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.