{"paper":{"title":"Relative Kubo-Ando Means of Completely Positive Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Relative and intrinsic Kubo-Ando means extend operator means to completely positive maps on C*-algebras via Arveson's Radon-Nikodym theorem.","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Mohsen Kian","submitted_at":"2026-05-12T07:58:03Z","abstract_excerpt":"We develop a Kubo--Ando theory on order intervals of completely positive maps. Using Arveson's Radon--Nikodym theorem as a structural tool, we define relative Kubo--Ando means \\(\\Phi\\sigma_\\Omega\\Psi\\) for completely positive maps dominated by a common ambient map \\(\\Omega\\). The special choice \\(\\Omega=\\Phi+\\Psi\\) yields an intrinsic mean of two completely positive maps.\n  We prove that these means are independent of the chosen Stinespring representation and satisfy the expected order-theoretic properties, including monotonicity, transformer inequalities, Jensen-type inequalities, data proces"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We introduce relative and intrinsic Kubo--Ando means for completely positive maps on C*-algebras. These means extend the usual Kubo--Ando means of positive operators and are defined using Arveson's Radon--Nikodym theorem for completely positive maps. We prove their basic order-theoretic properties, including monotonicity, transformer and Jensen inequalities, data processing, and monotonicity with respect to the ambient map.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Arveson's Radon-Nikodym theorem applies directly to the pairs of completely positive maps under consideration and yields a well-defined derivative that can be used to construct the means without additional restrictions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Relative and intrinsic Kubo-Ando means are introduced for completely positive maps, satisfying order properties and reducing to prior means on matrix algebras and common domains.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Relative and intrinsic Kubo-Ando means extend operator means to completely positive maps on C*-algebras via Arveson's Radon-Nikodym theorem.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bcc226909dac90dbc794514c229290050a488699299aff3342f38dbe389eb834"},"source":{"id":"2605.11701","kind":"arxiv","version":2},"verdict":{"id":"e77a508b-f9d2-411b-b10c-02b2f08d59bb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T04:46:45.663768Z","strongest_claim":"We introduce relative and intrinsic Kubo--Ando means for completely positive maps on C*-algebras. These means extend the usual Kubo--Ando means of positive operators and are defined using Arveson's Radon--Nikodym theorem for completely positive maps. We prove their basic order-theoretic properties, including monotonicity, transformer and Jensen inequalities, data processing, and monotonicity with respect to the ambient map.","one_line_summary":"Relative and intrinsic Kubo-Ando means are introduced for completely positive maps, satisfying order properties and reducing to prior means on matrix algebras and common domains.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Arveson's Radon-Nikodym theorem applies directly to the pairs of completely positive maps under consideration and yields a well-defined derivative that can be used to construct the means without additional restrictions.","pith_extraction_headline":"Relative and intrinsic Kubo-Ando means extend operator means to completely positive maps on C*-algebras via Arveson's Radon-Nikodym theorem."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.11701/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T11:39:35.526611Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T09:31:17.327218Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T08:12:54.381545Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c6e3def9bacbad6e7cddd40447a22a59d3ff826f73bbc54117e1b11ce10a8cb9"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"d6c37904036a002609c8da5bb6f8fda223a76814b990415465d9f3042428a35e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}