Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras

TL;DR

This paper characterizes when theories of tracial von Neumann algebras admit quantifier elimination and explores conditions under which their theories are not model complete, using tools like random matrices and quantum expanders.

Contribution

It provides a complete characterization of quantifier elimination for tracial von Neumann algebras and identifies conditions for non-model completeness involving quantum expanders.

Findings

Characterization of theories with quantifier elimination.

Identification of non-model completeness conditions.

Use of random matrices and quantum expanders in proofs.

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.