Submission date: 20 August 2021

Abstract:

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 III_1 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 III_1 factors forms a ∀_2-axiomatizable class. We show that for λ in (0,1), the class of III_λ 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 III_λ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any λ in (0,1), there is a family of pairwise non-elementarily equivalent III_λ factors of size continuum. While we cannot prove the same result for III_1 factors, we show that there are at least three pairwise non-elementarily equivalent III_1 factors by showing that the class of full factors is preserved under elementary equivalence.!

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Full text arXiv 2108.09223: pdf, ps.