Modular properties of type I locally compact quantum groups

Abstract

The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on $\mathbb{G}$ and $\hat{\mathbb{G}}$ act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group $\widehat{{\mathrm{SU}}_q(2)}$ and the quantum $az+b$ group.

33 pages