2000 Mathematics Subject Classification: Primary 46B20. Secondary 47A99, 46B42. It was shown in [2] that the most natural equalities valid for every rank-one operator T in real Banach spaces lead either to the Daugavet equation ||I+T|| = 1 + ||T|| or to the equation ||I − T|| = ||I+T||. We study if the spaces where the latter condition is satisfied for every finite-rank operator inherit the properties of Daugavet spaces.
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)...
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.
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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