Resumen de Lipschitz Structure of Metric and Banach Spaces
Andrés Quilis Sandemetrio
Since the inception of Banach Space Theory, the study of complemented and uncomplemented subspaces of Banach spaces has been one of the main themes of the area. Specifically, in non-separable Banach spaces, there have been many efofrts in constructing a theoretical framework to describe the linear complementation structure of Banach spaces. Classical concepts such as the Separable Complementation Property, Projectional Resolutions of the Identity, and the Plichko Property have been and continue to be studied in this area. Similarly, Lipschitz maps between Banach spaces have also played a main role in the development of the theory. Questions such as the Lipschitz classification of Banach spaces, difefrentiability of Lipschitz maps, or the existence of Lipschitz retractions onto subsets and subspaces of Banach spaces, have been and continue to be active topics of research with a wealth of results and applications. In this thesis we analyse the Lipschitz retractional structure of non-separable metric and Banach spaces, as an analogous theory to the linear complementation one in Banach spaces. We also discuss the connection of this topic with the ongoing program to study the structure of Lipschitz-free Banach spaces, and to the problem of finding bounded linear extension operators for Lipschitz functions. First, we generalize some classical tools of the linear theory to the non-linear setting: We define the concept of Lipschitz retractional skeletons as a generalization of Projectional skeletons. As applications of these concepts, we show that the Lipschitz-free space of a Plichko Banach space is again Plichko. We also use Lipschitz retractional skeletons to characterize metric spaces whose Lipschitz-free spaces enjoy the Plichko property witnessed by Dirac measures, and we show that the Lipschitz-free space of any R-tree is 1-Plichko witnessed by molecules. Next, we pass on to defining the (ß) Lipschitz Retraction Property (Lipschitz RP( ß) for short) for a pair of infinite cardinals ß. These are the non-linear analogues to the classical Complementation Properties. We observe that C(K) spaces enjoy the Lipschitz RP(0, 0), which in turn implies that their associated Lipschitz-free space satisfy the Separable Complementation Property. As a continuation of the previous study, we construct, for every infinite cardinal , a complete metric space which fails the Lipschitz RP(). In the countable case, we are able to produce a complete metric space, called the skein space, with a stronger property than the negation of the Lipschitz RP(0, 0): Every separable subset of the skein space with at least two points fails to be a Lipschitz retract. Finally, we generalize a result of Heinrich and Mankiewicz to the non-linear setting, by showing that for any metric space M, every subset is contained in another subset of the same density character which admits a bounded linear extension operator for the space of Lipschitz functions.
