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The notion of a $*$-law or $*$-distribution in free probability is also knownas the quantifier-free type in Farah, Hart, and Sherman's model theoreticframework for tracial von Neumann algebras. However, the full type can also beconsidered an analog of a classical probability distribution (indeed, BenYaacov showed that in the classical setting, atomless probability spaces admitquantifier elimination and hence there is no difference between the full typeand the quantifier-free type). We therefore develop a notion of Voiculescu'sfree 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 ofVoiculescu. Furthermore, we sketch some open problems and challenges fordeveloping model-theoretic versions of free independence and free Gibbs laws.

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