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

TL;DR

This paper characterizes the trace spaces of full free product $C^*$-algebras, showing they are typically the Poulsen simplex unless specific obstructions occur, with implications for group $C^*$-algebras and von Neumann algebra perturbations.

Contribution

It provides a complete characterization of trace spaces for free product $C^*$-algebras and introduces a new perturbation technique for von Neumann subalgebras.

Findings

Trace space is the Poulsen simplex under general conditions.

Extreme points of the trace space are dense in Wasserstein topology.

For group $C^*$-algebras, trace space is Poulsen unless the trivial character is isolated.

Abstract

We study the space of traces associated with arbitrary full free products of unital, separable $C^*$-algebras. We show that, unless certain basic obstructions (which we fully characterize) occur, the space of traces always results in the same object: the Poulsen simplex, that is, the unique infinite-dimensional metrizable Choquet simplex whose extreme points are dense. Moreover, we show that whenever such a trace space is the Poulsen simplex, the extreme points are dense in the Wasserstein topology. Concretely for the case of groups, we find that, unless the trivial character is isolated in the space of characters, the space of traces of any free product of non-trivial countable groups is the 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 $M$, providing necessary conditions under which $M_{1}$ and a small unitary perturbation of $M_{2}$ generate a II$_{1}$ factor.