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

Abstract

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. 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 ∗ \mathrm {C}^{*} -algebra of a countable discrete quantum group \mathbbl {\Gamma } implies that \mathbbl {\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.