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.