A theorem for random Fourier series on compact quantum groups

TL;DR

This paper extends Helgason's theorem to compact quantum groups, showing that certain Fourier series conditions imply measure regularity, and applies this to the operator algebra representation of convolution algebras.

Contribution

It generalizes a classical harmonic analysis result to the quantum group setting and investigates the operator algebra structure of convolution algebras.

Findings

Random Fourier series in quantum groups characterize measure regularity.

Results enable analysis of convolution algebra representations as operator algebras.

Extension of classical Fourier analysis theorems to quantum group frameworks.

Abstract

Helgason showed that a given measure $f\in M(G)$ on a compact group $G$ should be in $L^2(G)$ automatically if all random Fourier series of $f$ are in $M(G)$. We explore a natural analogue of the theorem in the framework of compact quantum groups and apply the obtained results to study complete representability problem for convolution algebras of compact quantum groups as an operator algebra.