Banach spaces & Banach lattices Seminar
An introduction to homological algebra for Banach spacers
Speaker: Nazaret Trejo (ICMAT-UCM)Date: Wednesday, 05 March 2025 - 12:30Place: Aula Gris 1, ICMAT
Abstract:
In this talk we have a simple objective: learning to derive a functor defined on the category of Banach spaces. But... what is a category? And what is a functor? We will start by defining the basic notions of category theory and homological algebra that we will need to carry on our goal.
It is common in the classical theory of homological algebra to study the derived functors of Hom(·,·), the functors related to the morphisms in the category. So we will compute them for Banach spaces and we will give a representation as exact sequences. But... is it that easy? Well, since Banach spaces and operators form a somewhat delicate category, during the derivation process we will have some troubles we will need to solve, but finally we can make everything work. The natural question now is: can we compute the derived functors of another Banach-functors? In a joint work with J.M.F Castillo, F. Cabello Sánchez and A. Salguero-Alarcón, we have derived the compact-operator functor and obtain two types of results: some about the interlacing exact sequences and compact-exact sequences and others about extensions of compact-operator in the line of Lindenstrauss' extensions theorems.
