{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:NSBVMMU3WUXTLWMORRZXNPERQS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3c19377e35f527a1cbe2857583408711769fee6f531f416add2f8e7f4a2c133e","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2008-05-30T15:22:21Z","title_canon_sha256":"b13000181afb459b193663742fbd3e7f75f2f9df3abc84f5f089277b798207f8"},"schema_version":"1.0","source":{"id":"0805.4788","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0805.4788","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"arxiv_version","alias_value":"0805.4788v2","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0805.4788","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"pith_short_12","alias_value":"NSBVMMU3WUXT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"NSBVMMU3WUXTLWMO","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"NSBVMMU3","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:1941eb433766a33169db3c17c9547be2070c56249a263651b5d253ec78cefb07","target":"graph","created_at":"2026-05-18T04:14:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Spectral morphisms between Banach algebras are useful for comparing their K-theory and their \"noncommutative dimensions\" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.","authors_text":"Bogdan Nica","cross_cats":["math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2008-05-30T15:22:21Z","title":"Relatively spectral morphisms and applications to K-theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0805.4788","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:99c6a9753b71dfa1defc88f82559776ee6928dadaef0424be6f4642311981266","target":"record","created_at":"2026-05-18T04:14:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3c19377e35f527a1cbe2857583408711769fee6f531f416add2f8e7f4a2c133e","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2008-05-30T15:22:21Z","title_canon_sha256":"b13000181afb459b193663742fbd3e7f75f2f9df3abc84f5f089277b798207f8"},"schema_version":"1.0","source":{"id":"0805.4788","kind":"arxiv","version":2}},"canonical_sha256":"6c8356329bb52f35d98e8c7376bc9184a48eaea8e77067a5cac8a13fa09f4372","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c8356329bb52f35d98e8c7376bc9184a48eaea8e77067a5cac8a13fa09f4372","first_computed_at":"2026-05-18T04:14:55.277267Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:55.277267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N3BKqQajgpLo5W8R7oTEaMC9XsQecaVy5BS6JF4zRf/S5abFy5+gBeSa4n9wsc9riocJhuEUl9eO5eLPGQCdAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:55.277722Z","signed_message":"canonical_sha256_bytes"},"source_id":"0805.4788","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99c6a9753b71dfa1defc88f82559776ee6928dadaef0424be6f4642311981266","sha256:1941eb433766a33169db3c17c9547be2070c56249a263651b5d253ec78cefb07"],"state_sha256":"84dd4fbf03dd280d7b9372e11c5b89a6e605c6fc14075589b9f9862df57e4635"}