Model theory of operator algebras III: elementary equivalence and II1 factors

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II1 factors and their ultrapowers. Among other things, we show that for any II1 factor M, there are continuum many nonisomorphic separable II1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II1 factor such that all II1 factors embed into one of its ultrapowers.