Quasidiagonality of nuclear C$^{*}$-algebras

Abstract

We prove that faithful traces on separable and nuclear $\mathrm{C}^*$-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear $\mathrm{C}^*$-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms–Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal $\mathrm{C}^*$-algebras.