I will talk about pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, I will introduce the main theorem: a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. By using this criterion, 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, I will show a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. I will also talk about the Pointwise convergence of Fejér and Bochner-Riesz means in the noncommutative setting. Finally, I will mention a byproduct-- the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

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ID : 356 501 3138
PW : 471247