The theory of tracial von Neumann algebras does not have a model companion. (English) Zbl 1316.03019

This paper is a contribution to the model theory of operator algebras. The authors begin by noting that the theory \(T_0\) of tracial von Neumann algebras is universally axiomatizable.
Using the crossed product construction for von Neumann algebras, the authors prove that \(\mathrm{Th}(\mathcal {R})\), where \(\mathcal{R}\) is the hyperfinite \(\Pi_1\) factor, does not have quantifier elimination. This leads to their main result: \(T_0\) does not have a model companion.
Finally, the authors consider the possibility that there is a model-complete theory of \(\Pi_1\) factors. They show that if the CEP (Connes Embedding Problem) has a positive solution, then there is no model-complete theory of \(\Pi_1\) factors.


03C65 Models of other mathematical theories
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