QUANTUM EXPANDERS AND QUANTIFIER REDUCTION FOR TRACIAL VONNEUMANN ALGEBRAS
Ilijas Farah, David Jekel, Jennifer Pi · Journal of Symbolic Logic · 2025
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 script upper M $\mathcal {M}$ M is never model complete if its direct integral decomposition contains upper I upper I Subscript 1 $\mathrm {II}_1$ I I 1 factors script upper N $\mathcal {N}$ N such that upper M 2 left parenthesis script upper N right parenthesis $M_2(\mathcal {N})$ M 2 ( N ) embeds into an ultrapower of script upper N $\mathcal {N}$ N . The proof in the case of upper I upper I Subscript 1 $\mathrm {II}_1$ I I 1 factors uses an explicit construction based on random matrices and quantum expanders.