On twisted Fourier analysis and convergence of Fourier series on discrete groups

Details On twisted Fourier analysis and convergence of Fourier series on discrete groups On twisted Fourier analysis and convergence of Fourier series on discrete groups

2006-05-05

arxiv.org/abs/math/0605145v2

J. Fourier Anal. Appl. (2009) 15, 336-365

Mathematics

Operator Algebras

35 pages; abridged, revised and updated

Scientific paper

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.

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