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

Let $Γ$ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For $f$ belonging to the reduced group $C^*$ algebra $C_r^*(Γ)$, we construct a rearrangement of its individual Fourier terms with a subsequence converging to $f$ in the operator norm. For every $2\le p<\inft