On the range of the Elliott invariant
Let \(A\) be a unital \(C^*\)-algebra. For any tracial state \(\omega\) on \(A\) there is a natural way to define a state \(r_ A(\omega)\) of the \(K_ 0\)-group of \(A\). \(r_ A\) is an affine continuous map from the tracial state space of \(A\) to that of \(K_ 0(A)\). 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 \(C[0,1]\). The author shows that for such \(C^*\)-algebras (and many others) the map \(r_ A\) must preserve extreme points, and that any continuous affine surjection between metrizable Choquet simplices which preserves extreme points and is open can be realized as the \(r_ A\)-map corresponding to a simple unital inductive limit \(C^*\)-algebra of a sequence of finite direct sums of matrix algebras over \(C[0,1]\).
inductive limits of sequences of finite direct sums of matrix algebras over \(C[0,1]\)
continuous affine surjection between metrizable Choquet simplices which preserves extreme points and is open
simple unital inductive limit \(C^*\)-algebra
scientific article; zbMATH DE number 599783