Around Property (T) for quantum groups

TL;DR

This paper advances the understanding of Property (T) in locally compact quantum groups by providing new characterisations, connecting it with ergodic theory, spectral gaps, and classical theorems, thereby broadening its theoretical framework.

Contribution

It introduces new characterisations of Property (T) for quantum groups, generalises existing structural results, and links Property (T) to classical theorems and ergodic properties in the quantum setting.

Findings

Property (T) characterized via Kazhdan pairs

Equivalence of Property (T) and Property (T)^{1,1} for certain quantum groups

Connections established between Property (T), spectral gaps, and ergodic actions

Abstract

We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with low duals Property (T) is shown to be equivalent to Property (T)$^{1,1}$ of Bekka and Valette. This is used to extend to this class of quantum groups classical theorems on 'typical' representations (due to Kerr and Pichot), and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Finally we discuss in the Appendix equivalent characterisations of the notion of a quantum group morphism with dense image.