Blackadar–Handelman conjecture on dimension functions

Let $A$ be a unital $C^*$-algebra. Define the classical Cuntz semigroup $\mathrm{W}(A)$ from positive elements in $M_\infty(A)$, and let $\mathrm{DF}(A)$ be the set of normalized dimension functions on $A$. A Choquet simplex is a compact convex set whose points admit unique representing probability measures on its extreme boundary. Blackadar–Handelman conjecture. The set $\mathrm{DF}(A)$ is a Choquet simplex.

The conjecture concerns the convex structure of dimension functions and their relation to states on the Grothendieck group of the classical Cuntz semigroup. The source explains that it is solved for unital C*-algebras of stable rank one, while presenting it historically as a conjecture in general.