{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:AMBJUZQSL6L3PMG2I25HE5EWFF","short_pith_number":"pith:AMBJUZQS","canonical_record":{"source":{"id":"1004.0851","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-06T13:43:02Z","cross_cats_sorted":[],"title_canon_sha256":"7bd22f20b74355b2e02f39faa8b15e571b2375895121631a471cfa7fd56bd155","abstract_canon_sha256":"a564af3ba9c6470cdbe3bcc84437d8ac6c6c7d8035bd25989679f3722d9f84b5"},"schema_version":"1.0"},"canonical_sha256":"03029a66125f97b7b0da46ba72749629420f3785e13c344b01406c81d9e22d9e","source":{"kind":"arxiv","id":"1004.0851","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0851","created_at":"2026-05-18T01:34:46Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0851v1","created_at":"2026-05-18T01:34:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0851","created_at":"2026-05-18T01:34:46Z"},{"alias_kind":"pith_short_12","alias_value":"AMBJUZQSL6L3","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"AMBJUZQSL6L3PMG2","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"AMBJUZQS","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:AMBJUZQSL6L3PMG2I25HE5EWFF","target":"record","payload":{"canonical_record":{"source":{"id":"1004.0851","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-06T13:43:02Z","cross_cats_sorted":[],"title_canon_sha256":"7bd22f20b74355b2e02f39faa8b15e571b2375895121631a471cfa7fd56bd155","abstract_canon_sha256":"a564af3ba9c6470cdbe3bcc84437d8ac6c6c7d8035bd25989679f3722d9f84b5"},"schema_version":"1.0"},"canonical_sha256":"03029a66125f97b7b0da46ba72749629420f3785e13c344b01406c81d9e22d9e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:46.002777Z","signature_b64":"afaj4NMcN3pssfAjESKT2kj1pYQKLbQdGOXAYaGS4FunfzHpimmfgX4w6G9qty93upTczcbfua0Raya46u7VBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03029a66125f97b7b0da46ba72749629420f3785e13c344b01406c81d9e22d9e","last_reissued_at":"2026-05-18T01:34:46.002156Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:46.002156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.0851","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:34:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tZ6ivj13EIBW5ETvagQexryGZOZCbMrFzeqvq1O0jAJOLlqQrP65xzkwpdpburECYx1aZVhaHLztZ5xUFwycBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:54:19.929794Z"},"content_sha256":"f1fff790046c9c7685e41d4b03ac78432ba6b7a17c800df9a17eacfe1e2e6748","schema_version":"1.0","event_id":"sha256:f1fff790046c9c7685e41d4b03ac78432ba6b7a17c800df9a17eacfe1e2e6748"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:AMBJUZQSL6L3PMG2I25HE5EWFF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Groupoid normalisers of tensor products: infinite von Neumann algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Junsheng Fang, Roger R. Smith, Stuart White","submitted_at":"2010-04-06T13:43:02Z","abstract_excerpt":"The groupoid normalisers of a unital inclusion $B\\subseteq M$ of von Neumann algebras consist of the set $\\mathcal{GN}_M(B)$ of partial isometries $v\\in M$ with $vBv^*\\subseteq B$ and $v^*Bv\\subseteq B$. Given two unital inclusions $B_i\\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\\ \\overline{\\otimes}\\ B_2\\subseteq M_1\\ \\overline{\\otimes}\\ M_2$ establishing the formula $$ \\mathcal{GN}_{M_1\\,\\overline{\\otimes}\\,M_2}(B_1\\ \\overline{\\otimes}\\ B_2)''=\\mathcal{GN}_{M_1}(B_1)''\\ \\overline{\\otimes}\\ \\mathcal{GN}_{M_2}(B_2)'' $$ when one i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0851","kind":"arxiv","version":1},"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":"2026-05-18T01:34:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nM4IkZC50nfm2mVneEI+9uVVjMyJOThIS+8WPFgsdpbkLNiA/exp63Pi8VQ/8tR6AmQKKODVij1W/OsUSwqyAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:54:19.930606Z"},"content_sha256":"f4339725497da883dbd207ce3a5e8a1f90deaa25f0d8ce6c95b36d994b136d33","schema_version":"1.0","event_id":"sha256:f4339725497da883dbd207ce3a5e8a1f90deaa25f0d8ce6c95b36d994b136d33"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AMBJUZQSL6L3PMG2I25HE5EWFF/bundle.json","state_url":"https://pith.science/pith/AMBJUZQSL6L3PMG2I25HE5EWFF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AMBJUZQSL6L3PMG2I25HE5EWFF/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":"2026-05-26T11:54:19Z","links":{"resolver":"https://pith.science/pith/AMBJUZQSL6L3PMG2I25HE5EWFF","bundle":"https://pith.science/pith/AMBJUZQSL6L3PMG2I25HE5EWFF/bundle.json","state":"https://pith.science/pith/AMBJUZQSL6L3PMG2I25HE5EWFF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AMBJUZQSL6L3PMG2I25HE5EWFF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:AMBJUZQSL6L3PMG2I25HE5EWFF","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":"a564af3ba9c6470cdbe3bcc84437d8ac6c6c7d8035bd25989679f3722d9f84b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-06T13:43:02Z","title_canon_sha256":"7bd22f20b74355b2e02f39faa8b15e571b2375895121631a471cfa7fd56bd155"},"schema_version":"1.0","source":{"id":"1004.0851","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0851","created_at":"2026-05-18T01:34:46Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0851v1","created_at":"2026-05-18T01:34:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0851","created_at":"2026-05-18T01:34:46Z"},{"alias_kind":"pith_short_12","alias_value":"AMBJUZQSL6L3","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"AMBJUZQSL6L3PMG2","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"AMBJUZQS","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:f4339725497da883dbd207ce3a5e8a1f90deaa25f0d8ce6c95b36d994b136d33","target":"graph","created_at":"2026-05-18T01:34:46Z","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 groupoid normalisers of a unital inclusion $B\\subseteq M$ of von Neumann algebras consist of the set $\\mathcal{GN}_M(B)$ of partial isometries $v\\in M$ with $vBv^*\\subseteq B$ and $v^*Bv\\subseteq B$. Given two unital inclusions $B_i\\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\\ \\overline{\\otimes}\\ B_2\\subseteq M_1\\ \\overline{\\otimes}\\ M_2$ establishing the formula $$ \\mathcal{GN}_{M_1\\,\\overline{\\otimes}\\,M_2}(B_1\\ \\overline{\\otimes}\\ B_2)''=\\mathcal{GN}_{M_1}(B_1)''\\ \\overline{\\otimes}\\ \\mathcal{GN}_{M_2}(B_2)'' $$ when one i","authors_text":"Junsheng Fang, Roger R. Smith, Stuart White","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-06T13:43:02Z","title":"Groupoid normalisers of tensor products: infinite von Neumann algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0851","kind":"arxiv","version":1},"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:f1fff790046c9c7685e41d4b03ac78432ba6b7a17c800df9a17eacfe1e2e6748","target":"record","created_at":"2026-05-18T01:34:46Z","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":"a564af3ba9c6470cdbe3bcc84437d8ac6c6c7d8035bd25989679f3722d9f84b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-06T13:43:02Z","title_canon_sha256":"7bd22f20b74355b2e02f39faa8b15e571b2375895121631a471cfa7fd56bd155"},"schema_version":"1.0","source":{"id":"1004.0851","kind":"arxiv","version":1}},"canonical_sha256":"03029a66125f97b7b0da46ba72749629420f3785e13c344b01406c81d9e22d9e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03029a66125f97b7b0da46ba72749629420f3785e13c344b01406c81d9e22d9e","first_computed_at":"2026-05-18T01:34:46.002156Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:46.002156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"afaj4NMcN3pssfAjESKT2kj1pYQKLbQdGOXAYaGS4FunfzHpimmfgX4w6G9qty93upTczcbfua0Raya46u7VBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:46.002777Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.0851","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f1fff790046c9c7685e41d4b03ac78432ba6b7a17c800df9a17eacfe1e2e6748","sha256:f4339725497da883dbd207ce3a5e8a1f90deaa25f0d8ce6c95b36d994b136d33"],"state_sha256":"6b030477a24675cde6e74459939bceeb91280fec8a397c7bd7678992c48edf2c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ElWS14/HxMxDkSX0fntWy26QRfrRmlXFWBGzucpnuo16IuPvjPoFp4e5jDPvaH/UacoAsHzfdbVcaInfGfnGBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T11:54:19.934919Z","bundle_sha256":"fbc72a708e2671b4967e583274e15e6205f5aac3e6424f6430e4a2ebbc17cc8a"}}