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

2012/07/01 by Isaac Goldbring, Goldbring, Isaac, Bradd Hart +3
Mathematics · #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA) #math.LO #math.OA

paper · pdf · doi:10.48550/arxiv.1207.0169

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

arxiv created 2012/07/26 · arxiv updated 2012/07/30

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 II₁ 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 II₁ factors.

Related

• A survey on the model theory of tracial von Neumann algebras
• Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras
• The Universal Theory of Locally Universal Tracial von Neumann Algebras is not Computable
• Correspondences, Ultraproducts and Model Theory
• Quantifier elimination in II₁ factors