{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2001:AKOWJTCMMD6BZNL6BPOYMYJNBZ","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":"f75ae950de8ffebc511c26765410024b747f477d9a7ebb8525193e642249d221","cross_cats_sorted":["math.OA"],"license":"","primary_cat":"math.FA","submitted_at":"2001-11-20T16:21:14Z","title_canon_sha256":"eee6b5ece09ce498a713f4b9e4c97042251d2b6d51680d40f8e0ccdbbc73c019"},"schema_version":"1.0","source":{"id":"math/0111227","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0111227","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"arxiv_version","alias_value":"math/0111227v3","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0111227","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"pith_short_12","alias_value":"AKOWJTCMMD6B","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"AKOWJTCMMD6BZNL6","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"AKOWJTCM","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:f6171f6c5556dad42241f72b4e9499129f5de8423d3ff7c8f85d3c264cbc47ae","target":"graph","created_at":"2026-05-18T01:05:37Z","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 prove that the following are equivalent for a locally compact group $G$: (i) $G$ is amenable; (ii) $M(G)$ is Connes-amenable; (iii) $M(G)$ has a normal, virtual diagonal.","authors_text":"Volker Runde","cross_cats":["math.OA"],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"2001-11-20T16:21:14Z","title":"Connes-amenability and normal, virtual diagonals for measure algebras, II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0111227","kind":"arxiv","version":3},"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":"sha256:1b7f9006aec6b0fc44aa57bb066c1abd200185aa7c41fe4d824bcca2212fa756","target":"record","created_at":"2026-05-18T01:05:37Z","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":"f75ae950de8ffebc511c26765410024b747f477d9a7ebb8525193e642249d221","cross_cats_sorted":["math.OA"],"license":"","primary_cat":"math.FA","submitted_at":"2001-11-20T16:21:14Z","title_canon_sha256":"eee6b5ece09ce498a713f4b9e4c97042251d2b6d51680d40f8e0ccdbbc73c019"},"schema_version":"1.0","source":{"id":"math/0111227","kind":"arxiv","version":3}},"canonical_sha256":"029d64cc4c60fc1cb57e0bdd86612d0e5cbd81708e297a0390eb432ab6139330","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"029d64cc4c60fc1cb57e0bdd86612d0e5cbd81708e297a0390eb432ab6139330","first_computed_at":"2026-05-18T01:05:37.452994Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:37.452994Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z1dJNO07RzzSeBK728l2CgN6NWm9iw+QZELoioeYQXp73IhZDcQFAAN79nEn+Q9H6c81M1t7ljcXmDzvUDp8DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:37.453506Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0111227","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b7f9006aec6b0fc44aa57bb066c1abd200185aa7c41fe4d824bcca2212fa756","sha256:f6171f6c5556dad42241f72b4e9499129f5de8423d3ff7c8f85d3c264cbc47ae"],"state_sha256":"d7a18c0a6bf2fda89bc953e6e0f4b4c3a81063e357e40c7e9bf187cae5ef94f9"}