{"paper":{"title":"Kaplansky's second test problem in operator algebras","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Chunlan Jiang, Minghui Ma, Rui Shi, Yuanhang Zhang","submitted_at":"2026-07-22T15:49:53Z","abstract_excerpt":"Kaplansky's second test problem on similarity asks: if $T$ and $S$ are elements in a unital Banach algebra $\\mathcal{B}$ and $T\\oplus T$ is similar to $S\\oplus S$ in $\\mathbb{M}_2(\\mathcal{B})$, is $T$ similar to $S$ in $\\mathcal{B}$?\n  We answer this problem affirmatively if $T$ is an operator with property $(J)$ in a type $\\mathrm{I}_n$ von Neumann algebra $\\mathcal{M}$, i.e., $\\{T\\}'\\cap\\mathcal{M}$ contains a bounded maximal abelian family of idempotents.\n  Moreover, the condition of property $(J)$ can be removed for $1\\leqslant n\\leqslant 3$.\n  A similar result is proved if $T$ is an elem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20593","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.20593/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}