Trace spaces of full free product $C^*$-algebras

Abstract

We prove that the space of traces $\text{T}(A)$ of the unital full free product $A=A_1*A_2$ of two unital, separable $C^*$-algebras $A_1$ and $A_2$ is typically a Poulsen simplex, i.e., a simplex whose extreme points are dense. We deduce that $\text{T}(A)$ is a Poulsen simplex whenever $A_1$ and $A_2$ have no $1$-dimensional representations, e.g., if $A_1$ and $A_2$ are finite dimensional with no $1$-dimensional direct summands. Additionally, we characterize when the space of traces of a free product of two countable groups is a Poulsen simplex. Our main technical contribution is a new perturbation result for pairs of von Neumann subalgebras $(M_1,M_2)$ of a tracial von Neumann algebra $M$ which gives necessary conditions ensuring that $M_1$ and a small unitary perturbation of $M_2$ generate a II$_1$ factor.