{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7MFWVPK5VLCPFGLIMDWY74AV7Y","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":"c12af1f8ba8726f0c88887cf9749e7ba193f77ae6ea97aadc9b26226dba2e746","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-01T21:39:17Z","title_canon_sha256":"62b4ff50365aa81f85e60f8dd427c90ada49fdc080847b5c343860655ea206dd"},"schema_version":"1.0","source":{"id":"1609.00409","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.00409","created_at":"2026-05-18T00:42:32Z"},{"alias_kind":"arxiv_version","alias_value":"1609.00409v4","created_at":"2026-05-18T00:42:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00409","created_at":"2026-05-18T00:42:32Z"},{"alias_kind":"pith_short_12","alias_value":"7MFWVPK5VLCP","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7MFWVPK5VLCPFGLI","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7MFWVPK5","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:877b61d29e6eb572b0c2cbd5b307c35a05607c7310416daf7da2b35f504213b1","target":"graph","created_at":"2026-05-18T00:42:32Z","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":"Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier-Whitehead $K$-duality, which is defined on the category of $C^*$-algebras whose $K$-theory is finitely generated and that satisfy the UCT with morphisms the $KK$-groups. We explore what happens when these assumptions are relaxed in various ways. In particular, we consider the relationship between Paschke duality and Spanier-Whitehead $K$-duality.","authors_text":"Claude L. Schochet, Jerome Kaminker","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-01T21:39:17Z","title":"Spanier-Whitehead K-duality for $C^*$-algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00409","kind":"arxiv","version":4},"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":"sha256:2c05061680d6fcface368f1f19b3e810c033a8107c5949f32aa10ddcb4dc99f5","target":"record","created_at":"2026-05-18T00:42:32Z","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":"c12af1f8ba8726f0c88887cf9749e7ba193f77ae6ea97aadc9b26226dba2e746","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-01T21:39:17Z","title_canon_sha256":"62b4ff50365aa81f85e60f8dd427c90ada49fdc080847b5c343860655ea206dd"},"schema_version":"1.0","source":{"id":"1609.00409","kind":"arxiv","version":4}},"canonical_sha256":"fb0b6abd5daac4f2996860ed8ff015fe1c2b544b13edf6d02ee1f04ef37810eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb0b6abd5daac4f2996860ed8ff015fe1c2b544b13edf6d02ee1f04ef37810eb","first_computed_at":"2026-05-18T00:42:32.572627Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:32.572627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q3Y6NQFfS0X3+PpfLX6MWP2Sbjbd0ZukpfLasqSrwvvCuJAwy4o1UcbAtyIEnsII6VUAr0FLPV73a2SYziN/DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:32.573340Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.00409","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c05061680d6fcface368f1f19b3e810c033a8107c5949f32aa10ddcb4dc99f5","sha256:877b61d29e6eb572b0c2cbd5b307c35a05607c7310416daf7da2b35f504213b1"],"state_sha256":"d2d5d1615c3bbc6aff355ab405ff6fa443365e8f5ddcecd979fa8faf09cca7ca"}