{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:5Y2ZU4TCIFTYT7PBGJHBX552IN","short_pith_number":"pith:5Y2ZU4TC","schema_version":"1.0","canonical_sha256":"ee359a7262416789fde1324e1bf7ba4358b7663698f13e9b0ea04ae94d3a5f27","source":{"kind":"arxiv","id":"1605.08843","version":1},"attestation_state":"computed","paper":{"title":"Balanced pairs of operators and their relative index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.KT","authors_text":"Vladimir Manuilov","submitted_at":"2016-05-28T04:28:30Z","abstract_excerpt":"We show that the K_1 group of a C*-algebra $A$ can be defined as homotopy classes of pairs, called balanced, of not necessarily unitary matrices over $A$ that have equal defects from being unitary. We also consider pairs of order zero pseudodifferential operators, not necessarily elliptic, with symbols being a balanced pair. A relative index is defined for such pairs of operators and it equals the topological index of the pair of their symbols."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1605.08843","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-05-28T04:28:30Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"e37128859e3e25a9f60baa5ca27af2be1d58ccf6b0303f63f72c294ce99cff42","abstract_canon_sha256":"4602b70c9379064741ae10c3b8304f85206e51879c38eb0f0a64022a937b2c6b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:22.937464Z","signature_b64":"dp8F51Raru5/CwoQIV6UOyZ5tjtZPmr8nmTOCxlQP+dBpgTKM5lW9GUz2N6ET7jQ6DsQkkhc8Vu/u6pouOGUBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee359a7262416789fde1324e1bf7ba4358b7663698f13e9b0ea04ae94d3a5f27","last_reissued_at":"2026-05-18T01:13:22.936790Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:22.936790Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Balanced pairs of operators and their relative index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.KT","authors_text":"Vladimir Manuilov","submitted_at":"2016-05-28T04:28:30Z","abstract_excerpt":"We show that the K_1 group of a C*-algebra $A$ can be defined as homotopy classes of pairs, called balanced, of not necessarily unitary matrices over $A$ that have equal defects from being unitary. We also consider pairs of order zero pseudodifferential operators, not necessarily elliptic, with symbols being a balanced pair. A relative index is defined for such pairs of operators and it equals the topological index of the pair of their symbols."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.08843","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.08843","created_at":"2026-05-18T01:13:22.936892+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.08843v1","created_at":"2026-05-18T01:13:22.936892+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.08843","created_at":"2026-05-18T01:13:22.936892+00:00"},{"alias_kind":"pith_short_12","alias_value":"5Y2ZU4TCIFTY","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"5Y2ZU4TCIFTYT7PB","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"5Y2ZU4TC","created_at":"2026-05-18T12:30:01.593930+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN","json":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN.json","graph_json":"https://pith.science/api/pith-number/5Y2ZU4TCIFTYT7PBGJHBX552IN/graph.json","events_json":"https://pith.science/api/pith-number/5Y2ZU4TCIFTYT7PBGJHBX552IN/events.json","paper":"https://pith.science/paper/5Y2ZU4TC"},"agent_actions":{"view_html":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN","download_json":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN.json","view_paper":"https://pith.science/paper/5Y2ZU4TC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.08843&json=true","fetch_graph":"https://pith.science/api/pith-number/5Y2ZU4TCIFTYT7PBGJHBX552IN/graph.json","fetch_events":"https://pith.science/api/pith-number/5Y2ZU4TCIFTYT7PBGJHBX552IN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN/action/storage_attestation","attest_author":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN/action/author_attestation","sign_citation":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN/action/citation_signature","submit_replication":"https://pith.science/pith/5Y2ZU4TCIFTYT7PBGJHBX552IN/action/replication_record"}},"created_at":"2026-05-18T01:13:22.936892+00:00","updated_at":"2026-05-18T01:13:22.936892+00:00"}