On the Range of the Elliott Invariant

Let A be a unital C*-algebra. For any tracial state ω on A there is natural way to define a state rA(ω) of the K0-group of A. rA is an affine continuous map from the tracial state space of A to that of K0(A). This map enters as a crucial ingredient in the invariant used by Elliott to classify the simple unital C*-algebras that arise as inductive limits of sequences of finite direct sums of matrix algebras over C[0, 1]. It is shown here that for such C*-algebras (and many others) the map rA must preserve extreme points, and that any continuous affine surjection between metrisable Choquet simplices which preserves extreme points and is open can be realised as the rA-map corresponding to a simple unital inductive limit C*-algebra of a sequence of finite direct sums of matrix algebras over C[0,1].