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

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