The theory of tracial von Neumann algebras does not have a model companion (Q2869913)

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.NEWLINENEWLINEUsing 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.NEWLINENEWLINEFinally, 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.