Slicely Countably Determined Banach spaces

TL;DR

This paper introduces the class of slicely countably determined Banach spaces, exploring their properties, examples, and applications to various geometric and structural aspects of Banach space theory.

Contribution

It defines the new class of slicely countably determined Banach spaces and investigates their properties and applications in the context of Daugavet properties and numerical index.

Findings

Spaces with the RNP are included in the class

Duals of spaces with the alternative Daugavet property contain ℓ₁

Operators not fixing ℓ₁ satisfy the alternative Daugavet equation

Abstract

We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. 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. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.