Weak Module Amenability of Module Extension Banach Algebras

In this paper we study module and weak module amenability of the module extension Banach algebra $A\oplus X$ of a Banach algebra A by a Banach A-module X. As an example we show that for an inverse semigroup S with set of idempotents E, the module extension ${\ell ^{1}}(E)\oplus {\ell ^{1}}(S)$ is amenable as an ${\ell ^{1}}(E)$-module iff S is amenable. We also study module biflatness and module biprojectivity of module extensions.