Fields of locally compact quantum groups: continuity and pushouts

TL;DR

This paper demonstrates that certain classes of locally compact quantum groups form continuous fields over their subgroups and that free products of continuous fields of $C^*$-algebras preserve continuity, generalizing existing results.

Contribution

It establishes the continuity of locally compact quantum groups over their subgroups and shows free products of continuous $C^*$-fields remain continuous, extending previous work.

Findings

Discrete quantum groups with coamenable dual are continuous fields over their central subgroups.

Free products of continuous $C^*$-algebras are also continuous fields.

Generalization of Blanchard's result on $C^*$-field continuity.

Abstract

We prove that (a) discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and (b) the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of $C^*$-algebras are again free via a Fell-topology characterization for $C^*$-field continuity, recovering a result of Blanchard's in a somewhat more general setting.