Strict comparison implies pureness and Z-stability

Prove that, for unital, simple, separable, nuclear, and non-elementary $*$-algebras, strict comparison implies pureness, and consequently establish $Z$-stability.

Background

Pureness is presented as a regularity condition on the Cuntz semigroup that combines comparison and divisibility and serves as an analogue of ZZ-stability at the Cuntz-semigroup level. The discussion places pureness within the Toms–Winter regularity framework for unital, simple, separable, nuclear, and non-elementary ∗*-algebras.

The paper explicitly identifies the implication from strict comparison to pureness as the remaining open direction in this setting; pureness is already described as implying ZZ-stability. This unresolved implication concerns a central structural relationship in the classification program and is distinct from the path-connectedness theorem proved in the paper.

References

It therefore fits naturally into the Toms-Winter regularity picture underlying the classification program: in the unital, simple, separable, nuclear, and non-elementary setting, the only direction that remains open in general would show that strict comparison implies pureness, which in turn implies $Z$-stability.

— Homotopies of constant Cuntz classes in pure $C^*$-algebras  (2610.12219 - Ouseph et al., 8 Oct 2026) in Section 1, Introduction