Schur multipliers and rigidity phenomena

Objetivo

The classification of group von Neumann algebras associated with higher rank lattices —considered in Connes' rigidity conjecture— remains one of the most important problems in operator algebras, functional analysis and geometric group theory. Lafforgue, de Laat and de la Salle's rigidity theorem sparked a powerful revival of Cowling/Haagerup's approximation methods. Their analysis of Lp-approximations and the key use of Schur multipliers opens a door to new harmonic analysis techniques that complement the groundbreaking achievements surrounding Popa's deformation/rigidity theory. Over the past few years I have delved into these techniques, resulting in the recent solution of several open problems on Fourier/Schur multipliers and very promising rigidity-related calculations.

My goals revolve around the intriguing relationship between trigonometric and matrix unit systems. I will explore a long-expected identity connecting Fourier and Schur multipliers over higher rank simple Lie groups and their lattices, along with an exciting idea to address Connes' rigidity for PSLn(Z) via unexpected Kakeya methods. This naturally leads to key new inequalities for nonToeplitz extensions of maximal functions and Bochner-Riesz means with far-reaching applications. Last, my recent work is crucial and timely to solve two long-standing noncommutative forms of Fefferman's ball multiplier theorem and Bourgain's solution of the Lambda(p)-set problem, with significant implications for free group factors and unconditionality in Banach spaces.

A thorough strategy with solid supporting evidence bring these highly ambitious goals within reach. The interdisciplinary nature of this proposal —encompassing geometric group theory, noncommutative Lp and operator space techniques, quantum/free probability, and Kakeya/Bochner-Riesz methods— is essential for a successful outcome, which will ensure a great impact at the interface of operator algebras and harmonic analysis.