# Central Beurling algebras: Weak amenability of the central Beurling algebras on [FC]$^-$ groups

## TL;DR

This paper investigates the weak amenability of central Beurling algebras on [FC]$^-$ groups, extending known results from commutative cases and establishing conditions based on group properties and weights.

## Contribution

It provides necessary and sufficient conditions for weak amenability of central Beurling algebras on specific groups, including polynomial weights on compactly generated [FC]$^-$ groups.

## Key findings

- Weak amenability characterized for [FC]$^-$ groups.
- Necessary conditions for [FD]$^-$ groups.
- Weak amenability of $ZL^1(G, ext{poly})$ if and only if $ ext{degree} < 1/2$.

## Abstract

We study weak amenability of central Beurling algebras $ZL^1(G,\omega)$. The investigation is a natural extension of the known work on the commutative Beurling algebra $L^1(G,\omega)$. For [FC]$^-$ groups we establish a necessary condition and for [FD]$^-$ groups we give sufficient conditions for the weak amenability of $Z\L1o$. For a compactly generated [FC]$^-$ group with the polynomial weight $\omega_\alpha(x) = (1 + |x|)^\alpha$, we prove that $ZL^1(G,\omega_\alpha)$ is weakly amenable if and only if $\alpha < 1/2$.