{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2000:AXG3MX7X5KULLMWYLK6J5AETGH","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":"34534e019966082744fc213e89132fee24d1a5c337e9788257ed730d7d453359","cross_cats_sorted":["math.OA"],"license":"","primary_cat":"math.FA","submitted_at":"2000-09-06T14:06:32Z","title_canon_sha256":"fb106d0a3dd11c9dabb6ce0153390e533067ea1a2bdc6a1e921e44b6ab233167"},"schema_version":"1.0","source":{"id":"math/0009052","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0009052","created_at":"2026-07-04T14:34:41Z"},{"alias_kind":"arxiv_version","alias_value":"math/0009052v2","created_at":"2026-07-04T14:34:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0009052","created_at":"2026-07-04T14:34:41Z"},{"alias_kind":"pith_short_12","alias_value":"AXG3MX7X5KUL","created_at":"2026-07-04T14:34:41Z"},{"alias_kind":"pith_short_16","alias_value":"AXG3MX7X5KULLMWY","created_at":"2026-07-04T14:34:41Z"},{"alias_kind":"pith_short_8","alias_value":"AXG3MX7X","created_at":"2026-07-04T14:34:41Z"}],"graph_snapshots":[{"event_id":"sha256:08a1c27b97a184e679ec5ea9e58c5ea04f5fe739e8aeab2a48009ecd4ca5800a","target":"graph","created_at":"2026-07-04T14:34:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0009052/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The \"similarity\" degree of a unital operator algebra $A$ was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the \"length\" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type $II_1$ factor with property $\\Gamma$ is at most 5, improving on a previous bound ($\\le 44$) due to E. Christensen.","authors_text":"Gilles Pisier","cross_cats":["math.OA"],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"2000-09-06T14:06:32Z","title":"Remarks on the similarity degree of an operator algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0009052","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":"sha256:01088664e7a7ddefc7b11d8c35c7c197d2b00369ea0b88559eccb07cf43bb97d","target":"record","created_at":"2026-07-04T14:34:41Z","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":"34534e019966082744fc213e89132fee24d1a5c337e9788257ed730d7d453359","cross_cats_sorted":["math.OA"],"license":"","primary_cat":"math.FA","submitted_at":"2000-09-06T14:06:32Z","title_canon_sha256":"fb106d0a3dd11c9dabb6ce0153390e533067ea1a2bdc6a1e921e44b6ab233167"},"schema_version":"1.0","source":{"id":"math/0009052","kind":"arxiv","version":2}},"canonical_sha256":"05cdb65ff7eaa8b5b2d85abc9e809331c3bd8a27ff23584c42a3913f843979e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"05cdb65ff7eaa8b5b2d85abc9e809331c3bd8a27ff23584c42a3913f843979e4","first_computed_at":"2026-07-04T14:34:41.074040Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:34:41.074040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JHj/5cHHayUjMxK60x8LUthE/u6rtCGkaMSLkPe/J6eQVo4cjMzQORoe1WYLDtGzymoEcdA2zTGyQevlRMM/Ag==","signature_status":"signed_v1","signed_at":"2026-07-04T14:34:41.074411Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0009052","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01088664e7a7ddefc7b11d8c35c7c197d2b00369ea0b88559eccb07cf43bb97d","sha256:08a1c27b97a184e679ec5ea9e58c5ea04f5fe739e8aeab2a48009ecd4ca5800a"],"state_sha256":"2f7ac2e44c83f0b9d2624ab3fbcb87e1e784172b3c00ed4d58b92f2c5cbe1296"}