Isomorphism of matrix ultraproducts under the continuum hypothesis
If the continuum hypothesis fails, some of the ultraproducts $\prod_{n \to \mathcal U} \mathbb M_n(\mathbb C)$ are not isomorphic. When it holds, the question below is equivalent to whether they are all elementarily equivalent in the sense of continuous model theory (Farah, Hart and Sherman).
Problem. Assume the continuum hypothesis. Is $\prod_{n \to \mathcal U} \mathbb M_n(\mathbb C) \cong \prod_{n \to \mathcal V} \mathbb M_n(\mathbb C)$ for all nonprincipal ultrafilters $\mathcal U$ and $\mathcal V$ on $\mathbb N$?
Source: S. Popa, as a question recorded by Farah, Hart and Sherman (2013, 2014); listed as U.8 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- OA — Operator Algebras
- Secondary
- LO
- Keywords
- continuous model theory, continuum hypothesis, matrix algebras, Peterson's operator algebra problems, ultraproducts
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{farah2013,
author = {Farah, Ilijas and Hart, Bradd and Sherman, David},
title = {Model theory of operator algebras {I}: stability},
journal = {Bull. Lond. Math. Soc.},
volume = {45},
number = {4},
pages = {825--838},
year = {2013},
doi = {10.1112/blms/bdt014},
}
@article{farah2014,
author = {Farah, Ilijas and Hart, Bradd and Sherman, David},
title = {Model theory of operator algebras {III}: elementary equivalence and {II}$_1$ factors},
journal = {Bull. Lond. Math. Soc.},
volume = {46},
number = {3},
pages = {609--628},
year = {2014},
doi = {10.1112/blms/bdu012},
}
@misc{peterson-u8,
author = {Peterson, Jesse},
title = {Open problems in operator algebras},
howpublished = {Web page},
year = {2026},
url = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
note = {Problem U.8; version of 15 May 2026},
}
@misc{kunnawalkam2025,
author = {Kunnawalkam Elayavalli, Srivatsav},
title = {50 open problems: ultraproduct {II}$_1$ factors},
year = {2025},
eprint = {2511.20377},
archivePrefix = {arXiv},
note = {Problem 1; v3, 25 Aug 2026},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00972, "Isomorphism of matrix ultraproducts under the continuum hypothesis", clopen.solutions.