Axiomatizability of the stable rank of C*-algebras

TL;DR

This paper proves that the class of C*-algebras with a stable rank above a certain threshold can be characterized by logical axioms, and explores the stability properties of stable rank under ultrapowers and perturbations.

Contribution

It establishes the axiomatizability of stable rank in the logic of metric structures and demonstrates its continuity and stability under ultrapowers and Kadison--Kastler perturbations.

Findings

Stable rank classes are axiomatizable in metric logic.

Stable rank is continuous under ultrapower formation.

Stable rank exhibits Kadison--Kastler stability.

Abstract

We show that the class of C*-algebras with stable rank greater than a given positive integer is axiomatizable in logic of metric structures. As a consequence we show that the stable rank is continuous with respect to forming ultrapowers of C*-algebras, and that stable rank is Kadison--Kastler stable.