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

Abstract

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.

16 pages

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• Pseudofinite and pseudocompact metric structures
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• Categorical semantics of metric spaces and continuous logic
• Sequential commutation in tracial von Neumann algebras
• Elementary equivalence and disintegration of tracial von Neumann algebras
• Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras