)$ is the Brandt inverse semigroup‎, ‎then $l^1(S)$ is (boundedly) approximately biprojective if and only if $G$ is amenable‎. ‎Moreover‎, ‎we study biflatness and (bounded) approximate biprojectivity of the measure algebra $M(S)$ of a topological Brandt semigroup‎.