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

Abstract

We show that the space of traces of the free group $F_d$ on $2\le d\le \infty$ 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 $C(X_1)\ast C(X_2)$ 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 $M_n(\mathbb{C})\ast M_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R\ordam 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.

arXiv commentWe added a more general theorem concerning free products of abelian C*-algebras. We also improved the statements regarding faces of the trace simplex we made