The theory of tracial von Neumann algebras does not have a model companion

TL;DR

This paper proves that the theory of tracial von Neumann algebras lacks a model companion, linking the absence to properties of McDuff II_1 factors and the Connes Embedding Problem.

Contribution

It establishes that the theory of tracial von Neumann algebras does not admit a model companion, connecting this to quantifier elimination and the Connes Embedding Problem.

Findings

No model companion exists for the theory of tracial von Neumann algebras.

Quantifier elimination fails for any locally universal, McDuff II_1 factor.

A positive solution to the Connes Embedding Problem would imply no model-complete theory for II_1 factors.

Abstract

In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II_1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II_1 factors.