Modular properties of type I locally compact quantum groups

TL;DR

This paper investigates the modular properties of type I locally compact quantum groups, analyzing operator actions on direct integrals and exploring implications for unimodularity and traciality, with detailed examples of specific quantum groups.

Contribution

It provides new insights into the modular theory of type I quantum groups and links properties like unimodularity and traciality through operator analysis.

Findings

Operators related to modular theory act on direct integrals.

Implications between unimodularity and traciality are established.

Detailed analysis of quantum groups $ ext{SU}_q(2)$ and $az+b$.

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 $\widehat{\mathbb{G}}$ 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 $\widehat{\mathrm{SU}_q(2)}$ and the quantum $az+b$ group.