Weak amenability of weighted group algebras

TL;DR

This paper characterizes the weak amenability of weighted group algebras by linking it to the boundedness of certain quasi-additive functions, providing new insights into the structure of Beurling algebras.

Contribution

It introduces inner quasi-additive functions and establishes a necessary and sufficient condition for weak amenability of weighted group algebras.

Findings

Weak amenability of $L^1(G, ext{omega})$ is characterized by unboundedness of non-inner quasi-additive functions.

Provides an answer to the open question on weak amenability of weighted group algebras.

Improves existing results on the conditions for weak amenability.

Abstract

In this paper, we study weak amenability of Beurling algebras. To this end, we introduce the notion inner quasi-additive functions and prove that for a locally compact group $G$, the Banach algebra $L^1(G, \omega)$ is weakly amenable if and only if every non-inner quasi-additive function in $L^\infty(G, 1/\omega)$ is unbounded. This provides an answer to the question concerning weak amenability of $L^1(G, \omega)$ and improve some known results in connection with it.