Model theory of operator algebras III: Elementary equivalence and II_1 factors

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 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 II_1 factor such that all II_1 factors embed into one of its ultrapowers.Comment: 16 page

Axiomatizability of reducts of algebras of relations

Submitted versio

An invariant for difference field extensions

In this paper we introduce a new invariant (the distant degree) for difference field extensions of finite transcendence degree, and we explore some of its properties. We also discuss a generalisation of this invariant and of the limit degree to groups with an automorphism.Comment: After posting the previous version, we discovered the work of Willis on totally disconnected locally compact groups. Over a large area of overlap, our "distant degree" invariant of an automorphism agrees with Willis' {\em scale}. In the new version we describe the relations between the two frameworks. 14pp. July 2011: Small change