On the isomorphism problem for ultraproducts of $\mathrm{C}^*$-algebras in continuous model theory

TL;DR

This paper explores whether ultraproducts of $ ext{C}^*$-algebras are isomorphic when they are elementarily equivalent, revealing set-theoretic dependencies and providing a continuous model theory analogue of classical results.

Contribution

It demonstrates that, under the negation of the continuum hypothesis, elementarily equivalent $ ext{C}^*$-algebras can have non-isomorphic ultrapowers, highlighting set-theoretic influences in continuous model theory.

Findings

Existence of elementarily equivalent $ ext{C}^*$-algebras with non-isomorphic ultrapowers under CH negation

Ultrapower isomorphism depends on set-theoretic assumptions like the continuum hypothesis

Continuous model theory behavior of $ ext{C}^*$-algebras relates to classical ultraproduct theorems

Abstract

In classical model theory, the Keisler--Shelah theorem establishes a fundamental connection between the elementary equivalence of structures and the isomorphism of their ultrapowers. Motivated by this, one may ask whether an analogous relationship holds in the framework of continuous model theory, which naturally encompasses metric structures such as $\mathrm{C}^\ast$-algebras. In this paper, we investigate the isomorphism problem for ultraproducts of operator algebras from a model-theoretic perspective. We prove that, assuming the negation of the continuum hypothesis, there exist two elementarily equivalent infinite-dimensional unital $\mathrm{C}^\ast$-algebras $A$ and $B$, whose density characters are at most $\mathfrak c$, such that for all non-principal ultrafilters $\mathcal U, \mathcal V$ on $\omega$, the ultrapowers $A^{\mathcal U}$ and $B^{\mathcal V}$ are not isomorphic. This result provides a continuous analogue of certain classical theorems concerning ultraproducts and demonstrates that the model-theoretic behavior of $\mathrm{C}^\ast$-algebras is closely related to set-theoretic principles such as the continuum hypothesis.