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

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.

6 pages; this version no longer relies on the Connes Embedding Problem to establish the main result