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We prove that if $\\text{\\bf Q}\\subset \\text{\\bf M}$ is either an ultraproduct $\\text{\\bf Q}=\\Pi_\\omega Q_n$ of subalgebras $Q_n\\subset M_n$, with $Q_n \\not\\prec_{M_n} Q_n'\\cap M_n$, $\\forall n$, or the centralizer $\\text{\\bf Q}=B'\\cap \\text{\\bf M}$ of a separable amenable *-subalgebra $B\\subset \\text{\\bf M}$, then for any separable subspace $X\\subset \\text{\\bf M}\\ominus (\\text{\\bf Q}'\\cap \\text{\\bf M})$, there exists a diffus"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1308.3982","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-08-19T11:06:15Z","cross_cats_sorted":[],"title_canon_sha256":"3a1fbfb35425c1bb93bae7334da363d02ec2142e64b650e2fd2c24bb47cb09a9","abstract_canon_sha256":"61ff2e547fbfbcc4e9e241fdcf66868e0b17dec64ac9c70068edeb6b661848f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:39.317120Z","signature_b64":"t8RxR+Oae5tSdUnMjRvI2wGoQoLC0+KIyTfiDzo0Ly9k74dpa6I4FFKjfVxlxDqdLsHWEtXL/EMzp0SbSl1LDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c849cb472cbb88acbbe91789e7278ce7743e6753d1fe44920191c621c4273cd6","last_reissued_at":"2026-05-18T03:00:39.316290Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:39.316290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Independence properties in subalgebras of ultraproduct II$_1$ factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Sorin Popa","submitted_at":"2013-08-19T11:06:15Z","abstract_excerpt":"Let $M_n$ be a sequence of finite factors with $\\dim(M_n)\\rightarrow \\infty$ and denote $\\text{\\bf M}=\\Pi_\\omega M_n$ their ultraproduct over a free ultrafilter $\\omega$. We prove that if $\\text{\\bf Q}\\subset \\text{\\bf M}$ is either an ultraproduct $\\text{\\bf Q}=\\Pi_\\omega Q_n$ of subalgebras $Q_n\\subset M_n$, with $Q_n \\not\\prec_{M_n} Q_n'\\cap M_n$, $\\forall n$, or the centralizer $\\text{\\bf Q}=B'\\cap \\text{\\bf M}$ of a separable amenable *-subalgebra $B\\subset \\text{\\bf M}$, then for any separable subspace $X\\subset \\text{\\bf M}\\ominus (\\text{\\bf Q}'\\cap \\text{\\bf M})$, there exists a diffus"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3982","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1308.3982","created_at":"2026-05-18T03:00:39.316446+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.3982v3","created_at":"2026-05-18T03:00:39.316446+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.3982","created_at":"2026-05-18T03:00:39.316446+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZBE4WRZMXOEK","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZBE4WRZMXOEKZO7J","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZBE4WRZM","created_at":"2026-05-18T12:28:09.283467+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45","json":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45.json","graph_json":"https://pith.science/api/pith-number/ZBE4WRZMXOEKZO7JC6E6OJ4M45/graph.json","events_json":"https://pith.science/api/pith-number/ZBE4WRZMXOEKZO7JC6E6OJ4M45/events.json","paper":"https://pith.science/paper/ZBE4WRZM"},"agent_actions":{"view_html":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45","download_json":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45.json","view_paper":"https://pith.science/paper/ZBE4WRZM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.3982&json=true","fetch_graph":"https://pith.science/api/pith-number/ZBE4WRZMXOEKZO7JC6E6OJ4M45/graph.json","fetch_events":"https://pith.science/api/pith-number/ZBE4WRZMXOEKZO7JC6E6OJ4M45/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45/action/storage_attestation","attest_author":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45/action/author_attestation","sign_citation":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45/action/citation_signature","submit_replication":"https://pith.science/pith/ZBE4WRZMXOEKZO7JC6E6OJ4M45/action/replication_record"}},"created_at":"2026-05-18T03:00:39.316446+00:00","updated_at":"2026-05-18T03:00:39.316446+00:00"}