II1 factors with nonisomorphic ultrapowers

Abstract

We prove that there exist uncountably many separable ${\mathrm{II}}_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are nonisomorphic. In fact, we prove that the families of nonisomorphic ${\mathrm{II}}_1$ factors originally introduced by McDuff are such examples. This entails the existence of a continuum of nonelementarily equivalent ${\mathrm{II}}_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.