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

TL;DR

This paper applies continuous model theory to analyze II_1 factors, revealing the abundance of nonisomorphic factors with isomorphic ultrapowers and providing a simplified perspective on the Connes Embedding Problem.

Contribution

It demonstrates the existence of many nonisomorphic II_1 factors sharing ultrapower isomorphisms and offers a basic approach to the Connes Embedding Problem.

Findings

Existence of continuum many nonisomorphic separable II_1 factors with isomorphic ultrapowers.

A simplified resolution of the Connes Embedding Problem.

All II_1 factors embed into ultrapowers of a specific separable II_1 factor.

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.