{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ERRI5MNSFFR4L7OA7XRFHDXUN5","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":"6b51b57a12437837e66caae7336feed7cef1e77c4341bbd675df01afc5ab39dd","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-10-13T21:04:46Z","title_canon_sha256":"fc361bb9ab9ce90b4f205ea5e8cb056cc8ef2f3f6c6eb9516f4b3802bba9635a"},"schema_version":"1.0","source":{"id":"1510.03896","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03896","created_at":"2026-05-18T01:05:12Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03896v1","created_at":"2026-05-18T01:05:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03896","created_at":"2026-05-18T01:05:12Z"},{"alias_kind":"pith_short_12","alias_value":"ERRI5MNSFFR4","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"ERRI5MNSFFR4L7OA","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"ERRI5MNS","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:872064e28b32c547aff39805f5c7de20cb9cb0a4eaae0d5792ca73e27668a0e2","target":"graph","created_at":"2026-05-18T01:05:12Z","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, operator-valued bi-free distributions are investigated. Given a subalgebra $D$ of a unital algebra $B$, it is established that a two-faced family $Z$ is bi-free from $(B, B^{\\mathrm{op}})$ over $D$ if and only if certain conditions relating the $B$-valued and $D$-valued bi-free cumulants of $Z$ are satisfied. Using this, we verify that a two-faced family of matrices is $R$-cyclic if and only if they are bi-free from the scalar matrices over the scalar diagonal matrices. Furthermore, the operator-valued bi-free partial $R$-, $S$-, and $T$-transforms are constructed. New proofs of","authors_text":"Paul Skoufranis","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-10-13T21:04:46Z","title":"On Operator-Valued Bi-Free Distributions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03896","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":"sha256:866d7eedddc6080f03a009747d207b2cb60e6ef77c7e32e85dc96807d199259d","target":"record","created_at":"2026-05-18T01:05:12Z","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":"6b51b57a12437837e66caae7336feed7cef1e77c4341bbd675df01afc5ab39dd","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-10-13T21:04:46Z","title_canon_sha256":"fc361bb9ab9ce90b4f205ea5e8cb056cc8ef2f3f6c6eb9516f4b3802bba9635a"},"schema_version":"1.0","source":{"id":"1510.03896","kind":"arxiv","version":1}},"canonical_sha256":"24628eb1b22963c5fdc0fde2538ef46f45cf9cf47d323537db33928a7bdb4e57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24628eb1b22963c5fdc0fde2538ef46f45cf9cf47d323537db33928a7bdb4e57","first_computed_at":"2026-05-18T01:05:12.688094Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:12.688094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8BE3T0Uo7v2WUXIAbvKxCdDI6ox2k/+CQ5b+0+AAfkKKUYvxQvyu6X9wLm0sdnUKsi7jPlXnZsCghqsEdQz1DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:12.688731Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.03896","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:866d7eedddc6080f03a009747d207b2cb60e6ef77c7e32e85dc96807d199259d","sha256:872064e28b32c547aff39805f5c7de20cb9cb0a4eaae0d5792ca73e27668a0e2"],"state_sha256":"1f1f9d0d6d59b0ac2d6cc8a4ac58589e9323ff9934878a2491ae2a9e93164068"}