{"paper":{"title":"On Multiplicity of Uniform Norms and Maximal Spectral Substructures in Commutative Banach Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Semisimple commutative Banach algebras have either a unique uniform norm or uncountably many, plus a largest weakly regular closed subalgebra and largest closed ideals with UUNP and SEP.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jekwin J. Dabhi, Prakash A. Dabhi","submitted_at":"2026-05-18T10:21:19Z","abstract_excerpt":"Let $\\mathcal A$ be a semisimple commutative Banach algebra. It is shown that either $\\mathcal A$ has exactly one uniform norm or it admits uncountably many uniform norms. Further, it is shown that there always exists a largest closed subalgebra of $\\mathcal A$ which is weakly regular, and that there always exist largest closed ideals in $\\mathcal A$ having unique uniform norm property (UUNP) and spectral extension property (SEP) respectively."},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"Let A be a semisimple commutative Banach algebra. It is shown that either A has exactly one uniform norm or it admits uncountably many uniform norms.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The algebra A is assumed to be semisimple (Jacobson radical is zero) and commutative, which is the setting stated at the start of the abstract for all subsequent claims about uniform norms and maximal substructures.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Semisimple commutative Banach algebras have either a unique uniform norm or uncountably many, plus a largest weakly regular closed subalgebra and largest closed ideals with UUNP and SEP.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"1038e8cdc7135bdb305b7aa19929cd8399dad74f05d0d04273f4f71c0acd4398"},"source":{"id":"2605.18179","kind":"arxiv","version":1},"verdict":{"id":"9c56e230-3b1b-4005-bc1f-25c3351a4925","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-20T00:05:16.976290Z","strongest_claim":"Let A be a semisimple commutative Banach algebra. It is shown that either A has exactly one uniform norm or it admits uncountably many uniform norms.","one_line_summary":"Semisimple commutative Banach algebras have either a unique uniform norm or uncountably many, plus a largest weakly regular closed subalgebra and largest closed ideals with UUNP and SEP.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The algebra A is assumed to be semisimple (Jacobson radical is zero) and commutative, which is the setting stated at the start of the abstract for all subsequent claims about uniform norms and maximal substructures.","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18179/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T23:41:59.032029Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.344536Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"88a65b254aed5f689980d5eb15d8558b95378af7403af56d878569a5717d5fa1"},"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"}