Pointwise Convergence of Noncommutative Fourier Series (Memoirs of the American Mathematical Society, 302). Dieser Artikel ist nicht verfügbar.Pointwise Convergence of Noncommutative Fourier Series (Memoirs of the American Mathematical Society, 302)

"This paper is devoted to the study of 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, this paper 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 approximate identities of noncommutative 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 Fejer and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructionsof Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy- Littlewood maximal inequalities associated with convex bodies"-- Provided by publisher.

Guixiang Hong, Harbin Institute of Technology, People's Republic of China, Simeng Wang, Harbin Institute of Technology, People's Republic of China, and Xuemin Wang, Seoul National University, Republic of Korea