Elementary equivalence and disintegration of tracial von Neumann algebras

TL;DR

This paper establishes that elementary equivalence of tracial von Neumann algebras' direct integrals implies fiberwise elementary equivalence, extending ultraproduct techniques to a continuous setting and confirming a conjecture by Farah and Ghasemi.

Contribution

It introduces a continuous analog of ultraproducts for von Neumann algebras, proving a disintegration theorem under elementary equivalence and extending key ultraproduct results.

Findings

Elementary equivalence of direct integrals implies fiberwise elementary equivalence.

Develops a continuous ultraproduct framework using characters on commutative von Neumann algebras.

Verifies a conjecture of Farah and Ghasemi regarding disintegration in this setting.

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 {\L}o\'s's theorem and countable saturation, to this more general setting.