Noncommutative Fourier Analysis through von Neumann Algebras

Date: Tuesday 22 Nov 2022

Abstract

We are interested in the natural extension from Fourier analysis for abelian groups and the corresponding function spaces to non-commutative groups. Unfortunately, this quickly becomes difficult, as the study of the duality of such groups leads to Tannaka-Krein duality and quantum groups. Here we consider only profinite groups, and restrict our attention to those for which the left regular representation generates a semifinite von Neumann algebra. In this setting, we have powerful functional analytic tools built from the spectral theory of these operators which allow us to extend the ideas of classical Fourier analysis. We will discuss convergence of Fourier series in this setting, possible extensions of the Carleson-Hunt theorem, and the difficulties that arise from non-unimodular groups and non-doubling metrics.