Slicely countably determined Banach spaces

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 ℓ 1 \ell _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 1 . In particular, we show that the dual of a real infinite-dimensional Banach space with the alternative Daugavet property contains ℓ 1 \ell _1 and that operators which do not fix copies of ℓ 1 \ell _1 on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.