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

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.Comment: 35 pages; abridged, revised and update

Skew products of finitely aligned left cancellative small categories and Cuntz-Krieger algebras

Given a group cocycle on a finitely aligned left cancellative small category (LCSC), we investigate the associated skew product category and its Cuntz–Krieger algebra, which we describe as the crossed product of the Cuntz–Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz–Krieger algebras arising from free actions of groups on finitely aligned LCSCs, and to construct coactions of groups on Exel–Pardo algebras. Finally, we discuss the universal group of a small category and connectedness of skew product categories