Weak amenability of weighted group algebras on some discrete groups
Volume 230 / 2015
Abstract
Weak amenability of $\ell ^1(G,\omega )$ for commutative groups $G$ was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of $\ell ^1(G,\omega )$ for two important non-commutative locally compact groups $G$: the free group $\mathbb {F}_2$, which is non-amenable, and the amenable $\boldsymbol {(ax+b)}$-group. We show that the condition that characterizes weak amenability of $\ell ^1(G,\omega )$ for commutative groups $G$ remains necessary for the non-commutative case, but it is sufficient neither for $\ell ^1(\mathbb {F}_2,\omega )$ nor for $\ell ^1(\boldsymbol {(ax+b)},\omega )$ to be weakly amenable. We prove that for several important classes of weights $\omega $ the algebra $\ell ^1(\mathbb {F}_2,\omega )$ is weakly amenable if and only if the weight $\omega $ is diagonally bounded. In particular, the polynomial weight $\omega _{\alpha }(x)=(1+|x|)^{\alpha }$, where $|x|$ denotes the length of the element $x\in \mathbb {F}_2$ and $\alpha \gt 0$, never makes $\ell ^1(\mathbb {F}_2,\omega _{\alpha })$ weakly amenable.
We also study weak amenability of an Abelian algebra $\ell ^1(\mathbb {Z}^2,\omega )$. We give an example showing that weak amenability of $\ell ^1(\mathbb {Z}^2,\omega )$ does not necessarily imply weak amenability of $\ell ^1(\mathbb {Z},\omega _i)$, where $\omega _i$ denotes the restriction of $\omega $ to the $i$th coordinate ($i=1,2$). We also provide a simple procedure for verification whether $\ell ^1(\mathbb {Z}^2,\omega )$ is weakly amenable.
