{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:CLROYBLFHH5TWLKFHPJABVGNHP","short_pith_number":"pith:CLROYBLF","schema_version":"1.0","canonical_sha256":"12e2ec056539fb3b2d453bd200d4cd3bf11fa143daa4392242ffa2f33f38cd03","source":{"kind":"arxiv","id":"2601.08779","version":2},"attestation_state":"computed","paper":{"title":"Uniqueness for embeddings of nuclear $C^*$-algebras into type II$_{1}$ factors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Unital full nuclear maps from separable exact C*-algebras satisfying the UCT into ultraproducts of finite factors are unitarily equivalent whenever they agree on traces and total K-theory.","cross_cats":[],"primary_cat":"math.OA","authors_text":"Shanshan Hua, Stuart White","submitted_at":"2026-01-13T18:09:47Z","abstract_excerpt":"Let $A$ be a separable, unital and exact $C^*$-algebra satisfying the universal coefficient theorem. We prove uniqueness theorems up to unitary conjugacy for unital, full and nuclear maps from $A$ into ultraproducts of finite von Neumann factors: any two such maps agreeing on traces and total $K$-theory are unitarily equivalent. There are two consequences. Firstly if one takes the factors to be a sequence $(M_{k_n})_{n}$ of matrix algebras, we obtain a uniqueness result for quasidiagonal approximations of $A$. Secondly, when $(\\mathcal M,\\tau_{\\calM})$ is a II$_1$ factor, a pair $\\phi,\\psi:A\\t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2601.08779","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2026-01-13T18:09:47Z","cross_cats_sorted":[],"title_canon_sha256":"59cd0c4eff394e28330d78342bcc4d697e4cb2c910719f2cbb1c03780c308860","abstract_canon_sha256":"e532b73ba7aecf8352cb0200774825b53de0ff14d58ad4a57ea4c4a2d54ff4c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:16.665131Z","signature_b64":"IT5BE4LHYii2/k1EHe/WCLT3kLW0Vij4AX/ROkRJSkYgZJ4BK/6q83pmvqjvRiDPwu10aYyhLRxRFDVTrGziDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12e2ec056539fb3b2d453bd200d4cd3bf11fa143daa4392242ffa2f33f38cd03","last_reissued_at":"2026-05-17T23:39:16.664284Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:16.664284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness for embeddings of nuclear $C^*$-algebras into type II$_{1}$ factors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Unital full nuclear maps from separable exact C*-algebras satisfying the UCT into ultraproducts of finite factors are unitarily equivalent whenever they agree on traces and total K-theory.","cross_cats":[],"primary_cat":"math.OA","authors_text":"Shanshan Hua, Stuart White","submitted_at":"2026-01-13T18:09:47Z","abstract_excerpt":"Let $A$ be a separable, unital and exact $C^*$-algebra satisfying the universal coefficient theorem. We prove uniqueness theorems up to unitary conjugacy for unital, full and nuclear maps from $A$ into ultraproducts of finite von Neumann factors: any two such maps agreeing on traces and total $K$-theory are unitarily equivalent. There are two consequences. Firstly if one takes the factors to be a sequence $(M_{k_n})_{n}$ of matrix algebras, we obtain a uniqueness result for quasidiagonal approximations of $A$. Secondly, when $(\\mathcal M,\\tau_{\\calM})$ is a II$_1$ factor, a pair $\\phi,\\psi:A\\t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"any two such maps agreeing on traces and total K-theory are unitarily equivalent","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"A is a separable, unital and exact C*-algebra satisfying the universal coefficient theorem; the maps are unital, full and nuclear","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Uniqueness up to unitary conjugacy holds for nuclear maps from separable exact C*-algebras satisfying the UCT into ultraproducts of finite factors when the maps agree on traces and total K-theory.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Unital full nuclear maps from separable exact C*-algebras satisfying the UCT into ultraproducts of finite factors are unitarily equivalent whenever they agree on traces and total K-theory.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b5139bf4f147776c14409486273cbacbc48fa2ac3d222d2b1cc9ea0b9f251f10"},"source":{"id":"2601.08779","kind":"arxiv","version":2},"verdict":{"id":"09164201-eefc-4e91-848a-7f38ac7263a9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T14:32:47.188938Z","strongest_claim":"any two such maps agreeing on traces and total K-theory are unitarily equivalent","one_line_summary":"Uniqueness up to unitary conjugacy holds for nuclear maps from separable exact C*-algebras satisfying the UCT into ultraproducts of finite factors when the maps agree on traces and total K-theory.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"A is a separable, unital and exact C*-algebra satisfying the universal coefficient theorem; the maps are unital, full and nuclear","pith_extraction_headline":"Unital full nuclear maps from separable exact C*-algebras satisfying the UCT into ultraproducts of finite factors are unitarily equivalent whenever they agree on traces and total K-theory."},"references":{"count":82,"sample":[{"doi":"","year":2011,"title":"R. Antoine, J. Bosa, and F. Perera. Completions of monoids with applications to the Cuntz semigroup.Internat. J. Math., 22(6):837–861, 2011","work_id":"aa1b5db1-55f2-40e7-be36-c65379689bba","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1977,"title":"W. Arveson. Notes on extensions ofC∗-algebras.Duke Math. J., 44(2):329–355, 1977","work_id":"116b4de9-8315-46ff-8bbf-165347cf2fc8","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"S. Barlak and X. Li. Cartan subalgebras and the UCT problem.Adv. Math., 316:748– 769, 2017","work_id":"72c06e8f-e2b1-4dbd-9e9e-ef5526b0b9e9","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Blackadar.K-theory for operator algebras, volume 5 ofMathematical Sciences Re- search Institute Publications","work_id":"6f4d392a-7e69-492c-ac58-f4173fe0f6b2","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"Blackadar.Operator algebras, volume 122 ofEncyclopaedia of Mathematical Sci- ences","work_id":"c54dd820-f49c-40e0-8e4a-8235d2bc56f2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":82,"snapshot_sha256":"955d3435eeee4b8a128a9acf1037eaf6aa0226da506da0e9372cd50601a877d9","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"077e71ecfb573a72eb5b0cd0a5f06ab0ae671ee8e50ef7f3d238577f0737ca37"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2601.08779","created_at":"2026-05-17T23:39:16.664441+00:00"},{"alias_kind":"arxiv_version","alias_value":"2601.08779v2","created_at":"2026-05-17T23:39:16.664441+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.08779","created_at":"2026-05-17T23:39:16.664441+00:00"},{"alias_kind":"pith_short_12","alias_value":"CLROYBLFHH5T","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"CLROYBLFHH5TWLKF","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"CLROYBLF","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP","json":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP.json","graph_json":"https://pith.science/api/pith-number/CLROYBLFHH5TWLKFHPJABVGNHP/graph.json","events_json":"https://pith.science/api/pith-number/CLROYBLFHH5TWLKFHPJABVGNHP/events.json","paper":"https://pith.science/paper/CLROYBLF"},"agent_actions":{"view_html":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP","download_json":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP.json","view_paper":"https://pith.science/paper/CLROYBLF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2601.08779&json=true","fetch_graph":"https://pith.science/api/pith-number/CLROYBLFHH5TWLKFHPJABVGNHP/graph.json","fetch_events":"https://pith.science/api/pith-number/CLROYBLFHH5TWLKFHPJABVGNHP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP/action/storage_attestation","attest_author":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP/action/author_attestation","sign_citation":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP/action/citation_signature","submit_replication":"https://pith.science/pith/CLROYBLFHH5TWLKFHPJABVGNHP/action/replication_record"}},"created_at":"2026-05-17T23:39:16.664441+00:00","updated_at":"2026-05-17T23:39:16.664441+00:00"}