II1 Factors with non-isomorphic ultrapowers

Rémi Boutonnet (1) , Ionut Chifan (2) , Adrian Ioana (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 University of Iowa [Iowa City]
3 Mathematics Department; University of California San Diego

We prove that there exist uncountably many separable II1 factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II1 factors originally introduced by McDuff [MD69a, MD69b] are such examples. This entails the existence of a continuum of non-elementarily equivalent II1 factors, thus settling a well-known open problem in the continuous model theory of operator algebras.

Domaines: Algèbres d'opérateurs [math.OA]

Rémi Boutonnet, Ionut Chifan, Adrian Ioana. II1 Factors with non-isomorphic ultrapowers. Duke Mathematical Journal, 2017. ⟨hal-01450072⟩