Weak amenability of commutative Beurling algebras

Abstract

For a locally compact Abelian group $G$ and a continuous weight function $ω$ on $G$ we show that the Beurling algebra $L^1(G, ω)$ is weakly amenable if and only if there is no nontrivial continuous group homomorphism $ϕ$: $G\to \mathbb{C}$ such that $\sup_{t\in G}\frac{|ϕ(t)|}{ω(t)ω(t^{-1})} < \infty$. Let $\hatω(t) = \limsup_{s\to \infty}ω(ts)/ω(s)$ ($t\in G$). Then $L^1(G, ω)$ is 2-weakly amenable if there is a constant $m> 0$ such that $\liminf_{n\to \infty}\frac{ω(t^n)\hatω(t^{-n})}{n} \leq m$ for all $t\in G$.