Tesis
THESIS DEFENSE -- Banach lattices and free objects
Ponente: Enrique García Sánchez ()Director/es: Pedro Tradacete (ICMAT-CSIC)Fecha: miércoles 17 de diciembre de 2025 - 16:00Lugar: Aula B8, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid
Resumen:
The free Banach lattice generated by a Banach space is a free object that assigns to every Banach space E a Banach lattice, denoted FBL[E], and a linear embedding from E into FBL[E] with the universal property that every operator from E to a Banach lattice extends uniquely to a lattice homomorphism on FBL[E]. In this dissertation, we study the connections of this free object with several Banach lattice notions and universal constructions. Namely, we provide a solution to the problem of finding an explicit representation for the free Banach lattice with upper p-estimates, which is an analogue of the free Banach lattice in the category of Banach lattices with upper p-estimates, and we construct push-outs in the categories of p-convex Banach lattices and Banach lattices with upper p-estimates, exploring some fundamental differences between these two relevant classes of Banach lattices along the way. We also investigate how biduals, that behave as free objects between the categories of Banach spaces and dual Banach spaces, interact with free Banach lattices, which leads us to introduce the free dual Banach lattice generated by a Banach space. Next, we apply interpolation and extrapolation techniques in order to better understand the spaces of positively homogeneous functions in which free Banach lattices are constructed. Finally, we focus on the more applied topic of stable phase retrieval, a problem from X-ray crystallography consisting in recovering stably a signal from the measurement of its modulus. Banach lattices have been recently suggested as the suitable framework to model this problem. Following this approach, we provide a characterization of the subspaces of complex Banach lattices doing stable phase retrieval and study the existence of infinite dimensional subspaces doing stable phase retrieval in spaces of continuous functions.
