Spectral finiteness, quantum norm continuity and classical points

TL;DR

This paper explores the conditions under which representations of compact quantum groups exhibit uniform continuity, extending classical results and utilizing decay properties of Fourier coefficients in tensor products.

Contribution

It generalizes classical uniform continuity notions to quantum group representations with finite spectrum, introducing new decay-based analytical techniques.

Findings

Equivalence of various uniform continuity notions for quantum group representations

Extension of classical results to the quantum setting

Use of Riemann-Lebesgue-type decay properties in analysis

Abstract

We prove various notions of uniform continuity for compact-quantum-group representations on Hilbert or Banach spaces equivalent to having finite spectrum, i.e. finitely many isotypic components. This generalizes the classical analogue for compact-group representations on Banach spaces, and relies in part on Riemann-Lebesgue-type decay properties for Fourier coefficients of elements in minimal tensor products with compact-quantum-group function algebras.