Daugavet- and delta-points in Banach spaces with unconditional bases

Trond Abrahamsen,André Martiny,Vegard Lima,Stanimir Troyanski

doi:10.1090/btran/68

Abstract

We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a 1 1 -unconditional basis. A norm one element x x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 2 from x x . A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than 2 2 . We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a 1 1 -unconditional basis with delta-points, but with no Daugavet-points, and a Banach space with a 1 1 -unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

Highlights

Let X be a Banach space with unit ball BX, unit sphere SX, and topological dual X∗
We show that no Banach space with a subsymmetric basis can have deltapoints
We say that X has the (i) Daugavet property if for every x ∈ SX and every ε > 0 we have BX = convΔε(x); (ii) diametral local diameter two property if for every x ∈ SX and every ε > 0 we have x ∈ convΔε(x)

Summary

Introduction

Let X be a Banach space with unit ball BX , unit sphere SX , and topological dual X∗. The main goal of this section is to prove that Banach spaces with a subsymmetric basis fail to have delta-points Before we start this mission, let us point out some results and concepts that we will need. Our tool to investigate the existence of slices of this type in a Banach space with a 1-unconditional basis, are certain families of subsets of the support of the elements in the space. The result is the key ingredient in our proof that there are no delta-points in Banach spaces with subsymmetric bases. Let X be a Banach space with 1-unconditional basis (ei)i∈N. As x is constant on A B, we cannot have |A| < |B| since a subset of B would be in M (x) contradicting the definition of M (x)

A space with 1-unconditional basis and delta-points

A space with 1-unconditional basis and daugavet-points