Existentially closed W*-probability spaces

Abstract

We study several model-theoretic aspects of W$^*$-probability spaces, that is, $\sigma$-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 $R_\infty$. We also study the existentially closed objects in the restricted class of W$^*$-probability spaces with Kirchberg's QWEP property, proving that $R_\infty$ 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 $\forall_2$-axiomatizable class. We show that for $\lambda\in (0,1)$, the class of III$_\lambda$ factors is not $\forall_2$-axiomatizable but is $\forall_3$-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$_\lambda$ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $\lambda\in (0,1)$, there is a family of pairwise non-elementarily equivalent III$_\lambda$ 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.