In this paper we defined the concept of MODULE AMENABILITY of Banach algebras and MODULE connes AMENABILITY of MODULE dual Banach algebras. Also we assert the concept of MODULE Arens regularity that is different with [1] and investigate the relation between MODULE AMENABILITY of Banach algebras and connes MODULE AMENABILITY of MODULE second dual Banach algebras. In the following we study the relation between MODULE AMENABILITY, weak MODULE AMENABILITY and MODULE approximate AMENABILITY of Banach algebra. The notation of AMENABILITY of Banach algebras was introduced by B. Johnson in [7]. A Banach algebra A is amenable if every bounded derivation from A into any dual Banach A-biMODULE is inner, equivalently if H(A; X) = 0 for any Banach A-biMODULE X, where H(A; X) is the first Hochschild co-homology group of A with coefficient in X. Also, a Banach algebra A is weakly amenable if H(A; A) = 0. Bade, Curtis and Dales introduced the notion of weak AMENABILITY on Banach algebras in [4]. They considered this concept only for commutative Banach algebras. After that Johnson defined the weak AMENABILITY for arbitrary Banach algebras.

Let A be a Banach algebra and E be a Banach A -biMODULE then S=A ÅE, the l1-direct sum of A and E becomes a MODULE extension Banach algebra when equipped with the algebras product (a, x): (a', x')=(aa', a.x'+x.a'). In this paper, we investigate D-AMENABILITY for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the MODULE extension Banach algebra S=l1 (ES) Å l1 (S) is D-amenable as a l1 (ES) -MODULE if and only if l1 (ES) is amenable as Banach algebra.

In this paper we define a congruence ~ on inverse semigroup S such that AMENABILITY of S is equivalent to AMENABILITY of S/ ~. We study MODULE AMENABILITY of semigroup algebra i1(S/ ~) when S is an inverse semigroup with idempotents E and prove that it is equivalent to MODULE AMENABILITY of i1 (S). The main difference of this action with the more studied trivial action is that in this case the corresponding homomorphic image is a Clifford semigroup rather than a discrete group.

In the present paper, the concepts of MODULE (uni- form) approximate AMENABILITY and contractibility of Banach al- gebras 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 l1 (S) as an l1 (E) -MODULE, for an inverse semigroup S with the set of idem- potents E, it is shown that l1 (S) is MODULE approximately amenable (contractible) if and only if it is MODULE uniformly approximately amenable if and only if S is amenable. Moreover, l1 (S)** is MODULE (uniformly) approximately amenable if and only if a maximal group homomorphic image of S is finite.

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, l1(S) is MODULE boundedly approximately amenable as l1(E)-MODULE if and only if S is amenable. Further examples are given of l1-semigroup Banach algebras which are MODULE boundedly approximately amenable but are not amenable.

Let S be an inverse semigroup with an upward directed set of idempotents E. In this paper we prove that if S is amenable, then l1(S) Ä^ l1(S) is MODULE amenable as an l1(E)-MODULE. Also we show that L1(S) Ä^ l1 (S) is MODULE super-amenable if an appropriate group homomorphic image of S is finite.

In this paper we define φ-Connes MODULE AMENABILITY of a dual Banach algebra A where φ is a bounded wk∗-MODULE homomorphism from A to A. We are mainly concerned with the study of φ-MODULE normal virtual diagonals. We show that if S is a weakly cancellative inverse semigroup with subsemigroup E of idempotents, χ is a bounded wk∗-MODULE homomorphism from l1(S) to l1(S) and l1(S) as a Banach MODULE over l1(E) is χ-Connes MODULE amenable, then it has a χ-MODULE normal virtual diagonal. In the case χ =id, the converse holds