Weak amenability and 2-weak amenability of Beurling algebras

TL;DR

This paper investigates conditions on weights in Beurling algebras on abelian groups that determine their weak amenability and 2-weak amenability, introducing vector-valued algebras and analyzing derivations.

Contribution

It introduces vector-valued Beurling algebras and links derivation properties to the augmentation ideal, providing new criteria for weak amenability and 2-weak amenability.

Findings

Identifies conditions for vanishing derivations in Beurling algebras.

Provides examples of Beurling algebras with various amenability properties.

Connects the augmentation ideal to derivation behavior.

Abstract

Let $L^1_\om(G)$ be a Beurling algebra on a locally compact abelian group $G$. We look for general conditions on the weight which allows the vanishing of continuous derivations of $L^1_\om(G)$. This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2-weakly amenable.