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

Akihiko Arai

TL;DR

This work analyzes how ultrapower isomorphism behaves for ${C}^{*}$‑algebras within continuous model theory. It develops the metric model‑theoretic framework, including languages, theories, ultraproducts, Łoś' theorem, and saturation, to study when elementary equivalence implies isomorphism of ultrapowers. The main contribution shows that, under $\neg CH$, there exist elementarily equivalent infinite‑dimensional unital ${C}^{*}$‑algebras $A,B$ with density ≤ ${\frak c}$ such that $A^{\mathcal U}\not\cong B^{\mathcal V}$ for all nonprincipal ultrafilters on $\omega$, providing a continuous analogue of classical results and highlighting the set‑theoretic sensitivity of ultrapower behavior. Under CH, these phenomena differ, as countable ultrapowers of certain algebras become isomorphic and relative commutants align, illustrating a sharp CH‑dependent dichotomy in the model theory of operator algebras. The results connect logical saturation, definable sets, and CH to the structure and classification of ${C}^{*}$‑algebras, with implications for how ultrapowers encode set‑theoretic assumptions in noncommutative contexts.

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 $B$ of size $\le \mathfrak c$ such that for all non-principal ultrafilters $\mathcal U, \mathcal V$ on $ω$, 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.

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

TL;DR

This work analyzes how ultrapower isomorphism behaves for ‑algebras within continuous model theory. It develops the metric model‑theoretic framework, including languages, theories, ultraproducts, Łoś' theorem, and saturation, to study when elementary equivalence implies isomorphism of ultrapowers. The main contribution shows that, under , there exist elementarily equivalent infinite‑dimensional unital ‑algebras with density ≤ such that for all nonprincipal ultrafilters on , providing a continuous analogue of classical results and highlighting the set‑theoretic sensitivity of ultrapower behavior. Under CH, these phenomena differ, as countable ultrapowers of certain algebras become isomorphic and relative commutants align, illustrating a sharp CH‑dependent dichotomy in the model theory of operator algebras. The results connect logical saturation, definable sets, and CH to the structure and classification of ‑algebras, with implications for how ultrapowers encode set‑theoretic assumptions in noncommutative contexts.

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 -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 -algebras and of size such that for all non-principal ultrafilters on , the ultrapowers and are not isomorphic. This result provides a continuous analogue of certain classical theorems concerning ultraproducts and demonstrates that the model-theoretic behavior of -algebras is closely related to set-theoretic principles such as the continuum hypothesis.

Paper Structure

This paper contains 14 sections, 29 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Languages and Structures
  4. Terms, Formulas and Their Interpretations
  5. Theories, Elementary Equivalences and Elementary Embeddings
  6. The Downward Löwenheim-Skolem-Tarski Theorem and Elementary chains
  7. Ultraproducts and Related Concepts
  8. Construction of the Ultraproducts
  9. Compactness Theorem
  10. Types and Saturation
  11. Definable Predicates and Definable Sets
  12. Some Results under Continuum Hypothesis
  13. Main Result
  14. Conclusion

Key Result

Proposition 1

Assume that the continuum hypothesis holds. Let $A$ be a ${\mathrm{C}^{\ast}}$-algebra with cardinality $\le {\mathfrak c}$. Then, for all nonprincipal ultrafilters $\mathcal{U}, \mathcal{V}$ on $\omega$, $A^\mathcal{U}$ and $A^\mathcal{V}$ are isomorphic. Moreover, if $A$ is separable, there is an

Theorems & Definitions (89)

  • Proposition 1
  • Theorem 1
  • Definition 1
  • Definition 2
  • Example 1: unital ${\mathrm{C}^{\ast}}$-algebras
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 2
  • Definition 6
  • ...and 79 more