A generalization of amenability for topological semigroups and semigroup algebras

Abstract

In this paper for two topological semigroups S and T , and a continuous homomorphism ϕ from S into T , we introduce and study the concept of (ϕ, T)-derivations on S and ϕ-amenability of T and investigate the relations between these two concepts. For two Banach algebras A and B and a continuous homomorphism ϕ from A into B we also introduce the notion of (ϕ, B)-amenability of A and show that a foundation semigroup T with identity is ϕ-amenable whenever the Banach algebra Ma(S) is (φ, Ma(T))-amenable, whereφ : M (S) −→ M (T) denotes the unique extension of ϕ. An example is given to show that the converse is not true.