The Space of Traces of the Free Group and Free Products of Matrix Algebras

TL;DR

This paper proves that the space of traces of free groups and certain free products of matrix algebras forms a Poulsen simplex, meaning all traces can be approximated by extreme traces, with implications for understanding their structure.

Contribution

It establishes that the trace spaces of free groups and specific free products of matrix algebras are Poulsen simplices, extending known results and answering open questions.

Findings

Trace spaces of free groups are Poulsen simplices.

Trace spaces of certain free products of matrix algebras are Poulsen simplices.

Results apply to faces like finite-dimensional and amenable traces.

Abstract

We show that the space of traces of the free group $F_d$ on $2\leq d \leq \infty $ generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form $C(X_1)*C(X_2)$ where $X_1$ and $X_2$ are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $M_n(\mathbb{C}) * M_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R\ordam for $n \geq 4$. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.