{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:WOLCXTQ2ZI3NR34TSAD6COV5CB","short_pith_number":"pith:WOLCXTQ2","schema_version":"1.0","canonical_sha256":"b3962bce1aca36d8ef939007e13abd1053f50c2536e311f56c2be5ad8df9aaec","source":{"kind":"arxiv","id":"0904.4383","version":5},"attestation_state":"computed","paper":{"title":"Unbounded bivariant $K$-theory and correspondences in noncommutative geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.KT","authors_text":"Bram Mesland","submitted_at":"2009-04-28T12:50:08Z","abstract_excerpt":"By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The theory of operator spaces provides the required tools. Finally, the above mentioned $KK$-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples."},"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":"0904.4383","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2009-04-28T12:50:08Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"64099a1af23c0200994431f474b810ad2f2e191f655972095ba8b4aebaf68b2c","abstract_canon_sha256":"9ed8dbc251b6bd64b2a20e7619ff00910ede20f1fb5bb2894dfaef5c180c837f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:05.536859Z","signature_b64":"nhRaT4nadkqHym0o83DeS380UtCpkfCZ1450chAZ8DDDcMvfLrIPwrVMsRHg0d6pEq0X4tdF3TeKzpAOyR0FBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3962bce1aca36d8ef939007e13abd1053f50c2536e311f56c2be5ad8df9aaec","last_reissued_at":"2026-05-18T02:54:05.536139Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:05.536139Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unbounded bivariant $K$-theory and correspondences in noncommutative geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.KT","authors_text":"Bram Mesland","submitted_at":"2009-04-28T12:50:08Z","abstract_excerpt":"By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The theory of operator spaces provides the required tools. Finally, the above mentioned $KK$-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.4383","kind":"arxiv","version":5},"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":"0904.4383","created_at":"2026-05-18T02:54:05.536251+00:00"},{"alias_kind":"arxiv_version","alias_value":"0904.4383v5","created_at":"2026-05-18T02:54:05.536251+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.4383","created_at":"2026-05-18T02:54:05.536251+00:00"},{"alias_kind":"pith_short_12","alias_value":"WOLCXTQ2ZI3N","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"WOLCXTQ2ZI3NR34T","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"WOLCXTQ2","created_at":"2026-05-18T12:26:02.257875+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/WOLCXTQ2ZI3NR34TSAD6COV5CB","json":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB.json","graph_json":"https://pith.science/api/pith-number/WOLCXTQ2ZI3NR34TSAD6COV5CB/graph.json","events_json":"https://pith.science/api/pith-number/WOLCXTQ2ZI3NR34TSAD6COV5CB/events.json","paper":"https://pith.science/paper/WOLCXTQ2"},"agent_actions":{"view_html":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB","download_json":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB.json","view_paper":"https://pith.science/paper/WOLCXTQ2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0904.4383&json=true","fetch_graph":"https://pith.science/api/pith-number/WOLCXTQ2ZI3NR34TSAD6COV5CB/graph.json","fetch_events":"https://pith.science/api/pith-number/WOLCXTQ2ZI3NR34TSAD6COV5CB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB/action/storage_attestation","attest_author":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB/action/author_attestation","sign_citation":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB/action/citation_signature","submit_replication":"https://pith.science/pith/WOLCXTQ2ZI3NR34TSAD6COV5CB/action/replication_record"}},"created_at":"2026-05-18T02:54:05.536251+00:00","updated_at":"2026-05-18T02:54:05.536251+00:00"}