2021/08/20 by Isaac Goldbring, Goldbring, Isaac, Cyril Houdayer +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2108.09223
openalex publication_date 2021/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study several model-theoretic aspects of W^*-probability spaces, that is, σ-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W^*-spaces and prove several structural results about such spaces, including that they are type III1 factors that tensorially absorb the Araki-Woods factor R_∞. We also study the existentially closed objects in the restricted class of W^*-probability spaces with Kirchberg's QWEP property, proving that R_∞ itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III1 factors forms a ∀2-axiomatizable class. We show that for λ∈ (0,1), the class of IIIλ factors is not ∀2-axiomatizable but is ∀3-axiomatizable; this latter result uses a version of Keisler's Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of IIIλ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any λ∈ (0,1), there is a family of pairwise non-elementarily equivalent IIIλ factors of size continuum. While we cannot prove the same result for III1 factors, we show that there are at least three pairwise non-elementarily equivalent III1 factors by showing that the class of full factors is preserved under elementary equivalence.