A recent paper shows that for any unital $C^*$‑algebra $A$, the set of normalized dimension functions—functions that assign a non‑negative real number to each positive matrix over $A$, add under direct sums and respect Cuntz comparison—is either empty or a Choquet simplex. This resolves a long‑standing question posed by Blackadar and Handelman.

The authors prove the result by combining an exact factorization for Cuntz comparison with Birkhoff’s classical decomposition of lattice valuations. The factorization gives a strong structural property of the Cuntz semigroup of $A$, namely that the submonoid $W(A)$ of classes represented by positive elements in matrix algebras over $A$ is hereditary inside $ ext{Cu}(A)$. From this property it follows that the cone of dimension functions is a hereditary subcone of the cone of Sylvester matrix rank functions on $A$, and thus its image is a closed face of the well‑known simplex of rank functions.

Importantly, the result is not just abstract: the authors prove that the simplex is non‑empty precisely when no matrix algebra over $A$ has a properly infinite unit. In practice this means that if there exists an $n$ such that the unit of $M_n(A)$ satisfies $1_{M_n(A)} ot riangleleft (n+1) frac{}{}{1}_{M_n(A)}$, then $DF(A)$ is empty. Otherwise, $DF(A)$ is a Choquet simplex.

Hence, for every unital $C^*$‑algebra the theory of dimension functions is completely clarified: they either form a Choquet simplex or do not exist at all, in line with the underlying infiniteness properties of the algebra.