{"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. 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