POINTWISE CONVERGENCE OF NONCOMMUTATIVE FOURIER SERIES
Résumé
This paper is devoted to the study of pointwise convergence of Fourier series for non-abelian compact groups, group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as sum-mation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this work studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximative identities of non-commutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 pointwise, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence for all p > 1. In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér means and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Even back to the Fourier analysis on Eu-clidean spaces and non-abelian compact groups, our results are novel and yield new insights and problems. On the other hand, we obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
Loading...