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

We show that the space of traces of free products of the form C(X₁)⁎C(X₂), where X₁ and X₂ 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≤d≤∞ 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(C)⁎M_n(C) is a Poulsen simplex as well, answering a question of Musat and Rørdam for n≥4. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.