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.

Abstract. The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c 0 can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (R n , · ) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.

We introduce and study the notion of generating operators as those norm-one operators G : X −→ Y such that for every 0 < δ < 1, the set {x ∈ X : x 1, Gx > 1 − δ} generates the unit ball of X by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of ℓ1 into c0 and of L∞[0, 1] into L1[0, 1]. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on c0-, ℓ1-, and ℓ∞-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces X and Y generates all non-expansive operators by closed convex hull. We show that this is the case when X = L1(µ) and Y has the Radon-Nikodým property with respect to µ. Therefore, when X = ℓ1(Γ), this is the case for every target space Y . Conversely, we also show that a real finite-dimensional space X satisfies that generating operators from X to Y generate all non-expansive operators by closed convex hull only in the case that X is an ℓ1-space.

We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of c 0 -, 1 -and ∞ -sums and giving as a new result the case of E-sums where E has the RNP and n(E) = 1 (in particular for finite-dimensional E with n(E) = 1). We also show that the numerical index of a Banach space Z which contains a dense union of increasing one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that, for a fixed p, the numerical indices of all infinite-dimensional L p (μ)-spaces coincide.

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