On twisted Fourier analysis and convergence of Fourier series on discrete groups (Q731268)

It is a basic problem in classical Fourier analysis that the Fourier series of a continuous function on the circle group does not necessarily converge uniformly (to that function). To study this problem various types of summation processes, e.g. Fejér summation and Abel-Poisson summation, are introduced and extensively studied in the classical harmonic analysis. In the paper under review the authors consider an abstract form of the problem in the setting of a reduced twisted group \(C^*\)-algebra \(C^*_r(G,\sigma)\) associated to a discrete group \(G\). They prove that in the case where \(G\) has either polynomial or subexponential H-growth (with respect to some proper positive function on \(G\)), then the formal Fourier series (associated to the elements of \(C^*_r(G,\sigma)\)) converges in the operator norm. The operator norm convergence of various types of summations is also studied; in this direction they show that the Abel-Poisson summation holds for a wide variety of groups including Coxeter groups and Gromov hyperbolic groups. The paper is very carefully written and contains many deep interesting results.