Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska

Studia Mathematica, Tome 231 (2015), p. 189-214 / Harvested from The Polish Digital Mathematics Library

Résumé

Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for ℓ¹((ax + b),ω) to be weakly amenable. We prove that for several important classes of weights ω the algebra ℓ¹(₂,ω) is weakly amenable if and only if the weight ω is diagonally bounded. In particular, the polynomial weight $ω_{α}(x) = (1 + |x|)^{α}$, where |x| denotes the length of the element x ∈ ₂ and α > 0, never makes $ℓ¹(₂,ω_{α})$ weakly amenable. We also study weak amenability of an Abelian algebra ℓ¹(ℤ²,ω). We give an example showing that weak amenability of ℓ¹(ℤ²,ω) does not necessarily imply weak amenability of $ℓ¹(ℤ,ω_{i})$, where $ω_{i}$ denotes the restriction of ω to the ith coordinate (i = 1,2). We also provide a simple procedure for verification whether ℓ¹(ℤ²,ω) is weakly amenable.

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Publié le : 2015-01-01

Zbl 06545404

EUDML-ID : urn:eudml:doc:285545