Different types of weak amenability for Banach algebras

Abstract

In this paper, we introduce and investigate the concepts of cyclically weakly amenable and point amenable. Then, we compare these concepts with the concepts of weakly amenable and cyclically amenable and find the relation between them. For example, we prove that a Banach algebra is weakly amenable if and only if it is both cyclically amenable and cyclically weakly amenable. In the case where $A$ is commutative, the weak amenability and cyclically weak amenability of $A$ are equivalent. We also show that if $A$ is a Banach algebra with $\Delta (A)\ne \varnothing$, then $A$ is cyclically weakly amenable if and only if $A$ is point amenable and essential. For a unital, commutative Banach algebra $A$, the notions of weakly amenable, cyclically weakly amenable and point amenable coincide. In this case, these are equivalent to the fact that every maximal ideal of $A$ is essential.