The space of Dimension functions on a C^*-algebra

In this paper, we prove that the space of normalized dimension functions on a unital C^*-algebra is either empty or a Choquet simplex. As a byproduct, we show that $W(A)$ is hereditary in Cu(A) for every C^*-algebra A. The argument combines an exact factorization for Cuntz comparison with Birkhoff's decomposition of lattice valuations. Moreover, we show that the space of normalized rank functions on a unital ring is either empty or a Choquet simplex.