{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:OHSHQGMDMYDEQPYQSX6EGBIAMJ","short_pith_number":"pith:OHSHQGMD","canonical_record":{"source":{"id":"1407.6929","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-07-25T15:09:10Z","cross_cats_sorted":[],"title_canon_sha256":"abf21e86d0295c2a2856753f9e4d3806cf8139f252c52ca340d3cbb5e4346bc5","abstract_canon_sha256":"ce020f59882aa2cfc2127bd4ea06fc3f79efd9b38ee5ab569b0a49a09f42f7f8"},"schema_version":"1.0"},"canonical_sha256":"71e47819836606483f1095fc430500627b72015fea36354bec541de5f6152ec4","source":{"kind":"arxiv","id":"1407.6929","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6929","created_at":"2026-05-18T02:19:42Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6929v4","created_at":"2026-05-18T02:19:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6929","created_at":"2026-05-18T02:19:42Z"},{"alias_kind":"pith_short_12","alias_value":"OHSHQGMDMYDE","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OHSHQGMDMYDEQPYQ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OHSHQGMD","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:OHSHQGMDMYDEQPYQSX6EGBIAMJ","target":"record","payload":{"canonical_record":{"source":{"id":"1407.6929","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-07-25T15:09:10Z","cross_cats_sorted":[],"title_canon_sha256":"abf21e86d0295c2a2856753f9e4d3806cf8139f252c52ca340d3cbb5e4346bc5","abstract_canon_sha256":"ce020f59882aa2cfc2127bd4ea06fc3f79efd9b38ee5ab569b0a49a09f42f7f8"},"schema_version":"1.0"},"canonical_sha256":"71e47819836606483f1095fc430500627b72015fea36354bec541de5f6152ec4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:42.610608Z","signature_b64":"VIYp7eeN4Ir3KLocLJHC4xmO21kuf6+3zJBS6ByvSuGGRS0xNe99AirbPheHr+Tcjcw62YK5aVqLza6d6WszAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71e47819836606483f1095fc430500627b72015fea36354bec541de5f6152ec4","last_reissued_at":"2026-05-18T02:19:42.610085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:42.610085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.6929","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:19:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GB1qGNo7qV6iJLu/JI2nfeXXlA7gD7v8bji2jO5jSTZXqsTrd4uwv6fuSAwR6uyYFkTmaVNkvb8Uvste0xwXAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T10:33:27.514216Z"},"content_sha256":"aa5a6e2c3e4fb3d9c5c0a84dedc68dc938c016bb07a0e131ec1c95cecedbfd5a","schema_version":"1.0","event_id":"sha256:aa5a6e2c3e4fb3d9c5c0a84dedc68dc938c016bb07a0e131ec1c95cecedbfd5a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:OHSHQGMDMYDEQPYQSX6EGBIAMJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Similarity degree of type II$_1$ von Neumann algebras with Property $\\Gamma$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Don Hadwin, Junhao Shen, Wenhua Qian","submitted_at":"2014-07-25T15:09:10Z","abstract_excerpt":"In this paper, we discuss some equivalent definitions of Property $\\Gamma$ for a type II$_1$ von Neumann algebra. Using these equivalent definitions, we prove that the Pisier's similarity degree of a type II$_1$ von Neumann algebra with Property $\\Gamma$ is equal to $3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6929","kind":"arxiv","version":4},"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":"2026-05-18T02:19:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/Xqh5zK+6/w3CBhrvcnudV71x8b0ZsABExltqIsBMCs3yp0JMiXgmDyJp2SKib9EtbqH6SUq1btZZoEGmMp0Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T10:33:27.515047Z"},"content_sha256":"352a94694f80bc6415385fd120122a74c56234665a0f1c4af7b2947d27626aac","schema_version":"1.0","event_id":"sha256:352a94694f80bc6415385fd120122a74c56234665a0f1c4af7b2947d27626aac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OHSHQGMDMYDEQPYQSX6EGBIAMJ/bundle.json","state_url":"https://pith.science/pith/OHSHQGMDMYDEQPYQSX6EGBIAMJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OHSHQGMDMYDEQPYQSX6EGBIAMJ/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":"2026-05-31T10:33:27Z","links":{"resolver":"https://pith.science/pith/OHSHQGMDMYDEQPYQSX6EGBIAMJ","bundle":"https://pith.science/pith/OHSHQGMDMYDEQPYQSX6EGBIAMJ/bundle.json","state":"https://pith.science/pith/OHSHQGMDMYDEQPYQSX6EGBIAMJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OHSHQGMDMYDEQPYQSX6EGBIAMJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OHSHQGMDMYDEQPYQSX6EGBIAMJ","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":"ce020f59882aa2cfc2127bd4ea06fc3f79efd9b38ee5ab569b0a49a09f42f7f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-07-25T15:09:10Z","title_canon_sha256":"abf21e86d0295c2a2856753f9e4d3806cf8139f252c52ca340d3cbb5e4346bc5"},"schema_version":"1.0","source":{"id":"1407.6929","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6929","created_at":"2026-05-18T02:19:42Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6929v4","created_at":"2026-05-18T02:19:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6929","created_at":"2026-05-18T02:19:42Z"},{"alias_kind":"pith_short_12","alias_value":"OHSHQGMDMYDE","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OHSHQGMDMYDEQPYQ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OHSHQGMD","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:352a94694f80bc6415385fd120122a74c56234665a0f1c4af7b2947d27626aac","target":"graph","created_at":"2026-05-18T02:19:42Z","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 discuss some equivalent definitions of Property $\\Gamma$ for a type II$_1$ von Neumann algebra. Using these equivalent definitions, we prove that the Pisier's similarity degree of a type II$_1$ von Neumann algebra with Property $\\Gamma$ is equal to $3$.","authors_text":"Don Hadwin, Junhao Shen, Wenhua Qian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-07-25T15:09:10Z","title":"Similarity degree of type II$_1$ von Neumann algebras with Property $\\Gamma$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6929","kind":"arxiv","version":4},"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:aa5a6e2c3e4fb3d9c5c0a84dedc68dc938c016bb07a0e131ec1c95cecedbfd5a","target":"record","created_at":"2026-05-18T02:19:42Z","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":"ce020f59882aa2cfc2127bd4ea06fc3f79efd9b38ee5ab569b0a49a09f42f7f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-07-25T15:09:10Z","title_canon_sha256":"abf21e86d0295c2a2856753f9e4d3806cf8139f252c52ca340d3cbb5e4346bc5"},"schema_version":"1.0","source":{"id":"1407.6929","kind":"arxiv","version":4}},"canonical_sha256":"71e47819836606483f1095fc430500627b72015fea36354bec541de5f6152ec4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71e47819836606483f1095fc430500627b72015fea36354bec541de5f6152ec4","first_computed_at":"2026-05-18T02:19:42.610085Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:42.610085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VIYp7eeN4Ir3KLocLJHC4xmO21kuf6+3zJBS6ByvSuGGRS0xNe99AirbPheHr+Tcjcw62YK5aVqLza6d6WszAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:42.610608Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6929","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa5a6e2c3e4fb3d9c5c0a84dedc68dc938c016bb07a0e131ec1c95cecedbfd5a","sha256:352a94694f80bc6415385fd120122a74c56234665a0f1c4af7b2947d27626aac"],"state_sha256":"64ddbc42a6e755f464c32e0d97405458d032c9d097ec4a160fac1b0220fb6aa1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MtMbzy2A3zRKIkBKRjPwIL5Pm5mtRJRJIiT1SQZ22ZoIY2ark9NexkPOVLK6pwehkF9daOEabGmi6JvO9F00CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T10:33:27.518613Z","bundle_sha256":"fad0623d45d3772f532bbad35879bc28a238f7fb6c85b683e0c06a3f9e046fd8"}}