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

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.

Asymptotic geometry and delta-points

“…But if (e i ) is the unit vector basis in S and (e * i ) the the biorthogonal functionals in the dual, then e * 2 + e * i ∈ S S * and converges weak * to e * 2 , but not in norm. Note however that S does admit ∆-point by Proposition 2.15 in [3].…”

“…From their introduction on, Daugavet-and ∆-points attracted a lot of attention and were intensively studied in classical Banach spaces ( [1,2,3,23]). In particular a strong emphasis was put on finding linear or geometric properties that would prevent norm one elements in a space to be Daugavetor ∆-points.…”

“…It soon appeared that even nice properties which on a global level prevent the space to have the Daugavet property or the DLD2P do not provide an obstruction to the existence of Daugavet-or ∆-points in the space. For example there exists a Banach space with a 1-unconditional basis such that the set of Daugavet-points are weakly dense in the unit ball [3,Theorem 4.7].…”

Stability of diametral diameter two properties

“…The space c 0 is an example of a space with the D2P and failing the Daugavet property. The existence of a space with the LD2P and without the D2P was first proven in [9], another example can be found in [4].…”

Daugavet points in projective tensor products

“…Therefore, the concept of Daugavet points can be faced as a local version of the Daugavet property. See [1,2,7,9] for background on Daugavet points and diverse relevant examples.…”