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.

For a normed space X, let JX : X ® X** denote the canonical embedding of X into X**, with the second adjoint (JX ) **: X**® X****. We defined MX (in Section 3 of the paper [1]) by MX = {x**Î 2 X** : JX**(x**) = (JX )**(x**)}….

In this paper we give some characterizations of TOPOLOGICAL extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative TOPOLOGICAL left invariant mean then T is TOPOLOGICAL left lumpy if and only if there is a multiplicative TOPOLOGICAL left invariant mean M on M(S)* such that M(XT)=1, where XT is the characteristic functional of T. Consequently if T is a TOPOLOGICAL left lumpy locally compact Borel subsemigroup of a locally compact semigroup S, then T is extremely TOPOLOGICAL left amenable if and only if S is.

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 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.

Quasi MODULE is a new algebraic structure, based on MODULE, which is composed of a semigroup structure and a partial order accompanied with an external ring multiplication. We proposed this structure in our paper [1] while we were studying the hyperspace C(M) consisting of all nonempty compact subsets of a TOPOLOGICAL MODULE M over some TOPOLOGICAL ring R. Quasi MODULE can be considered as a generalisation of MODULE in some sense. In the present paper we have defi ned TOPOLOGICAL quasi MODULE and given some examples of it. We have shown that the Cartesian product of arbitrary family of TOPOLOGICAL quasi MODULEs is again a TOPOLOGICAL quasi MODULE over some TOPOLOGICAL unitary ring. Finally we have defi ned projective system of TOPOLOGICAL quasi MODULEs and projective limit of this system. We have proved various TOPOLOGICAL properties of the projective limit of a projective system.

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.

Assume that A, B are Banach algebras and that m: A´B ® B, m¢: A´A ® B are bounded bilinear mappings. We study the relationships between Arens regularity of m, m¢ and the Banach algebras A, B. For a Banach A-biMODULE B, we show that B factors with respect to A if and only if B** is unital as an A**-MODULE. Let Ze¢¢ (B**) = B** where e¢¢ is a mixed unit of A**. Then B* factors on both sides with respect to A if and only if B** has a unit as A**-MODULE.

In this paper we introduce the concept of TOPOLOGICAL number for locally convex TOPOLOGICAL spaces and prove some of its properties. It gives some criterions to study locally convex TOPOLOGICAL spaces in a discrete approach.