{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:I6QOAIZT6LJXDGB3BYTPBRS3AF","short_pith_number":"pith:I6QOAIZT","canonical_record":{"source":{"id":"1405.5820","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-05-22T16:29:05Z","cross_cats_sorted":["math.FA","math.GR"],"title_canon_sha256":"16111f8a1ca67703705532323a12400816289ba076bf59db31d329f7b45e99b6","abstract_canon_sha256":"799064c416c3d8c147b90b37afebf903f18343947062c87962beff493ef0cfd8"},"schema_version":"1.0"},"canonical_sha256":"47a0e02333f2d371983b0e26f0c65b01426d0baa2e180a9e2e613d653d768903","source":{"kind":"arxiv","id":"1405.5820","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.5820","created_at":"2026-05-18T01:04:33Z"},{"alias_kind":"arxiv_version","alias_value":"1405.5820v2","created_at":"2026-05-18T01:04:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.5820","created_at":"2026-05-18T01:04:33Z"},{"alias_kind":"pith_short_12","alias_value":"I6QOAIZT6LJX","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"I6QOAIZT6LJXDGB3","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"I6QOAIZT","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:I6QOAIZT6LJXDGB3BYTPBRS3AF","target":"record","payload":{"canonical_record":{"source":{"id":"1405.5820","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-05-22T16:29:05Z","cross_cats_sorted":["math.FA","math.GR"],"title_canon_sha256":"16111f8a1ca67703705532323a12400816289ba076bf59db31d329f7b45e99b6","abstract_canon_sha256":"799064c416c3d8c147b90b37afebf903f18343947062c87962beff493ef0cfd8"},"schema_version":"1.0"},"canonical_sha256":"47a0e02333f2d371983b0e26f0c65b01426d0baa2e180a9e2e613d653d768903","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:33.595946Z","signature_b64":"OYIfoXiMBmFgc5t9O9FUv95VvUvpKw+LPy4r7qPVCrSZAIwH6mYVbvvdziYXRSGRlpXIDQPLCKKi/2kTHWCwBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47a0e02333f2d371983b0e26f0c65b01426d0baa2e180a9e2e613d653d768903","last_reissued_at":"2026-05-18T01:04:33.595164Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:33.595164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.5820","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:04:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"if7w1P5zkUhqOcbXYI7E4sv8mGCUqoQyIKjSH5bvQsiVw7J+zeke4BZeftomjHyotK8yxVlJYFJN45CaP41VAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:39:21.940091Z"},"content_sha256":"5cd7e38499327c3659e084420dd819f25c11d7368de1b48f6af3e39f86d8e9f4","schema_version":"1.0","event_id":"sha256:5cd7e38499327c3659e084420dd819f25c11d7368de1b48f6af3e39f86d8e9f4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:I6QOAIZT6LJXDGB3BYTPBRS3AF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The weak Haagerup property II: Examples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GR"],"primary_cat":"math.OA","authors_text":"S{\\o}ren Knudby, Uffe Haagerup","submitted_at":"2014-05-22T16:29:05Z","abstract_excerpt":"The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author and the Haagerup property introduced by Connes and Choda.\n  In this paper it is shown that a connected simple Lie group G has the weak Haagerup property if and only if the real rank of G is zero or one. Hence for connected simple Lie groups the weak Haagerup property coincides with weak amenability. Moreover, it turns out that for connected simple Lie groups t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5820","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:04:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cnjN0iLENQnvNb7GG7fklH7FsNeCVwz12T4Ammt5FLFXAlsL4X4bP/2MKmiLBZdO24rUnMtRfAvUraG9nvJ5AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:39:21.940722Z"},"content_sha256":"2a3162ca1a21aa2c6ebb1df99b97ba582d53dd186654564fbf733f61ba59ea97","schema_version":"1.0","event_id":"sha256:2a3162ca1a21aa2c6ebb1df99b97ba582d53dd186654564fbf733f61ba59ea97"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I6QOAIZT6LJXDGB3BYTPBRS3AF/bundle.json","state_url":"https://pith.science/pith/I6QOAIZT6LJXDGB3BYTPBRS3AF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I6QOAIZT6LJXDGB3BYTPBRS3AF/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":"2026-06-04T19:39:21Z","links":{"resolver":"https://pith.science/pith/I6QOAIZT6LJXDGB3BYTPBRS3AF","bundle":"https://pith.science/pith/I6QOAIZT6LJXDGB3BYTPBRS3AF/bundle.json","state":"https://pith.science/pith/I6QOAIZT6LJXDGB3BYTPBRS3AF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I6QOAIZT6LJXDGB3BYTPBRS3AF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:I6QOAIZT6LJXDGB3BYTPBRS3AF","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":"799064c416c3d8c147b90b37afebf903f18343947062c87962beff493ef0cfd8","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-05-22T16:29:05Z","title_canon_sha256":"16111f8a1ca67703705532323a12400816289ba076bf59db31d329f7b45e99b6"},"schema_version":"1.0","source":{"id":"1405.5820","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.5820","created_at":"2026-05-18T01:04:33Z"},{"alias_kind":"arxiv_version","alias_value":"1405.5820v2","created_at":"2026-05-18T01:04:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.5820","created_at":"2026-05-18T01:04:33Z"},{"alias_kind":"pith_short_12","alias_value":"I6QOAIZT6LJX","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"I6QOAIZT6LJXDGB3","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"I6QOAIZT","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:2a3162ca1a21aa2c6ebb1df99b97ba582d53dd186654564fbf733f61ba59ea97","target":"graph","created_at":"2026-05-18T01:04:33Z","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":"The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author and the Haagerup property introduced by Connes and Choda.\n  In this paper it is shown that a connected simple Lie group G has the weak Haagerup property if and only if the real rank of G is zero or one. Hence for connected simple Lie groups the weak Haagerup property coincides with weak amenability. Moreover, it turns out that for connected simple Lie groups t","authors_text":"S{\\o}ren Knudby, Uffe Haagerup","cross_cats":["math.FA","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-05-22T16:29:05Z","title":"The weak Haagerup property II: Examples"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5820","kind":"arxiv","version":2},"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":"sha256:5cd7e38499327c3659e084420dd819f25c11d7368de1b48f6af3e39f86d8e9f4","target":"record","created_at":"2026-05-18T01:04:33Z","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":"799064c416c3d8c147b90b37afebf903f18343947062c87962beff493ef0cfd8","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-05-22T16:29:05Z","title_canon_sha256":"16111f8a1ca67703705532323a12400816289ba076bf59db31d329f7b45e99b6"},"schema_version":"1.0","source":{"id":"1405.5820","kind":"arxiv","version":2}},"canonical_sha256":"47a0e02333f2d371983b0e26f0c65b01426d0baa2e180a9e2e613d653d768903","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47a0e02333f2d371983b0e26f0c65b01426d0baa2e180a9e2e613d653d768903","first_computed_at":"2026-05-18T01:04:33.595164Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:33.595164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OYIfoXiMBmFgc5t9O9FUv95VvUvpKw+LPy4r7qPVCrSZAIwH6mYVbvvdziYXRSGRlpXIDQPLCKKi/2kTHWCwBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:33.595946Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.5820","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5cd7e38499327c3659e084420dd819f25c11d7368de1b48f6af3e39f86d8e9f4","sha256:2a3162ca1a21aa2c6ebb1df99b97ba582d53dd186654564fbf733f61ba59ea97"],"state_sha256":"f9a26fbe96e6af207160adeb29704331427063df4c4f93503ed6a46f92f65fe9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VHRMmu1OrfT5lxBTLjFWYA33WADA2UX62IC/vVS/FgBM1j613o8V9CE03rdxB2pLXz/nnOE7RgWJxZa/ic/gDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T19:39:21.943796Z","bundle_sha256":"149c1d732fcc340bd47e7a760b6fb3885a528713d07b17d2376d68fcdcd08da4"}}