The trace simplex of a noncommutative Villadsen algebra

TL;DR

This paper constructs a noncommutative Villadsen algebra with a trace simplex that can be the Poulsen simplex, revealing complex tracial state structures in such algebras.

Contribution

It introduces a noncommutative Villadsen algebra with a trace simplex that can be the Poulsen simplex, expanding understanding of tracial state spaces.

Findings

The trace simplex of the constructed algebra can be the Poulsen simplex.

If the AF subalgebra has a unique trace, the entire trace space is the Poulsen simplex.

The tracial cone of certain AF-Villadsen algebras matches that of a modified algebra without point evaluations.

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.