Modular properties of type I locally compact quantum groups. (English) Zbl 1538.22012
Summary: 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 \(\widehat{\mathbb{G}}\) 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 \(\widehat{{\text{SU}}_q(2)}\) and the quantum \(az + b\) group.
MSC:
| 22D10 | Unitary representations of locally compact groups |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
| 46L51 | Noncommutative measure and integration |
