In this paper, we provide several characterisations for uniform amenability concerning a family of finitely generated groups. More precisely, show that the Hulanicki–Reiter condition can be weakened in directions, including cardinalities supports and certain operator norms.

Let A be a Banach algebra and I closed ideal of A. We say that is amenable relative to I, if A/I an algebra. study the amenability algebras investigate triangular associated locally compact groups. generalize some previous known results by applying concept algebras, especially, we present generalization Johnson?s theorem in amenability.

We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and Żuk. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.

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.