{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:HLMP6E3N5T7DLGL3UDTOTBAV4G","short_pith_number":"pith:HLMP6E3N","schema_version":"1.0","canonical_sha256":"3ad8ff136decfe35997ba0e6e98415e19c9f1f9e84303a3250660a3b43a83a96","source":{"kind":"arxiv","id":"1210.3133","version":2},"attestation_state":"computed","paper":{"title":"Pictures of KK-theory for real C*-algebras and almost commuting matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Efren Ruiz, Jeffrey L. Boersema, Terry A. Loring","submitted_at":"2012-10-11T06:03:39Z","abstract_excerpt":"We give a systematic account of the various pictures of KK-theory for real C*-algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of KK-theory, and we use CRT-structures to prove that a natural transformation from F(A) to G(A) between homotopy equivalent, stable, half-exact functors defined on real C*-algebras is an isomorphism provided it is an isomorphism on the smaller class of C*-algebras. Finally, we develop E-theory for real C*-algebras and use that to obtain new negative results regarding th"},"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":"1210.3133","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-11T06:03:39Z","cross_cats_sorted":[],"title_canon_sha256":"180eb55852368870a654958245a4944ca58aebc6289f490770bb95c0d0b5d5cb","abstract_canon_sha256":"52b39fc1acfcaf49a060f050969cc642628a0b35a1d72449a1b34fab3a96e566"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:08.993483Z","signature_b64":"kbSnl8/YPCvHT9noPvJ4BV04uVf7sEX6Ot2VepIu2J7MX41NVgt7tRPeNvvFRA+18ntQVzBLIw5jzkJHVeRFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ad8ff136decfe35997ba0e6e98415e19c9f1f9e84303a3250660a3b43a83a96","last_reissued_at":"2026-05-18T01:25:08.992981Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:08.992981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pictures of KK-theory for real C*-algebras and almost commuting matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Efren Ruiz, Jeffrey L. Boersema, Terry A. Loring","submitted_at":"2012-10-11T06:03:39Z","abstract_excerpt":"We give a systematic account of the various pictures of KK-theory for real C*-algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of KK-theory, and we use CRT-structures to prove that a natural transformation from F(A) to G(A) between homotopy equivalent, stable, half-exact functors defined on real C*-algebras is an isomorphism provided it is an isomorphism on the smaller class of C*-algebras. Finally, we develop E-theory for real C*-algebras and use that to obtain new negative results regarding th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3133","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":"1210.3133","created_at":"2026-05-18T01:25:08.993054+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.3133v2","created_at":"2026-05-18T01:25:08.993054+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.3133","created_at":"2026-05-18T01:25:08.993054+00:00"},{"alias_kind":"pith_short_12","alias_value":"HLMP6E3N5T7D","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_16","alias_value":"HLMP6E3N5T7DLGL3","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_8","alias_value":"HLMP6E3N","created_at":"2026-05-18T12:27:09.501522+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/HLMP6E3N5T7DLGL3UDTOTBAV4G","json":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G.json","graph_json":"https://pith.science/api/pith-number/HLMP6E3N5T7DLGL3UDTOTBAV4G/graph.json","events_json":"https://pith.science/api/pith-number/HLMP6E3N5T7DLGL3UDTOTBAV4G/events.json","paper":"https://pith.science/paper/HLMP6E3N"},"agent_actions":{"view_html":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G","download_json":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G.json","view_paper":"https://pith.science/paper/HLMP6E3N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.3133&json=true","fetch_graph":"https://pith.science/api/pith-number/HLMP6E3N5T7DLGL3UDTOTBAV4G/graph.json","fetch_events":"https://pith.science/api/pith-number/HLMP6E3N5T7DLGL3UDTOTBAV4G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G/action/storage_attestation","attest_author":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G/action/author_attestation","sign_citation":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G/action/citation_signature","submit_replication":"https://pith.science/pith/HLMP6E3N5T7DLGL3UDTOTBAV4G/action/replication_record"}},"created_at":"2026-05-18T01:25:08.993054+00:00","updated_at":"2026-05-18T01:25:08.993054+00:00"}