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Weak amenability of weighted group algebras on some discrete groups

Volume 230 / 2015

Varvara Shepelska Studia Mathematica 230 (2015), 189-214 MSC: Primary 46H20, 43A20; Secondary 43A10. DOI: 10.4064/sm8100-12-2015 Published online: 25 January 2016

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.

Authors

  • Varvara ShepelskaDepartment of Mathematics
    University of Manitoba
    420 Machray Hall, 186 Dysart Road
    Winnipeg, Manitoba
    R3T 2N2 Canada
    e-mail

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