Continuous model theories for von Neumann algebras

TL;DR

This paper develops a continuous logic framework to axiomatize and analyze von Neumann algebras, including their ultrapowers and preduals, providing new insights into their structural properties and isomorphisms.

Contribution

It introduces a novel axiomatization of von Neumann algebras and their ultrapowers within continuous logic, extending previous results and applying to type III and II factors.

Findings

Axiomatization of $\sigma$-finite $W^*$-probability spaces and preduals.

Axiomatizability of $III_ ext{lambda}$ factors and their preduals.

Application to ultrapower isomorphism problems for type $III$ and $II_ ext{infty}$ factors.

Abstract

We axiomatize in (first order finitary) continuous logic for metric structures $\sigma$-finite $W^*$-probability spaces and preduals of von Neumann algebras jointly with a weak-* dense $C^*$-algebra of its dual. This corresponds to the Ocneanu ultrapower and the Groh ultrapower of ($\sigma$-finite in the first case) von Neumann algebras. We give various axiomatizability results corresponding to recent results of Ando and Haagerup including axiomatizability of $III_\lambda$ factors for $0<\lambda\leq 1$ fixed and their preduals. We also strengthen the concrete Groh theory to an axiomatization result for preduals of von Neumann algebras in the language of tracial matrix-ordered operator spaces, a natural language for preduals of dual operator systems. We give an application to the isomorphism of ultrapowers of factors of type $III$ and $II_\infty$ for different ultrafilters.