2022/08/29 by David Jekel, Jekel, David
Mathematics · Physics and Astronomy · #03C66 #46L54 #60B20 #Advanced Operator Algebra Research #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA) #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.2208.13867
openalex publication_date 2022/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 X is a d-tuple in M with χU(X:M) > -∞ for a given ultrafilter U, then there exists an embedding ι of M into Q = ∏n → U Mn(ℂ) with χ(ι(X): Q) = χ(X:M); in particular, such an embedding will satisfy ι(X)' ∩ Q = 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.