Existentially closed W*-probability spaces

Abstract

AbstractWe study several model-theoretic aspects of W$$^*$$ ∗ -probability spaces, that is, $$\sigma $$ 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$$_1$$ 1 factors that tensorially absorb the Araki–Woods factor $$R_\infty $$ R ∞ . We also study the existentially closed objects in the restricted class of W$$^*$$ ∗ -probability spaces with Kirchberg’s QWEP property, proving that $$R_\infty $$ 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$$_1$$ 1 factors forms a $$\forall _2$$ ∀ 2 -axiomatizable class. We show that for $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , the class of III$$_\lambda $$ λ factors is not $$\forall _2$$ ∀ 2 -axiomatizable but is $$\forall _3$$ ∀ 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$$_\lambda $$ λ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , there is a family of pairwise non-elementarily equivalent III$$_\lambda $$ λ factors of size continuum. While we cannot prove the same result for III$$_1$$ 1 factors, we show that there are at least three pairwise non-elementarily equivalent III$$_1$$ 1 factors by showing that the class of full factors is preserved under elementary equivalence.