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.

Let A be a Banach algebra and X be a Banach A-bimodule. Then S = AÅX, the l1-direct sum of A and X becomes a module extension Banach algebra when equipped with the algebra product (a, x).(a’, x’) = (aa’, ax’+xa’). In this paper, we investigate biflatness and biprojectivity for these Banach algebras. We also discuss on automatic continuity of derivations on S = A Å A.

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.

We study the topological centers of some specific adjoints of a Banach module action. Then, we investigate the Arens regularity and strong irregularity of these actions.

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 A be a Banach algebra and X be a Banach A-bimodule. In this paper, we define a new product on A ÅX and generalize the module extension Banach algebras. We obtain characterizations of Arens regularity, commutativity, semisimplity, and study the ideal structure and derivations of this new Banach algebra.

For two algebras A and B, a linear map T: A® B is called separating, if x.y=0 implies Tx.Ty=0 for all x, yÎ A. The general form and the automatic continuity of separating maps between various Banach algebras have been studied extensively. In this paper, we first extend the notion of separating map for module case and then we give a description of a linear separating map T: B®X, where B is a unital commutative semisimple regular Banach algebra satisfying the Ditkin’s condition and X is a left Banach module over a unital commutative Banach algebra. We also show that if X is hyper semisimple and T is bijective, then T is automatically continuous and T-1 is separating as well.

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