The trace simplex of a noncommutative Villadsen algebra

Abstract

We construct a ``noncommutative'' Villadsen algebra $B$ and show that, given an extreme tracial state $\nu$ on its canonical AF subalgebra, the subset of $T(B)$ consisting of those tracial states that equal $\nu$ when restricted to the canonical AF subalgebra is the Poulsen simplex. In particular, if the canonical AF subalgebra has a unique trace, then $T(B)$ is the Poulsen simplex. We go on to show that in certain instances, the tracial cone of a ``classical'' AF-Villadsen algebra $D$ is isomorphic to the tracial cone of the algebra obtained from $D$ by deleting all point evaluations.