Abstract. We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the Radon-Nikodým property and all spaces without copies of 1 . We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach space with the alternative Daugavet property contains 1 and that operators which do not fix copies of 1 on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.
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 for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L1[0, 1] over an ℓ1subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pe lczyński in the negative.
We study weak amenability of central Beurling algebras ZL 1 (G, ω). The investigation is a natural extension of the known work on the commutative Beurling algebra L 1 (G, ω). For [FC] − groups we establish a necessary condition and for [FD] − groups we give sufficient conditions for the weak amenability of ZL 1 (G, ω). For a compactly generated [FC] − group with the polynomial weight ωα(x) = (1 + |x|) α , we prove that ZL 1 (G, ωα) is weakly amenable if and only if α < 1/2.
Let G be a discrete group, let p ≥ 1, and let ω be a weight on G. Using the approach from [9], we provide sufficient conditions on a weight ω for ℓ p (G, ω) to be a Banach algebra admitting a norm-controlled inversion in the reduced C * -algebra of G, namely C * r (G). We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-discrete case and study the existence of norm-controlled inversion in B(L 2 (G)) for some related convolution algebras.
We construct an example of a Banach space which is not lush, but whose dual space is lush. This example shows that lushness is not equivalent to numerical index one.A characterization of lushness for some quotient spaces of L 1 (μ) spaces and new results on C -rich subspaces of (scalar-or vector-valued) C (K ) spaces are also presented.
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