The Universal Theory of Locally Universal Tracial von Neumann Algebras Is Not Computable

Abstract Building on Lin’s breakthrough $\mathsf{MIP}^{\mathsf{co}} = \mathsf{coRE}$ and an encoding of non-local games as universal sentences in the language of tracial von Neumann algebras, we show that locally universal tracial von Neumann algebras have undecidable universal theories. This implies that no such algebra admits a computable presentation. Our results also provide, for the first time, examples of separable $\textrm{II}_{1}$ factors without computable presentations, and in fact yield a broad family of them, including McDuff factors, factors with neither property $\Gamma $ nor property (T), and property (T) factors. We also obtain analogous results for locally universal semifinite von Neumann algebras and tracial $C^{*}$-algebras. The latter provides strong evidence for a negative solution to the Kirchberg embedding problem. We discuss how these are obstructions to approximation properties in the class of tracial/semifinite von Neumann algebras.