{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:IIEFVGU7EL4ANUZRHCJRCBQSME","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":"68b74c45a240f3cd682bd1d03e5495381340e9f0eb8f2845259b0142f8f9e8f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-03-14T19:50:17Z","title_canon_sha256":"6b98b2a132d6f908c15ffe1a34474f9e71394a21261b546ac08b9d0a7dde1100"},"schema_version":"1.0","source":{"id":"1203.3186","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.3186","created_at":"2026-05-18T00:34:05Z"},{"alias_kind":"arxiv_version","alias_value":"1203.3186v3","created_at":"2026-05-18T00:34:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.3186","created_at":"2026-05-18T00:34:05Z"},{"alias_kind":"pith_short_12","alias_value":"IIEFVGU7EL4A","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"IIEFVGU7EL4ANUZR","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"IIEFVGU7","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:b7f25d2fbd84619fdb07d41f9a6b7bc4d5b005504ae126580b23efc5a70b63cf","target":"graph","created_at":"2026-05-18T00:34:05Z","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":"Using an inequality due to Ricard and Xu, we give a different proof of Paul Skoufranis's recent result showing that the strong convergence of possibly non-commutative random variables $X^{(k)}\\to X$ is stable under reduced free product with a fixed non-commutative random variable $Y$. In fact we obtain a more general fact: assuming that the families $X^{(k)}=\\{X_i^{(k)}\\}$ and $Y^{(k)}=\\{Y_j^{(k)}\\}$ are $*$-free as well as their limits (in moments) $X =\\{X_i \\}$ and $Y =\\{Y_j\\}$, the strong convergences $X^{(k)}\\to X$ and $Y^{(k)}\\to Y$ imply that of $\\{X^{(k)},Y^{(k)} \\}$ to $\\{X ,Y\\}$. Phra","authors_text":"Gilles Pisier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-03-14T19:50:17Z","title":"Strong convergence for reduced free products (Remarks on a result by Paul Skoufranis)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3186","kind":"arxiv","version":3},"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":"sha256:71fb43d376680db719a44823f0010ca7d8661ed5ac22e916eb0b9ffc4938ce81","target":"record","created_at":"2026-05-18T00:34:05Z","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":"68b74c45a240f3cd682bd1d03e5495381340e9f0eb8f2845259b0142f8f9e8f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-03-14T19:50:17Z","title_canon_sha256":"6b98b2a132d6f908c15ffe1a34474f9e71394a21261b546ac08b9d0a7dde1100"},"schema_version":"1.0","source":{"id":"1203.3186","kind":"arxiv","version":3}},"canonical_sha256":"42085a9a9f22f806d33138931106126107bd1374f311fa334881381e7c0b902c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"42085a9a9f22f806d33138931106126107bd1374f311fa334881381e7c0b902c","first_computed_at":"2026-05-18T00:34:05.474619Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:05.474619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JcB8GiqsKER5SbE7fLYw8b2FSnDEgI7rwSFaikWAZPF0SaeNtqF0/u9UbJhdsHClbNwuOfV3TTdYZ6hxHmY+Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:05.475307Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.3186","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71fb43d376680db719a44823f0010ca7d8661ed5ac22e916eb0b9ffc4938ce81","sha256:b7f25d2fbd84619fdb07d41f9a6b7bc4d5b005504ae126580b23efc5a70b63cf"],"state_sha256":"8ba12b22b6ae7548c8001b085489a14bccbdef3df861052f70de3f4598a5cf18"}