{"paper":{"title":"On the isomorphism problem for ultraproducts of $\\mathrm{C}^*$-algebras in continuous model theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Assuming the negation of the continuum hypothesis, there exist elementarily equivalent C*-algebras of density at most c whose ultrapowers are never isomorphic.","cross_cats":["math.LO"],"primary_cat":"math.OA","authors_text":"Akihiko Arai","submitted_at":"2025-11-19T20:44:46Z","abstract_excerpt":"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 elementa"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"assuming the negation of the continuum hypothesis, there exist two elementarily equivalent infinite-dimensional unital C*-algebras A and B, whose density characters are at most c, such that for all non-principal ultrafilters U, V on ω, the ultrapowers A^U and B^V are not isomorphic.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence of two elementarily equivalent C*-algebras A and B (with the stated size bound) whose ultrapowers remain non-isomorphic for every choice of non-principal ultrafilters, which the abstract presents as following from the negation of CH but whose concrete construction is not visible in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Assuming the negation of the continuum hypothesis, there exist elementarily equivalent C*-algebras A and B such that A^U and B^V are non-isomorphic for every pair of non-principal ultrafilters U and V on omega.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Assuming the negation of the continuum hypothesis, there exist elementarily equivalent C*-algebras of density at most c whose ultrapowers are never isomorphic.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e1cd29c8971f48c0591aa680c596ebcbbb0865ff717dec3592ab1150262672be"},"source":{"id":"2511.15867","kind":"arxiv","version":3},"verdict":{"id":"2c66e9dd-2f08-46e7-acd7-5477998a22c1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T21:04:02.116764Z","strongest_claim":"assuming the negation of the continuum hypothesis, there exist two elementarily equivalent infinite-dimensional unital C*-algebras A and B, whose density characters are at most c, such that for all non-principal ultrafilters U, V on ω, the ultrapowers A^U and B^V are not isomorphic.","one_line_summary":"Assuming the negation of the continuum hypothesis, there exist elementarily equivalent C*-algebras A and B such that A^U and B^V are non-isomorphic for every pair of non-principal ultrafilters U and V on omega.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence of two elementarily equivalent C*-algebras A and B (with the stated size bound) whose ultrapowers remain non-isomorphic for every choice of non-principal ultrafilters, which the abstract presents as following from the negation of CH but whose concrete construction is not visible in the abstract.","pith_extraction_headline":"Assuming the negation of the continuum hypothesis, there exist elementarily equivalent C*-algebras of density at most c whose ultrapowers are never isomorphic."},"references":{"count":32,"sample":[{"doi":"","year":2008,"title":"Ward Henson, and Alexander Usvyatsov","work_id":"e58ecc78-a402-4bd9-ac89-991d0f453178","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Theory ofC ∗-algebras and von Neumann alge- bras, volume 122 ofEncyclopaedia of Mathematical Sciences","work_id":"5d430532-fbcd-42d1-81a4-09996f542a85","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Operator Algebras and Non-commutative Geometry, III","work_id":"8c2c39fc-c4fa-40b0-b233-59bc02970933","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"C. C. Chang and H. Jerome Keisler.Model theory, volume 73 ofStudies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, third edition, 1990","work_id":"a974539d-5d5f-4e50-b806-466ee1f5355d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1966,"title":"Jerome Keisler.Continuous model theory, volume 58 of Annals of Mathematics Studies","work_id":"4d0ccdcd-b45a-4845-a2fb-d00704ad2e05","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"6e0ff4e5c8bc4cc82236904c9a23509149b21f566ec3bd3c627f1bd605a62474","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"639497b2ada20b71594d0ada42360f0762cf4fe15a3bdce5757fef6a1703488e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}