Uniformity and isotypic smallness for quantum-group representations

TL;DR

This paper explores the conditions under which quantum group representations exhibit uniformity and isotypic smallness, extending classical results to certain classes of compact quantum groups and introducing quantum-specific mechanisms.

Contribution

It generalizes classical uniformity and isotypic smallness equivalence to compact quantum groups with specific properties, introducing new quantum-specific mechanisms.

Findings

Equivalence holds for coamenable compact quantum groups.

Equivalence holds for compact quantum groups with bounded irreducible representations.

Provides two independent mechanisms for recovering classical properties in quantum setting.

Abstract

Compact-group representations on Banach spaces are known to be norm-continuous precisely when they have finite spectra. For a quantum group with continuous-function algebra $\mathcal{C}(\mathbb{G})$ norm continuity can be cast analogously as the bounded weak$^*$-norm continuity of the representation's attached map $\mathcal{C}(\mathbb{G})^*\to \mathrm{End}(E)$. While the uniformity/isotypic finiteness equivalence no longer holds generally, it does for compact quantum groups either coamenable or having dimension-bounded irreducible representations. This generalizes the aforementioned classical variant, providing two independent quantum-specific mechanisms of recovering it.