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arXiv · 2310.06197

Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras

Ilijas Farah·David Jekel·Jennifer Pi
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Abstract

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 $\mathrm{II}_1$ factors $\mathcal{M}$ such that $M_2(\mathcal{M})$ embeds into an ultrapower of $\mathcal{M}$. The proof in the case of $\mathrm{II}_1$ factors uses an explicit construction based on random matrices and quantum expanders.

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Ilijas Farah, David Jekel, Jennifer Pi. 2023-10-09. Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras. https://doi.org/10.1017/jsl.2025.10100

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