Pure C*-algebras

We show that every C*-algebra that is (m,n)-pure in the sense of Winter is already pure, that is, its Cuntz semigroup is almost unperforated and almost divisible. More generally, we show that even weaker comparison and divisibility properties automatically lead to pureness. We use this to show that, under a mild comparison assumption, pureness is automatic for C*-algebras that are either nowhere scattered with real rank zero or stable rank one, or simple, unital, non-elementary, with a unique quasitrace. As an application to the non-simple Toms-Winter conjecture, we show that every C*-algebra with the Global Glimm Property and finite nuclear dimension is pure. It follows that a separable, locally subhomogeneous C*-algebras with stable rank one and topological dimension zero is pure if and only if it is $\mathcal{Z}$-stable, if and only if it is nowhere scattered and has finite nuclear dimension.