## 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.

### MSC:

 03C65 Models of other mathematical theories
Full Text:

### References:

 [1] The Bulletin of the London Mathematical Society [2] C*-algebras and finite-dimensional approximations 88 (2008) [3] DOI: 10.1112/plms/pdm054 · Zbl 1158.46045 [4] DOI: 10.1007/s00208-006-0074-y · Zbl 1121.46052 [5] DOI: 10.2140/pjm.1995.171.309 · Zbl 0892.22004 [6] DOI: 10.1142/S0219061309000811 · Zbl 1211.03051 [7] DOI: 10.1016/j.aim.2011.04.003 · Zbl 1229.46041
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