Super weak compactness and its applications to Banach space theory

Zusammenfassung

The main objective of the thesis is the study of the localized version of superreflexivity: the super weak compactness. This notion is strongly related to the existence of uniformly convex functions and the first part of the paper is devoted to the study of these functions. In particular, we propose discrete methods, based on the nonexistence of dyadic trees, to construct uniformly convex functions whose properties will be improved later. As an important consequence we obtain an alternative proof of Enflo's theorem. Many papers contain estimates of the super weak compactness unknowingly. For example, the super weak compactness and its quantization cast light on the structure of subspaces of Banach spaces generated by a Hilbert space. New characterizations of SWC sets, in terms of fixed points or ergodic properties, are established, thus improving existing results. The super weak compactness is also strongly related to Banach-Saks properties. The notion of uniform Banach-Saks property is introduced and characterized in different ways. In particular, we show that it is equivalent to the p-Banach-Saks property and we determine precisely the value of this p-index. The following parts seem to depart from the previous topic, but in fact they are consequences of the study of the super weak compactness. We study the extremal structure of ultraproducts of bounded sets. We establish several stability results concerning the extremal structure. For example, we extend some results of Talponen by showing that a point of a bounded convex set is strongly extreme if and only if its canonical image in the ultraproduct is (strongly) extreme. Furthermore, we show that the extreme points and strongly extreme points of the ultraproduct coincide. Similar results are established for the exposed points. In the following sections, we focus on Lipschitz free spaces. First, we study Lipschitz free spaces over ultraproducts of metric spaces. In particular, we show that if a metric space is finitely representable in a Banach space then the corresponding free spaces verify a similar relation. Next, we obtain interesting results concerning the existence of cotype in Lipschitz free spaces. In the following paper, we show that several classical properties of Banach spaces are equivalent to separability for the class of Lipschitz free spaces. In particular, the question of whether nonseparable Lipschitz free spaces can have a sequentially compact sequentially weak∗ dual ball is undecidable. We also provide an example of a nonseparable dual Lipschitz free space that does not have the Radon-Nikodym property. The last chapter deals with approximations in Lp spaces. We show that the sets of simple functions taking a fixed number of values are proximal. We introduce and study a class of sets, called uniformly approximable sets, which is larger than the class of uniformly integrable sets. Different characterizations of these sets and different stability properties are established.