Kakeya conjecture and the structure of higher rank lattice von Neumann algebras

Given a von Neumann algebra M, we can consider the set of values of p such that Lp(M) has the approximation property: the identity on it is a limit of finite rank operators for a suitable topology. Apart from the case when p is infinite, which has been the subject of a lot of work initiated by Haagerup in the late 70s, this invariant has not been very much exploited so far. But ancient works in collaboration with Vincent Lafforgue and Tim de Laat suggest that, maybe, it can distinguish the factors of SL(n,Z) for different values of n. I will explain something that I realized only recently, and that explains why this is a difficult question: it implies some form of the classical Kakeya conjecture, which predicts the shape of sets in the Euclidean space in which a needle can be turned upside down. This talk from Mikael de la Salle will be an opportunity to discuss other connections between classical Fourier analysis and analysis in group von Neumann algebras, including in collaboration with Javier Parcet and Eduardo Tablate