The space of traces of the free group and free products of matrix algebras

Abstract

We show that the space of traces of free products of the form $C\left(X_1\right)⁎C\left(X_2\right)$, where $X_1$ and $X_2$ are compact metrizable spaces without isolated points, is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. In particular, the space of traces of the free group $F_d$ on $2\le d\le \infty$ generators is a Poulsen simplex, and we demonstrate that this is no longer true for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $M_n\left(C\right)⁎M_n\left(C\right)$ is a Poulsen simplex as well, answering a question of Musat and Rørdam for $n\ge 4$. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.