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

TL;DR

This paper explores the connection between free probability and model theory of tracial von Neumann algebras, developing a notion of free microstates entropy for full types and examining embeddings into ultrapowers.

Contribution

It introduces a full type-based free microstates entropy and demonstrates embeddings preserving entropy in ultrapower algebras, linking free probability with model theory.

Findings

Existence of embeddings with preserved free entropy.

Full type-based free microstates entropy developed.

Open problems in free independence and Gibbs laws.

Abstract

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.