Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X39620131215Module approximate amenability of Banach algebras11371158466ENH.Pourmahmood-AghababaTabriz University, Tabriz, IranA.BodaghiIslamic Azad University of Garmsar, Garmsar, IranJournal Article20111029In the present paper, the concepts of module (uniform) approximate amenability and contractibility of Banach algebras that are modules over another Banach algebra, are introduced. The general theory is developed and some hereditary properties are given. In analogy with the Banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same properties. It is also shown that module uniform approximate (contractibility) amenability and module (contractibility, respectively) amenability for commutative Banach modules are equivalent. Applying these results to l^1 (S) as an l^1 (E)-module, for an inverse semigroup S with the set of<br />idempotents E, it is shown that l^1(S) is module approximately amenable (contractible) if and only if it is module uniformly approximately amenable if and only if S is amenable.<br />Moreover, l^1(S)^{**} is module (uniformly) approximately amenable if and only if an appropriate group homomorphic image of S is finite.http://bims.iranjournals.ir/article_466_605a2ac8ad2936a6371f9bda251cbb65.pdf