Pointwise convergence of noncommutative Fourier series

In this talk I will present some recent progress on the study of pointwise convergence of Fourier series for 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 some summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, we study the refined counterpart of pointwise convergence of these Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result 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 1 < p < 1. In a similar fashion, we also obtain results for a large subclass of groups (as well as discrete quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejer means and Bochner-Riesz means in the noncommutative setting. Even back to the Fourier series of Lp-functions on Euclidean spaces and non-abelian compact groups, our results seem novel. On the other hand, we obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.