Weak amenability of commutative Beurling algebras

Abstract

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

To appear in Proc. Amer. Math. Soc

References in corpus (1)

• Approximate and pseudo-amenability of various classes of Banach algebras