Elementary equivalence and disintegration of tracial von Neumann algebras

Abstract

We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as Łoś's theorem and countable saturation, to this more general setting.

34 pages; v2, v3 updated citations and minor edits

References in corpus (3)

• Sequential commutation in tracial von Neumann algebras
• Preservation of elementarity by tensor products of tracial von Neumann algebras
• Extremal models and direct integrals in affine logic

Cited by in corpus (1)

• Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras