{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:467DWYADUQ4G72TPWHJ24XBU54","short_pith_number":"pith:467DWYAD","schema_version":"1.0","canonical_sha256":"e7be3b6003a4386fea6fb1d3ae5c34ef323e442480e8be0412b8975bc6a6874f","source":{"kind":"arxiv","id":"0712.1842","version":2},"attestation_state":"computed","paper":{"title":"A Note on Kasparov Product and Duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.OA","authors_text":"Hyun Ho Lee","submitted_at":"2007-12-11T22:53:13Z","abstract_excerpt":"Using Paschke-Higson duality, we can get a natural index pairing $K_{i}(A) \\times K_{i+1}(D_{\\Phi}) \\to \\boldsymbol{Z} \\quad (i=0,1) (\\mbox{mod}2)$, where $A$ is a separable $C\\sp*$-algebra, and $\\Phi$ is a representation of $A$ on a separable infinite dimensional Hilbert space $H$. It is proved that this is a special case of the Kasparov Product. As a step, we show a proof of Bott-periodicity for KK-theory asserting that $\\mathbb{C}_1$ and $S$ are $KK$-equivalent using the odd index pairing."},"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":"0712.1842","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2007-12-11T22:53:13Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"168f4c19720cc6bffa5c3fb6bb95499a0e7023d1c2af0d266f32aaacf385643e","abstract_canon_sha256":"6373a4dbc082ed53b95e1fed2bc40bf53a03958ee5da6c4baa5f54bdfef2d783"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:04.061720Z","signature_b64":"KS5rSXxp0CaSdCT0xVs4MIcPzPg/2MkMxipmO4ZsshoJjEzLYd3gxzSQwxdUlrDgcXeW9K86MxhlcMO6MQHbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7be3b6003a4386fea6fb1d3ae5c34ef323e442480e8be0412b8975bc6a6874f","last_reissued_at":"2026-05-18T04:41:04.061228Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:04.061228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Note on Kasparov Product and Duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.OA","authors_text":"Hyun Ho Lee","submitted_at":"2007-12-11T22:53:13Z","abstract_excerpt":"Using Paschke-Higson duality, we can get a natural index pairing $K_{i}(A) \\times K_{i+1}(D_{\\Phi}) \\to \\boldsymbol{Z} \\quad (i=0,1) (\\mbox{mod}2)$, where $A$ is a separable $C\\sp*$-algebra, and $\\Phi$ is a representation of $A$ on a separable infinite dimensional Hilbert space $H$. It is proved that this is a special case of the Kasparov Product. As a step, we show a proof of Bott-periodicity for KK-theory asserting that $\\mathbb{C}_1$ and $S$ are $KK$-equivalent using the odd index pairing."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.1842","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":""},"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":"0712.1842","created_at":"2026-05-18T04:41:04.061309+00:00"},{"alias_kind":"arxiv_version","alias_value":"0712.1842v2","created_at":"2026-05-18T04:41:04.061309+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.1842","created_at":"2026-05-18T04:41:04.061309+00:00"},{"alias_kind":"pith_short_12","alias_value":"467DWYADUQ4G","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"467DWYADUQ4G72TP","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"467DWYAD","created_at":"2026-05-18T12:25:54.717736+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/467DWYADUQ4G72TPWHJ24XBU54","json":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54.json","graph_json":"https://pith.science/api/pith-number/467DWYADUQ4G72TPWHJ24XBU54/graph.json","events_json":"https://pith.science/api/pith-number/467DWYADUQ4G72TPWHJ24XBU54/events.json","paper":"https://pith.science/paper/467DWYAD"},"agent_actions":{"view_html":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54","download_json":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54.json","view_paper":"https://pith.science/paper/467DWYAD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0712.1842&json=true","fetch_graph":"https://pith.science/api/pith-number/467DWYADUQ4G72TPWHJ24XBU54/graph.json","fetch_events":"https://pith.science/api/pith-number/467DWYADUQ4G72TPWHJ24XBU54/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54/action/timestamp_anchor","attest_storage":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54/action/storage_attestation","attest_author":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54/action/author_attestation","sign_citation":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54/action/citation_signature","submit_replication":"https://pith.science/pith/467DWYADUQ4G72TPWHJ24XBU54/action/replication_record"}},"created_at":"2026-05-18T04:41:04.061309+00:00","updated_at":"2026-05-18T04:41:04.061309+00:00"}