Model theory of operator algebras I: Stability

TL;DR

This paper investigates the model-theoretic properties of operator algebras, specifically focusing on the stability of ultrapowers and relative commutants, and provides definitive answers to longstanding questions in the field.

Contribution

It extends previous results to establish that ultrapowers and relative commutants of certain operator algebras are independent of ultrafilter choice, advancing the understanding of their model-theoretic stability.

Findings

Ultrapowers of C*-algebras and II_1 factors are ultrafilter-independent.

Relative commutants in these algebras do not depend on ultrafilter choice.

The results confirm stability properties for a broad class of operator algebras.

Abstract

Several authors have considered whether the ultrapower and the relative commutant of a C*-algebra or II_1 factor depend on the choice of the ultrafilter. We settle each of these questions, extending results of Ge-Hadwin and the first author.