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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We answer two open questions concerning the recently introduced notions of slicely countably determined (SCD) sets and SCD operators in Banach spaces. An application to narrow operators in spaces with the Daugavet property is given.
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We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak* analogue. We introduce and study analogues of narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L₁[0,1] by an ℓ₁-subspace need not have the Daugavet property. The latter answers in the negative a question posed to us by A. Pełczyński.
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