Model theory of operator algebras II: Model theory

TL;DR

This paper develops a logic framework for metric structures like C*-algebras and von Neumann algebras, proving a stability criterion related to ultrapowers and the Continuum Hypothesis.

Contribution

It introduces a new logical approach for metric structures and establishes a model-theoretic stability criterion independent of the Continuum Hypothesis.

Findings

The theory of a separable metric structure is stable iff all its ultrapowers are isomorphic.

The stability criterion holds regardless of the Continuum Hypothesis.

Provides a foundation for applying model theory to operator algebras.

Abstract

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on N are isomorphic even when the Continuum Hypothesis fails.