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2015 | 230 | 3 | 189-214
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Weak amenability of weighted group algebras on some discrete groups

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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 𝔽₂, 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(𝔽₂,ω) nor for ℓ¹((ax + b),ω) to be weakly amenable. We prove that for several important classes of weights ω the algebra ℓ¹(𝔽₂,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 ∈ 𝔽₂ and α > 0, never makes $ℓ¹(𝔽₂,ω_{α})$ 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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  • Department of Mathematics, University of Manitoba, 420 Machray Hall, 186 Dysart Road, Winnipeg, Manitoba, R3T 2N2 Canada
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm8100-12-2015
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