Free probability and model theory of tracial $\mathrm{W}^*$-algebras

The notion of a $*$-law or $*$-distribution in free probability is also known as the quantifier-free type in Farah, Hart, and Sherman's model theoretic framework for tracial von Neumann algebras. However, the full type can also be considered an analog of a classical probability distribution (indeed, Ben Yaacov showed that in the classical setting, atomless probability spaces admit quantifier elimination and hence there is no difference between the full type and the quantifier-free type). We therefore develop a notion of Voiculescu's free microstates entropy for a full type, and we show that if $\mathbf{X}$ is a $d$-tuple in $\mathcal{M}$ with $\chi^{\mathcal{U}}(\mathbf{X}:\mathcal{M}) > -\infty$ for a given ultrafilter $\mathcal{U}$, then there exists an embedding $\iota$ of $\mathcal{M}$ into $\mathcal{Q} = \prod_{n \to \mathcal{U}} M_n(\mathbb{C})$ with $\chi(\iota(\mathbf{X}): \mathcal{Q}) = \chi(\mathbf{X}:\mathcal{M})$; in particular, such an embedding will satisfy $\iota(\mathbf{X})' \cap \mathcal{Q} = \mathbb{C}$ by the results of Voiculescu. Furthermore, we sketch some open problems and challenges for developing model-theoretic versions of free independence and free Gibbs laws.