Compact quantum metric spaces from quantum groups of rapid decay

TL;DR

This paper extends the concept of property RD and spectral triples to non-unimodular quantum groups, analyzing their structure as compact quantum metric spaces, thus broadening the scope of quantum group analysis.

Contribution

It introduces a modified property RD definition for non-unimodular quantum groups and extends spectral triple constructions to quantum groups, linking them to quantum metric spaces.

Findings

Extended property RD to non-unimodular quantum groups

Constructed spectral triples for quantum groups of rapid decay

Analyzed spectral triples as compact quantum metric spaces

Abstract

We present a modified version of the definition of property RD for discrete quantum groups given by Vergnioux in order to accommodate examples of non-unimodular quantum groups. Moreover we extend the construction of spectral triples associated to discrete groups with length functions, originally due to Connes, to the setting of quantum groups. For quantum groups of rapid decay we study the resulting spectral triples from the point of view of compact quantum metric spaces in the sense of Rieffel.