Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras

We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra $\mathcal{N}$ is never model complete if its direct integral decomposition contains ${\mathrm{I}\mathrm{I}}_1$ factors $\mathcal{M}$ such that $M_2(\mathcal{M})$ embeds into an ultrapower of $\mathcal{M}$. The proof in the case of ${\mathrm{I}\mathrm{I}}_1$ factors uses an explicit construction based on random matrices and quantum expanders.