Seminario de Análisis  pre-lectura de tesis de Eduardo Tablate

Ponente: Eduardo Tablate Vila

Lugar: Aula 420, Módulo 17

Día y hora: Lunes 18 de Marzo, 11:30-12:30. SEMINARIO PRE-TESIS. Día Inusual 

Título: Smooth and idempotent Schur multipliers

Abstract:

Schur multipliers are linear maps defined on matrix algebras with profound applications on functional analysis, harmonic analysis, perturbation theory and in the study of the geometry of von Neumann algebras. In fact, these objects are strongly connected with the Grothendieck inequality, Krein's conjecture and with the theory about Λ_p and lacunary sets in the context of some nonabelian groups.
In addition, in the groundbreaking works of Cowling and Haagerup, and thereafter the works of de Laat, Lafforgue and de la Salle, in free groups and semisimiple lattices deep geometric properties of these groups are encapsulated in terms of the approximation properties of certain Schur multipliers defined on their matrix algebras. Despite all these applications there was a big gap regarding the understanding the boundedness properties of these maps in the natural metric spaces in which they act, the Schatten classes.

The purpose of this talk is to find noncommutative extensions of fundamental results of Harmonic Analysis in this context. In this sense we find three families of Schur multipliers: the smooth ones, the nonsmooth ones and an intermediate family between them. We will present several sharp results related to the two first families and their boundedness properties. Later we will present several applications in perturbation theory and in the context of harmonic analysis on group von Neumann algebras. The most important ones are a reinforcement of the Arazy's conjecture, a sharp analysis of smooth Fourier multipliers (on certain noncommutative L_p spaces) on simple Lie groups and a full characterisation of L_p-bounded idempotent Fourier multipliers on Lie groups.