{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:PFQMJVRRLVBPE7US3LJP5H3VC4","short_pith_number":"pith:PFQMJVRR","canonical_record":{"source":{"id":"1707.03740","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-07-12T14:36:14Z","cross_cats_sorted":[],"title_canon_sha256":"18c3658ee45c8cdaf93055a95aefa214544437c6c602d4ef9afee7c49c0ba0df","abstract_canon_sha256":"bf48cb965dc7dcff3bb07b83c85f4964bc663ff45a5c904d3a6108c895211853"},"schema_version":"1.0"},"canonical_sha256":"7960c4d6315d42f27e92dad2fe9f7517120460ba83b13078dc9f7b8fa102249f","source":{"kind":"arxiv","id":"1707.03740","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.03740","created_at":"2026-05-17T23:55:29Z"},{"alias_kind":"arxiv_version","alias_value":"1707.03740v2","created_at":"2026-05-17T23:55:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.03740","created_at":"2026-05-17T23:55:29Z"},{"alias_kind":"pith_short_12","alias_value":"PFQMJVRRLVBP","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PFQMJVRRLVBPE7US","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PFQMJVRR","created_at":"2026-05-18T12:31:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:PFQMJVRRLVBPE7US3LJP5H3VC4","target":"record","payload":{"canonical_record":{"source":{"id":"1707.03740","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-07-12T14:36:14Z","cross_cats_sorted":[],"title_canon_sha256":"18c3658ee45c8cdaf93055a95aefa214544437c6c602d4ef9afee7c49c0ba0df","abstract_canon_sha256":"bf48cb965dc7dcff3bb07b83c85f4964bc663ff45a5c904d3a6108c895211853"},"schema_version":"1.0"},"canonical_sha256":"7960c4d6315d42f27e92dad2fe9f7517120460ba83b13078dc9f7b8fa102249f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:29.896945Z","signature_b64":"2UkbS688pUH8JSVmIfprQHqQTc36XJgcHJq1YUT/fZi4dQTddXuSWF7t5N7VMf8kzEE2V2HVQUZVTPM+8ZAzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7960c4d6315d42f27e92dad2fe9f7517120460ba83b13078dc9f7b8fa102249f","last_reissued_at":"2026-05-17T23:55:29.896444Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:29.896444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.03740","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":"2026-05-17T23:55:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1lSCfOQgTV1tsvwC7GzvyiiqMfeOKNydD1n61UOUnOH5KCNpn7LxvNbKKAWxFVuYL7m44G4n5WN2ili0wMOHDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T09:15:55.991604Z"},"content_sha256":"f4858e3108b05a75c96427a7443d30da00de09eee791ada1113e40179b2903ad","schema_version":"1.0","event_id":"sha256:f4858e3108b05a75c96427a7443d30da00de09eee791ada1113e40179b2903ad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:PFQMJVRRLVBPE7US3LJP5H3VC4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ideal structure and pure infiniteness of ample groupoid $C^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Christian B\\\"onicke, Kang Li","submitted_at":"2017-07-12T14:36:14Z","abstract_excerpt":"In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to \\'etale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_r^*(G)$ and the open invariant subsets of the unit space $G^{(0)}$ of $G$. As a consequence, we show that if $G$ is an inner exact, essentially principal, ample groupoid, then $C_r^*(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in $C_r^*(G)$. We also establish a sufficient condition on the ample "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03740","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":"2026-05-17T23:55:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6aePlDEIjfAbmfOXVkPkpSt++ayqG1NSWYkIzczZ3HUx+Ih6ANXOQFo3eB1RsiN3QTMUJIYsFE8UvvrvQ5TfAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T09:15:55.991957Z"},"content_sha256":"b75c9217811c127d4fd0e7a30619370bf5929bde0ad6344472bac3b6d4333d0e","schema_version":"1.0","event_id":"sha256:b75c9217811c127d4fd0e7a30619370bf5929bde0ad6344472bac3b6d4333d0e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PFQMJVRRLVBPE7US3LJP5H3VC4/bundle.json","state_url":"https://pith.science/pith/PFQMJVRRLVBPE7US3LJP5H3VC4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PFQMJVRRLVBPE7US3LJP5H3VC4/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":"2026-06-28T09:15:55Z","links":{"resolver":"https://pith.science/pith/PFQMJVRRLVBPE7US3LJP5H3VC4","bundle":"https://pith.science/pith/PFQMJVRRLVBPE7US3LJP5H3VC4/bundle.json","state":"https://pith.science/pith/PFQMJVRRLVBPE7US3LJP5H3VC4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PFQMJVRRLVBPE7US3LJP5H3VC4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:PFQMJVRRLVBPE7US3LJP5H3VC4","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":"bf48cb965dc7dcff3bb07b83c85f4964bc663ff45a5c904d3a6108c895211853","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-07-12T14:36:14Z","title_canon_sha256":"18c3658ee45c8cdaf93055a95aefa214544437c6c602d4ef9afee7c49c0ba0df"},"schema_version":"1.0","source":{"id":"1707.03740","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.03740","created_at":"2026-05-17T23:55:29Z"},{"alias_kind":"arxiv_version","alias_value":"1707.03740v2","created_at":"2026-05-17T23:55:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.03740","created_at":"2026-05-17T23:55:29Z"},{"alias_kind":"pith_short_12","alias_value":"PFQMJVRRLVBP","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PFQMJVRRLVBPE7US","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PFQMJVRR","created_at":"2026-05-18T12:31:37Z"}],"graph_snapshots":[{"event_id":"sha256:b75c9217811c127d4fd0e7a30619370bf5929bde0ad6344472bac3b6d4333d0e","target":"graph","created_at":"2026-05-17T23:55:29Z","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":"In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to \\'etale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_r^*(G)$ and the open invariant subsets of the unit space $G^{(0)}$ of $G$. As a consequence, we show that if $G$ is an inner exact, essentially principal, ample groupoid, then $C_r^*(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in $C_r^*(G)$. We also establish a sufficient condition on the ample ","authors_text":"Christian B\\\"onicke, Kang Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-07-12T14:36:14Z","title":"Ideal structure and pure infiniteness of ample groupoid $C^*$-algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03740","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:f4858e3108b05a75c96427a7443d30da00de09eee791ada1113e40179b2903ad","target":"record","created_at":"2026-05-17T23:55:29Z","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":"bf48cb965dc7dcff3bb07b83c85f4964bc663ff45a5c904d3a6108c895211853","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-07-12T14:36:14Z","title_canon_sha256":"18c3658ee45c8cdaf93055a95aefa214544437c6c602d4ef9afee7c49c0ba0df"},"schema_version":"1.0","source":{"id":"1707.03740","kind":"arxiv","version":2}},"canonical_sha256":"7960c4d6315d42f27e92dad2fe9f7517120460ba83b13078dc9f7b8fa102249f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7960c4d6315d42f27e92dad2fe9f7517120460ba83b13078dc9f7b8fa102249f","first_computed_at":"2026-05-17T23:55:29.896444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:29.896444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2UkbS688pUH8JSVmIfprQHqQTc36XJgcHJq1YUT/fZi4dQTddXuSWF7t5N7VMf8kzEE2V2HVQUZVTPM+8ZAzAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:29.896945Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.03740","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4858e3108b05a75c96427a7443d30da00de09eee791ada1113e40179b2903ad","sha256:b75c9217811c127d4fd0e7a30619370bf5929bde0ad6344472bac3b6d4333d0e"],"state_sha256":"9e8ef27e3e010ed8f88b1feff1bc581ed520f5ebe92cf5249817bbfb1737ec8e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kQ2ECvYgiM8MEUkIK3XUcPxOFN7FegViEzuOG650PuSUQUM+kuv7HwpZLZmD4+bgP94TE+z758kfAN7f91ipBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T09:15:55.993822Z","bundle_sha256":"927cd3904d9554865a63477899de218e6fb7f02feeea2869ef1487ea89b2dff9"}}