Module approximate biprojectivity and module approximate biflatness of Banach algebras

Abstract

In this paper, in analogy with the classical case, we introduce module approximately biprojective and module approximately biflat Banach algebras. For an inverse semigroup S with the set of idempotents E, we show that the concepts of module approximate biprojectivity and module pseudo-amenability for $$l^1(S,\omega )$$ coincide (as an $$l^1(E)$$ -module). We also find necessary and sufficient conditions for the weighted semigroup algebra $$l^{1}(S,\omega )$$ to be module approximately biprojective and module approximately biflat. Finally, we study the module cohomological properties of projective tensor product of Banach algebras.