{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:JFWKDBPP5R5JH6QQOSS57REA3F","short_pith_number":"pith:JFWKDBPP","canonical_record":{"source":{"id":"0906.4825","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-06-26T00:50:35Z","cross_cats_sorted":[],"title_canon_sha256":"1f3492d834f656a24218393c7557f9b32f1347279157440d5ea42f3557fda22b","abstract_canon_sha256":"a93c363268f330e686f48620593cffa721ea711dec1104e99e4911feba1909e3"},"schema_version":"1.0"},"canonical_sha256":"496ca185efec7a93fa1074a5dfc480d979bc4dbca5ef7727de4f399d58b0aa7b","source":{"kind":"arxiv","id":"0906.4825","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.4825","created_at":"2026-05-18T04:00:38Z"},{"alias_kind":"arxiv_version","alias_value":"0906.4825v2","created_at":"2026-05-18T04:00:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.4825","created_at":"2026-05-18T04:00:38Z"},{"alias_kind":"pith_short_12","alias_value":"JFWKDBPP5R5J","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"JFWKDBPP5R5JH6QQ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"JFWKDBPP","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:JFWKDBPP5R5JH6QQOSS57REA3F","target":"record","payload":{"canonical_record":{"source":{"id":"0906.4825","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-06-26T00:50:35Z","cross_cats_sorted":[],"title_canon_sha256":"1f3492d834f656a24218393c7557f9b32f1347279157440d5ea42f3557fda22b","abstract_canon_sha256":"a93c363268f330e686f48620593cffa721ea711dec1104e99e4911feba1909e3"},"schema_version":"1.0"},"canonical_sha256":"496ca185efec7a93fa1074a5dfc480d979bc4dbca5ef7727de4f399d58b0aa7b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:38.265655Z","signature_b64":"1gKM2uXi4KhmFfDrAINbczZZh65LrA+ISewQWSfHdhK3cfOU++U6zLH20Gd9njrjNe4KP1vwzBFsw/pNEfcrCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"496ca185efec7a93fa1074a5dfc480d979bc4dbca5ef7727de4f399d58b0aa7b","last_reissued_at":"2026-05-18T04:00:38.264912Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:38.264912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.4825","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":"2026-05-18T04:00:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kj5pkIDCf6wYO4mejB28SO+CNVfjGXa/3fCob7GJU2Cf/vCMauKrnj/9b6bP+fNTzKEDpMxYc9spxvlVruCTBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:06:12.351221Z"},"content_sha256":"a154ca728f431c71e98a72cda1e5850f2ebef10a7ae040f57b0c3b1e404390f5","schema_version":"1.0","event_id":"sha256:a154ca728f431c71e98a72cda1e5850f2ebef10a7ae040f57b0c3b1e404390f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:JFWKDBPP5R5JH6QQOSS57REA3F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Aidan Sims, Nadia S. Larsen, Sean Vittadello, Toke Meier Carlsen","submitted_at":"2009-06-26T00:50:35Z","abstract_excerpt":"Let X be a product system over a quasi-lattice ordered group. Under mild hypotheses, we associate to X a C*-algebra which is co-universal for injective Nica covariant Toeplitz representations of X which preserve the gauge coaction. Under appropriate amenability criteria, this co-universal C*-algebra coincides with the Cuntz-Nica-Pimsner algebra introduced by Sims and Yeend. We prove two key uniqueness theorems, and indicate how to use our theorems to realise a number of reduced crossed products as instances of our co-universal algebras. In each case, it is an easy corollary that the Cuntz-Nica"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.4825","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":"2026-05-18T04:00:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5dphFXoZf8NSxYn0peZKfLVkRkcVZTi27o72lYVbVjG7Ms4K39TzHCTTdbpmRJZmkhbRyjiA0BaTyt5U4dv7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:06:12.351941Z"},"content_sha256":"7de1f767e2cd549d4d2571c360620fce48de050275c4121155b59a8c2c4bfd96","schema_version":"1.0","event_id":"sha256:7de1f767e2cd549d4d2571c360620fce48de050275c4121155b59a8c2c4bfd96"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JFWKDBPP5R5JH6QQOSS57REA3F/bundle.json","state_url":"https://pith.science/pith/JFWKDBPP5R5JH6QQOSS57REA3F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JFWKDBPP5R5JH6QQOSS57REA3F/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":"2026-05-26T18:06:12Z","links":{"resolver":"https://pith.science/pith/JFWKDBPP5R5JH6QQOSS57REA3F","bundle":"https://pith.science/pith/JFWKDBPP5R5JH6QQOSS57REA3F/bundle.json","state":"https://pith.science/pith/JFWKDBPP5R5JH6QQOSS57REA3F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JFWKDBPP5R5JH6QQOSS57REA3F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:JFWKDBPP5R5JH6QQOSS57REA3F","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":"a93c363268f330e686f48620593cffa721ea711dec1104e99e4911feba1909e3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-06-26T00:50:35Z","title_canon_sha256":"1f3492d834f656a24218393c7557f9b32f1347279157440d5ea42f3557fda22b"},"schema_version":"1.0","source":{"id":"0906.4825","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.4825","created_at":"2026-05-18T04:00:38Z"},{"alias_kind":"arxiv_version","alias_value":"0906.4825v2","created_at":"2026-05-18T04:00:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.4825","created_at":"2026-05-18T04:00:38Z"},{"alias_kind":"pith_short_12","alias_value":"JFWKDBPP5R5J","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"JFWKDBPP5R5JH6QQ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"JFWKDBPP","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:7de1f767e2cd549d4d2571c360620fce48de050275c4121155b59a8c2c4bfd96","target":"graph","created_at":"2026-05-18T04:00:38Z","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":"Let X be a product system over a quasi-lattice ordered group. Under mild hypotheses, we associate to X a C*-algebra which is co-universal for injective Nica covariant Toeplitz representations of X which preserve the gauge coaction. Under appropriate amenability criteria, this co-universal C*-algebra coincides with the Cuntz-Nica-Pimsner algebra introduced by Sims and Yeend. We prove two key uniqueness theorems, and indicate how to use our theorems to realise a number of reduced crossed products as instances of our co-universal algebras. In each case, it is an easy corollary that the Cuntz-Nica","authors_text":"Aidan Sims, Nadia S. Larsen, Sean Vittadello, Toke Meier Carlsen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-06-26T00:50:35Z","title":"Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.4825","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:a154ca728f431c71e98a72cda1e5850f2ebef10a7ae040f57b0c3b1e404390f5","target":"record","created_at":"2026-05-18T04:00:38Z","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":"a93c363268f330e686f48620593cffa721ea711dec1104e99e4911feba1909e3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-06-26T00:50:35Z","title_canon_sha256":"1f3492d834f656a24218393c7557f9b32f1347279157440d5ea42f3557fda22b"},"schema_version":"1.0","source":{"id":"0906.4825","kind":"arxiv","version":2}},"canonical_sha256":"496ca185efec7a93fa1074a5dfc480d979bc4dbca5ef7727de4f399d58b0aa7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"496ca185efec7a93fa1074a5dfc480d979bc4dbca5ef7727de4f399d58b0aa7b","first_computed_at":"2026-05-18T04:00:38.264912Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:38.264912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1gKM2uXi4KhmFfDrAINbczZZh65LrA+ISewQWSfHdhK3cfOU++U6zLH20Gd9njrjNe4KP1vwzBFsw/pNEfcrCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:38.265655Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.4825","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a154ca728f431c71e98a72cda1e5850f2ebef10a7ae040f57b0c3b1e404390f5","sha256:7de1f767e2cd549d4d2571c360620fce48de050275c4121155b59a8c2c4bfd96"],"state_sha256":"3cb11fa24d1499048d1328794efea8a2c377fed97bb80e7a34e6bb4cb7254fac"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RpuN8+b5qPHuqB9VSlGI9EO7tpxJkTRehBpJwkiKcnvxJEZbw5mPjTeKDJXg8BuT5qU3MmcxUJcKwZn7J9gnDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T18:06:12.355657Z","bundle_sha256":"3ddf12f4d1d7118a577e285c9ceb29405a1e8202dab5d081b86be3ac992c5212"}}