{"paper":{"title":"On the Nielsen-Thomsen sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Nielsen-Thomsen bases and rotation maps enable comparison of *-homomorphisms at the Hausdorffized algebraic K1 level and yield a new proof that two AT-algebras are non-isomorphic.","cross_cats":[],"primary_cat":"math.OA","authors_text":"Laurent Cantier","submitted_at":"2024-12-16T16:58:53Z","abstract_excerpt":"The Nielsen-Thomsen sequence plays a pivotal role in refining invariants for C$^*$-algebras beyond the Elliott classification framework. This paper revisits the sequence, introducing the concepts of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms, to better understand its unnatural splitting. These insights enable novel comparison methods for *-homomorphisms at the level of the Hausdorffized algebraic K$_1$-groups, and subsequently the Hausdorffized unitary Cuntz group. We apply our methods to classification via the Hausdorffized unitary Cuntz semigroup. In particular, we pre"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"These insights enable novel comparison methods for *-homomorphisms at the level of the Hausdorffized algebraic K1-groups, and subsequently the Hausdorffized unitary Cuntz group. We apply our methods to classification via the Hausdorffized unitary Cuntz semigroup. In particular, we present a new proof of the non-isomorphism between two AT-algebras constructed by Gong, Jiang and Li.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The newly introduced notions of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms correctly capture the properties of the sequence's unnatural splitting and thereby validate the claimed comparison methods (abstract, paragraph 2).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Develops new algebraic tools for the Nielsen-Thomsen sequence to compare *-homomorphisms at Hausdorffized K1 and Cuntz levels and applies them to C*-algebra classification.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Nielsen-Thomsen bases and rotation maps enable comparison of *-homomorphisms at the Hausdorffized algebraic K1 level and yield a new proof that two AT-algebras are non-isomorphic.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"36fae872cc7db5c05540fdd6b04714693c773433cc355adb0a5b1e0c11393089"},"source":{"id":"2412.11975","kind":"arxiv","version":4},"verdict":{"id":"27761fa4-08da-4ccc-a8bc-24d9ac6eccc2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T07:26:16.776540Z","strongest_claim":"These insights enable novel comparison methods for *-homomorphisms at the level of the Hausdorffized algebraic K1-groups, and subsequently the Hausdorffized unitary Cuntz group. We apply our methods to classification via the Hausdorffized unitary Cuntz semigroup. In particular, we present a new proof of the non-isomorphism between two AT-algebras constructed by Gong, Jiang and Li.","one_line_summary":"Develops new algebraic tools for the Nielsen-Thomsen sequence to compare *-homomorphisms at Hausdorffized K1 and Cuntz levels and applies them to C*-algebra classification.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The newly introduced notions of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms correctly capture the properties of the sequence's unnatural splitting and thereby validate the claimed comparison methods (abstract, paragraph 2).","pith_extraction_headline":"Nielsen-Thomsen bases and rotation maps enable comparison of *-homomorphisms at the Hausdorffized algebraic K1 level and yield a new proof that two AT-algebras are non-isomorphic."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.11975/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"}