Abraham Rueda Zoca (University of Murcia, Spain)
Δ- and Daugavet points in Lipschitz-free spaces
04.12.2020
Abstract
In this talk we will focus on giving necessary and sufficient conditions for a Lipschitz-free spaces F(M) to have Δ- and Daugavet points. As a consequence of our study, we will provide examples of metric spaces M and molecules in F(M) which are Δ-points but not Daugavet points, which is a completely different behaviour to the case of L_1-spaces. We end with some open questions.
Elisa Regina dos Santos (Federal University of Uberlândia, Brazil)
Polynomial Daugavet Centers
27.11.2020
Abstract
A polynomial Q:X→Y is called a polynomial Daugavet center if the equality
∥Q+P∥=∥Q∥+∥P∥
is satisfied for all rank-one polynomials P:X→Y. In this paper, we present geometric characterizations of polynomial Daugavet centers. We show that if Q is a polynomial Daugavet center, then every weakly compact polynomial P also satisfies this equation. Finally, we prove that if Y is a subspace of a Banach space E and Q:X→Y is a polynomial Daugavet center, then E can be equivalent renormed in such a way that the norm on Y is not modified and J∘Q:X→E is a polynomial Daugavet center. We also present some examples of polynomial Daugavet centers.
Olesia Zavarzina (V.N. Karazin Kharkiv National University, Ukraine)
Generalized-lush spaces and connected problems
20.11.2020
Abstract
The talk is devoted to geometric properties of GL-spaces. We will demonstrate that every finite-dimensional GL-space is polyhedral; We also characterise the spaces E=(Rn,∥⋅∥E)with absolute norm such that for every finite collection of GL-spaces their E-sum is a GL-space (GL-respecting spaces). We will also give the classification of GL- and GLR-spaces in dimention 2.
Sheldon Dantas (Czech Technical University in Prague, Czech Republic) and Gonzalo Martínez Cervantes (University of Murcia, Spain)
Octahedral norms in Free Banach Lattices
13.11.2020
Abstract
Our aim in this talk is twofold. First of all, we present an overview on basic facts about free Banach lattices generated by a given Banach space E, which will be denoted by FBL[E]. We present some properties and known results about such a space. This will give all the necessary background we need for the second part of the talk, which consists on the study of octahedral norms in FBL[E]. We provide some sufficient conditions so that such a space has an octahedral norm. We also discuss almost squareness and Fréchet differentiability on these spaces. We conclude the talk by presenting some natural questions related to the Daugavet property.
André Martiny (University of Agder, Norway)
Daugavet- and delta-points in Banach spaces with unconditional bases
23.10.2020
Abstract
We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a 1-unconditional basis. A norm one element x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from 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.
We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a 1-unconditional basis with delta-points, but with no Daugavet-points, and a Banach space with a 1-unconditional basis with a unit ball in which the Daugavet-points are weakly dense.