{"paper":{"title":"Almost-idempotent quantum channels and approximate $C^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","quant-ph"],"primary_cat":"math.OA","authors_text":"Alexei Kitaev","submitted_at":"2024-05-03T18:59:50Z","abstract_excerpt":"Let $\\Phi$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $\\Phi$ is $\\eta$-idempotent, namely, $\\|\\Phi^2-\\Phi\\|_{\\mathrm{cb}} \\le\\eta$, and construct an associated $\\varepsilon$-$C^*$ algebra (of almost-invariant observables) for $\\varepsilon=O(\\eta)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\\varepsilon$. We prove that any finite-dimensional $\\varepsilon$-$C^*$ algebra $A$ is $O(\\varepsilon)$-isomorphic to some genuine $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.02434","kind":"arxiv","version":2},"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/2405.02434/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"}