Fourier rearrangements on discrete groups: $L^p$ and operator-norm subsequence convergence

Abstract

Let $\Gamma$ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For $f$ belonging to the reduced group $C^*$ algebra $C_r^*(\Gamma)$, we construct a rearrangement of its individual Fourier terms with a subsequence converging to $f$ in the operator norm. For every $2\le p<\infty$ and $f$ belonging to the noncommutative $L^p$ space $ L^p(\hat \Gamma)$, we obtain convergence in the $L^p$ norm. The proof combines uniformly bounded finite-support Fourier cutoffs with an elementary Bernoulli even-moment estimate. We give a self-contained free-group argument, then establish a general criterion and verify it for the two classes above. The operator-norm conclusion establishes the R\'ev\'esz property for these groups. In particular, it confirms the polynomial-growth conjecture of Hamm, Hayes, and Petrosyan and disproves their conjecture that the free group $\F$ does not have this property.