Approximate Amenability of Matrix Algebras

Abstract

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