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 studythe...

in the present paper, the concepts of module (uniform) approximate amenability and contractibility of banach algebras 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 ...

We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One the main ingredients used this is general lemma which we call location lemma, allows us to “locate” conjugation invariant means on group $G$ relative given normal subgroup $N$ $G$. give several further applications beyond aforementioned amenable products.

In this paper, we study approximate amenability of matrix algebras. We show that every derivation from Mn(A) into Mn(E) is the sum of an inner derivation and a derivation induced by a derivation from A into E(m), where A is a Banach algebra and E is a Banach A-bimodule. By using this, we provide many results in approximate and permanent weak amenability of these algebras.

We define a Banach algebra A to be dual if A = (A∗) ∗ for a closed submodule A∗ of A∗. The class of dual Banach algebras includes all W ∗-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception...