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Journal of Operator Theory

Volume 87, Issue 2, Spring 2022  pp. 319-354.

Modular properties of type I locally compact quantum groups

Authors:  Jacek Krajczok
Author institution:Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00-656, Poland

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

Keywords: type I locally compact quantum group, Plancherel measure, Tomita--Takesaki theory, unimodularity

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