{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:I5UVCW5KZ4E7YYGD2PBTZGPOYN","short_pith_number":"pith:I5UVCW5K","canonical_record":{"source":{"id":"1304.6193","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-23T07:52:42Z","cross_cats_sorted":[],"title_canon_sha256":"8f57b00dced37878459e633ef9ab82868f3f3c08754d734f6de764422cb704df","abstract_canon_sha256":"642e7977caf0ee2ba7a4288c69043ab1a049e41809cc6af1ba24d83d777332a3"},"schema_version":"1.0"},"canonical_sha256":"4769515baacf09fc60c3d3c33c99eec36c1463234b763e956344e583478d04e0","source":{"kind":"arxiv","id":"1304.6193","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.6193","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"arxiv_version","alias_value":"1304.6193v5","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.6193","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"pith_short_12","alias_value":"I5UVCW5KZ4E7","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"I5UVCW5KZ4E7YYGD","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"I5UVCW5K","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:I5UVCW5KZ4E7YYGD2PBTZGPOYN","target":"record","payload":{"canonical_record":{"source":{"id":"1304.6193","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-23T07:52:42Z","cross_cats_sorted":[],"title_canon_sha256":"8f57b00dced37878459e633ef9ab82868f3f3c08754d734f6de764422cb704df","abstract_canon_sha256":"642e7977caf0ee2ba7a4288c69043ab1a049e41809cc6af1ba24d83d777332a3"},"schema_version":"1.0"},"canonical_sha256":"4769515baacf09fc60c3d3c33c99eec36c1463234b763e956344e583478d04e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:05.954766Z","signature_b64":"hYNXLMk2mttLVlMQiGGlg5dXkqQ9MoOzYO6IiIYRjpa1LUGqeFSDVz31Xd1GQTbcsZ2gEb0Nc9GHKiCct8M1Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4769515baacf09fc60c3d3c33c99eec36c1463234b763e956344e583478d04e0","last_reissued_at":"2026-05-18T03:15:05.953991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:05.953991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.6193","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:15:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JeZqVeQTP3fG6ER+uuZk5fC2ltt/ZDDJIn3DZFO34l/a2EN/ibV9I+IHHr4NnJ3W7+eEGSPIF4O6o/8ogg3iDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T20:30:36.400777Z"},"content_sha256":"69ad5d71e44e22847d1c5c5b2ae50305fdf61dba431dc2806ec35a72571408df","schema_version":"1.0","event_id":"sha256:69ad5d71e44e22847d1c5c5b2ae50305fdf61dba431dc2806ec35a72571408df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:I5UVCW5KZ4E7YYGD2PBTZGPOYN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Notes on C_0-representations and the Haagerup property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Paul Jolissaint","submitted_at":"2013-04-23T07:52:42Z","abstract_excerpt":"For any locally compact group $G$, we show the existence and uniqueness up to quasi-equivalence of a unitary $C_0$-representation $\\pi_0$ of $G$ such that all coefficient functions of $C_0$-representations of $G$ are coefficient functions of $\\pi_0$. The present work, strongly influenced by the work of N. Brown and E. Guentner (which dealt exclusively with discrete groups), leads to new characterizations of the Haagerup property: if $G$ is second countable, then it has that property if and only if the representation $\\pi_0$ induces a *-isomorphism of $C^*(G)$ onto $C^*_{\\pi_0}(G)$. When $G$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.6193","kind":"arxiv","version":5},"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":"2026-05-18T03:15:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wFSBDi8KB0QFZVF2ONuJ7os4tyfhnXMMG12lRscBKv+8q2nfpIHzaqdMUCUC+HW41ZkyOuvGwv0oTcOHOzgHCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T20:30:36.401111Z"},"content_sha256":"de95717cbace50fee9771bf135d3a3154ef53b018c6568dff9362aaeaec6479d","schema_version":"1.0","event_id":"sha256:de95717cbace50fee9771bf135d3a3154ef53b018c6568dff9362aaeaec6479d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I5UVCW5KZ4E7YYGD2PBTZGPOYN/bundle.json","state_url":"https://pith.science/pith/I5UVCW5KZ4E7YYGD2PBTZGPOYN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I5UVCW5KZ4E7YYGD2PBTZGPOYN/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":"2026-06-26T20:30:36Z","links":{"resolver":"https://pith.science/pith/I5UVCW5KZ4E7YYGD2PBTZGPOYN","bundle":"https://pith.science/pith/I5UVCW5KZ4E7YYGD2PBTZGPOYN/bundle.json","state":"https://pith.science/pith/I5UVCW5KZ4E7YYGD2PBTZGPOYN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I5UVCW5KZ4E7YYGD2PBTZGPOYN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:I5UVCW5KZ4E7YYGD2PBTZGPOYN","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":"642e7977caf0ee2ba7a4288c69043ab1a049e41809cc6af1ba24d83d777332a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-23T07:52:42Z","title_canon_sha256":"8f57b00dced37878459e633ef9ab82868f3f3c08754d734f6de764422cb704df"},"schema_version":"1.0","source":{"id":"1304.6193","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.6193","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"arxiv_version","alias_value":"1304.6193v5","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.6193","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"pith_short_12","alias_value":"I5UVCW5KZ4E7","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"I5UVCW5KZ4E7YYGD","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"I5UVCW5K","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:de95717cbace50fee9771bf135d3a3154ef53b018c6568dff9362aaeaec6479d","target":"graph","created_at":"2026-05-18T03:15:05Z","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":"For any locally compact group $G$, we show the existence and uniqueness up to quasi-equivalence of a unitary $C_0$-representation $\\pi_0$ of $G$ such that all coefficient functions of $C_0$-representations of $G$ are coefficient functions of $\\pi_0$. The present work, strongly influenced by the work of N. Brown and E. Guentner (which dealt exclusively with discrete groups), leads to new characterizations of the Haagerup property: if $G$ is second countable, then it has that property if and only if the representation $\\pi_0$ induces a *-isomorphism of $C^*(G)$ onto $C^*_{\\pi_0}(G)$. When $G$ is","authors_text":"Paul Jolissaint","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-23T07:52:42Z","title":"Notes on C_0-representations and the Haagerup property"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.6193","kind":"arxiv","version":5},"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":"sha256:69ad5d71e44e22847d1c5c5b2ae50305fdf61dba431dc2806ec35a72571408df","target":"record","created_at":"2026-05-18T03:15:05Z","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":"642e7977caf0ee2ba7a4288c69043ab1a049e41809cc6af1ba24d83d777332a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-23T07:52:42Z","title_canon_sha256":"8f57b00dced37878459e633ef9ab82868f3f3c08754d734f6de764422cb704df"},"schema_version":"1.0","source":{"id":"1304.6193","kind":"arxiv","version":5}},"canonical_sha256":"4769515baacf09fc60c3d3c33c99eec36c1463234b763e956344e583478d04e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4769515baacf09fc60c3d3c33c99eec36c1463234b763e956344e583478d04e0","first_computed_at":"2026-05-18T03:15:05.953991Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:05.953991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hYNXLMk2mttLVlMQiGGlg5dXkqQ9MoOzYO6IiIYRjpa1LUGqeFSDVz31Xd1GQTbcsZ2gEb0Nc9GHKiCct8M1Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:05.954766Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.6193","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69ad5d71e44e22847d1c5c5b2ae50305fdf61dba431dc2806ec35a72571408df","sha256:de95717cbace50fee9771bf135d3a3154ef53b018c6568dff9362aaeaec6479d"],"state_sha256":"4f9bd0aaa3e1bc040e5e49e0aad9e7546cab4e85eba54d2847545a96310013b9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GfQaNauMd80Xq02tXylU18xdRunZSsbKi3PzeoZ7YbCYvQ3iCSp4QQcess8gGPz1h2BUR9AccbxWu/q3DbAoAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T20:30:36.402965Z","bundle_sha256":"afb8d7499a215391cf92b5f2bbfda58c1ba93e5c4a61fdf5a2c6f4aa5c57f7c8"}}