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 $\varphi$ from $S$ into $T$, we introduce and study the concept of $(\varphi, T)$-derivations on $S$ and $\varphi$-amenability of $T$ and investigate the relations between these two concepts. For two Banach algebras $A$ and $B$ and a continuous homomorphism $\varphi$ from $A$ into $B$ we also introduce the notion of $(\varphi, B)$-amenability of $A$ and show that a foundation semigroup $T$ with identity is $\varphi$-amenable whenever the Banach algebra $M_a(S)$ is $(\tilde{\varphi},M_a(T))$-amenable, where $\tilde{\varphi}:M(S)\to M(T)$ denotes the unique extension of $\varphi$. An example is given to show that the converse is nottrue.