Property $T$ for general locally compact quantum groups

TL;DR

This paper characterizes property T for general locally compact quantum groups through equivalent formulations involving their full quantum group C*-algebras and representations, extending classical group results and providing a construction method for property T quantum groups.

Contribution

It introduces new equivalent formulations of property T for quantum groups and extends classical group results to the quantum setting, including a construction method for property T quantum groups.

Findings

Equivalent formulations of property T for quantum groups.

Characterization of property T in terms of spectrum of C*-algebras.

Construction of property T quantum groups via bicrossed products.

Abstract

In this short article, we obtained some equivalent formulations of property $T$ for a general locally compact quantum group $\mathbb{G}$, in terms of the full quantum group $C^*$-algebras $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ and the $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if $\mathbb{G}$ is of Kac type, we show that $\mathbb{G}$ has property $T$ if and only if every finite dimensional irreducible $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ is an isolated point in the spectrum of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property $T$ discrete quantum groups using bicrossed products.