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

Joav Orovitz, Raz Slutsky, Itamar Vigdorovich

Published 2023-08-21Version 1

We show that the space of traces of the free group is a Poulsen simplex, i.e., every trace is a pointwise limit of extremal traces. We prove that this fails for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $\mathbf{M}_n(\mathbb{C}) * \mathbf{M}_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R{\o}rdam for $n \geq 4$. Similar results are shown for certain faces of the simplices above, most notably, for the face of finite-dimensional traces.