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.

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 ...