Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska¹

¹ Department of Mathematics University of Manitoba 420 Machray Hall, 186 Dysart Road Winnipeg, Manitoba R3T 2N2 Canada

Studia Mathematica, Tome 230 (2015) no. 3, pp. 189-214

Résumé

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.

Détail

DOI : 10.4064/sm8100-12-2015

Keywords: weak amenability ell omega commutative groups completely characterized gronbaek paper study weak amenability ell omega important non commutative locally compact groups group mathbb which non amenable amenable boldsymbol group condition characterizes weak amenability ell omega commutative groups remains necessary non commutative sufficient neither ell mathbb omega nor ell boldsymbol omega weakly amenable prove several important classes weights omega algebra ell mathbb omega weakly amenable only weight omega diagonally bounded particular polynomial weight omega alpha alpha where denotes length element mathbb alpha never makes ell mathbb omega alpha weakly amenable study weak amenability abelian algebra ell mathbb omega example showing weak amenability ell mathbb omega does necessarily imply weak amenability ell mathbb omega where omega denotes restriction omega ith coordinate provide simple procedure verification whether ell mathbb omega weakly amenable