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

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.