On the Baum--Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg Conjecture

TL;DR

This paper explores the Baum--Connes conjecture and quantum Rosenberg Conjecture for discrete quantum groups, establishing conditions under which certain algebraic properties imply amenability and preserving the UCT in specific classes.

Contribution

It provides a decomposition of the equivariant Kasparov category for discrete quantum groups with torsion and proves the quantum Rosenberg Conjecture for a broad class of unimodular quantum groups.

Findings

Crossed product by certain discrete quantum groups preserves the UCT.

Quasidiagonality of reduced C*-algebras implies amenability for these quantum groups.

The quantum Rosenberg Conjecture holds for a large class of unimodular quantum groups.

Abstract

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.