{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:BGCCXEZYYVYLO2D6NPAUJTLPNA","short_pith_number":"pith:BGCCXEZY","canonical_record":{"source":{"id":"1210.5285","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-18T23:57:53Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"9cc5a0b11e1032099b41dc74d3a04e645ab377492ea96816802024b957937016","abstract_canon_sha256":"c8c9e393435c697aaac4848cebc26c32a8690ae5ed8421af6b8d82ef96b79145"},"schema_version":"1.0"},"canonical_sha256":"09842b9338c570b7687e6bc144cd6f6829802c77985aaa67d3a234881501a263","source":{"kind":"arxiv","id":"1210.5285","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5285","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5285v1","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5285","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"pith_short_12","alias_value":"BGCCXEZYYVYL","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"BGCCXEZYYVYLO2D6","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"BGCCXEZY","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:BGCCXEZYYVYLO2D6NPAUJTLPNA","target":"record","payload":{"canonical_record":{"source":{"id":"1210.5285","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-18T23:57:53Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"9cc5a0b11e1032099b41dc74d3a04e645ab377492ea96816802024b957937016","abstract_canon_sha256":"c8c9e393435c697aaac4848cebc26c32a8690ae5ed8421af6b8d82ef96b79145"},"schema_version":"1.0"},"canonical_sha256":"09842b9338c570b7687e6bc144cd6f6829802c77985aaa67d3a234881501a263","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:45.616173Z","signature_b64":"EmHJjuGQJYU5RojMcYxJGEvHXZe0OQ2RSRiWzjAkaUHcihRxVNsAq2wqw4/Q3Wdboc5OFieOFmMIViEq08w2Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09842b9338c570b7687e6bc144cd6f6829802c77985aaa67d3a234881501a263","last_reissued_at":"2026-05-18T03:42:45.615556Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:45.615556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.5285","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:42:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mQcUvZcp38Z+orwUe7QtqseCFSdV+P4XHPcP5SpfA+SgzWFjARh6L6UEPif16VTvZp4og+18OcyV7DuNMulgCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T13:15:10.941960Z"},"content_sha256":"c9f034978dcc7df36d3fbe9d76f95c74887af52f6aef3dd635e5791f667679ef","schema_version":"1.0","event_id":"sha256:c9f034978dcc7df36d3fbe9d76f95c74887af52f6aef3dd635e5791f667679ef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:BGCCXEZYYVYLO2D6NPAUJTLPNA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximate Double Commutants and Distance Formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Don Hadwin, Junhao Shen","submitted_at":"2012-10-18T23:57:53Z","abstract_excerpt":"We extend work of the first author concering relative double commutants and approximate double commutants of unital subalgebras of unital C*-algebras, including metric versions involving distance estimates. We prove metric results for AH subalgebras of von Neumann algebras or AF subalgebras of primitive C*-algebras. We prove other general results, including some for nonselfadjoint commutative subalgebras, using C*-algebraic versions of the Stone-Weierstrass and Bishop-Stone-Weierstrass theorems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5285","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:42:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5oD+fr0hHE19IWtbpRRPmM3Hqvu20EMn2mErrYqvgBPa4cpsjcESA07c5smUd8wxfsntaE+9BVRapqHi1vQcBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T13:15:10.942737Z"},"content_sha256":"dd391d54f424ff657e0d805cf522e184309d584e08bf4a62d01a0553047ea07a","schema_version":"1.0","event_id":"sha256:dd391d54f424ff657e0d805cf522e184309d584e08bf4a62d01a0553047ea07a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BGCCXEZYYVYLO2D6NPAUJTLPNA/bundle.json","state_url":"https://pith.science/pith/BGCCXEZYYVYLO2D6NPAUJTLPNA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BGCCXEZYYVYLO2D6NPAUJTLPNA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-23T13:15:10Z","links":{"resolver":"https://pith.science/pith/BGCCXEZYYVYLO2D6NPAUJTLPNA","bundle":"https://pith.science/pith/BGCCXEZYYVYLO2D6NPAUJTLPNA/bundle.json","state":"https://pith.science/pith/BGCCXEZYYVYLO2D6NPAUJTLPNA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BGCCXEZYYVYLO2D6NPAUJTLPNA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:BGCCXEZYYVYLO2D6NPAUJTLPNA","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":"c8c9e393435c697aaac4848cebc26c32a8690ae5ed8421af6b8d82ef96b79145","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-18T23:57:53Z","title_canon_sha256":"9cc5a0b11e1032099b41dc74d3a04e645ab377492ea96816802024b957937016"},"schema_version":"1.0","source":{"id":"1210.5285","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5285","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5285v1","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5285","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"pith_short_12","alias_value":"BGCCXEZYYVYL","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"BGCCXEZYYVYLO2D6","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"BGCCXEZY","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:dd391d54f424ff657e0d805cf522e184309d584e08bf4a62d01a0553047ea07a","target":"graph","created_at":"2026-05-18T03:42:45Z","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":"We extend work of the first author concering relative double commutants and approximate double commutants of unital subalgebras of unital C*-algebras, including metric versions involving distance estimates. We prove metric results for AH subalgebras of von Neumann algebras or AF subalgebras of primitive C*-algebras. We prove other general results, including some for nonselfadjoint commutative subalgebras, using C*-algebraic versions of the Stone-Weierstrass and Bishop-Stone-Weierstrass theorems.","authors_text":"Don Hadwin, Junhao Shen","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-18T23:57:53Z","title":"Approximate Double Commutants and Distance Formulas"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5285","kind":"arxiv","version":1},"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:c9f034978dcc7df36d3fbe9d76f95c74887af52f6aef3dd635e5791f667679ef","target":"record","created_at":"2026-05-18T03:42:45Z","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":"c8c9e393435c697aaac4848cebc26c32a8690ae5ed8421af6b8d82ef96b79145","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-18T23:57:53Z","title_canon_sha256":"9cc5a0b11e1032099b41dc74d3a04e645ab377492ea96816802024b957937016"},"schema_version":"1.0","source":{"id":"1210.5285","kind":"arxiv","version":1}},"canonical_sha256":"09842b9338c570b7687e6bc144cd6f6829802c77985aaa67d3a234881501a263","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09842b9338c570b7687e6bc144cd6f6829802c77985aaa67d3a234881501a263","first_computed_at":"2026-05-18T03:42:45.615556Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:45.615556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EmHJjuGQJYU5RojMcYxJGEvHXZe0OQ2RSRiWzjAkaUHcihRxVNsAq2wqw4/Q3Wdboc5OFieOFmMIViEq08w2Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:45.616173Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.5285","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c9f034978dcc7df36d3fbe9d76f95c74887af52f6aef3dd635e5791f667679ef","sha256:dd391d54f424ff657e0d805cf522e184309d584e08bf4a62d01a0553047ea07a"],"state_sha256":"328a82ad2bdd467bb97d4ebede1038a600c26e81f074f0a60dd85d31e30a01c3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O6t7QiplZf+TBk+XH/1OvKZUY9Vtqux/gBkt13UOBBADBW4iPKF0SSeqkrC4sY+aqHgi2kTJjkm57snQX8oMDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T13:15:10.946955Z","bundle_sha256":"cfd9c16f6496c76b9805f604dff5ca5782c09fe992c7f561df11778ec6b674da"}}