Module Bounded Approximate Amenability of Banach Algebras

In this study we continue an investigation of the notion of module approximate amenability of a Banach algebra A which is a module over another Banach algebra A. In fact we introduce the class of module boundedly approximately amenable Banach algebras (m.b.app.am.) . It is shown that the class of module boundedly approximately amenable Banach algebra is different from the class of amenable Banach algebras. Also, we show that for an inverse semigroup S with the set of idempotent E, l 1 (S) is module boundedly approximately amenable as l 1 (E)-module if and only if S is amenable. Further examples are given of l 1 -semigroup Banach algebras which are module boundedly approximately amenable but are not amenable.