Different types of weak amenability for Banach algebras

TL;DR

This paper explores various notions of weak amenability in Banach algebras, establishing their relationships and equivalences, especially in commutative and unital cases, advancing the theoretical understanding of algebraic structures.

Contribution

It introduces and compares new concepts of cyclically weakly amenable and point amenable, clarifying their relations with existing notions of weak and cyclically amenable Banach algebras.

Findings

Weakly amenable iff cyclically weakly amenable and cyclically amenable

In commutative case, weak and cyclically weak amenability are equivalent

For unital, commutative Banach algebra, all notions coincide

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 $\Delta(A)\neq\emptyset$, 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.