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.
"The theory of tracial von Neumann algebras does not have a model companion." J. Symbolic Logic 78 (3) 1000 - 1004, September 2013. https://doi.org/10.2178/jsl.7803170