Ranks of operators and pureness of C*-algebras

For simple, nonelementary C*-algebras, we show uniform density of functions realized as ranks of operators. It follows that strict comparison by traces forces almost divisibility, and hence pureness. As an application, we show that strict comparison implies $\mathcal{Z}$-stability for separable, simple, nonelementary C*-algebras with locally finite nuclear dimension. In particular, the Toms–Winter conjecture holds for approximately subhomogeneous C*-algebras. More generally, we obtain uniform density of ranks for C*-algebras with the Global Glimm Property. Consequently, an exact C*-algebra is pure if and only if it has strict comparison together with the Global Glimm Property.