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We prove that given any separable subalgebra $B$ of the ultrapower II$_1$ factor $M^\\omega$, for a non-principal ultrafilter $\\omega$ on $\\Bbb N$, there exists a unitary element $u\\in M^\\omega$ such that $uBu^*$ is orthogonal to $Q^\\omega$."},"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":"1707.07317","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-07-23T15:57:26Z","cross_cats_sorted":[],"title_canon_sha256":"aca8cc4082de7ec21a34ad7c76b65199b201a3b966af384c66b864fc90900fa5","abstract_canon_sha256":"e3850316c4e9ab14422220f76b231150d5cbe64d27896eab54c44619aa8bbece"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:49.831404Z","signature_b64":"Dkhhg7HF63bLTwLPzb0ItEdjGdL6K4sWCiQSQNcprm7xsEzKqd+dTgCteZcNnRNXM+GGahfLnVM35RlsZ97jCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89fe7c1dc5854e736f054e9601ea0dd3d06b98228bda4abb1b2989bd9f2b3983","last_reissued_at":"2026-05-18T00:02:49.830839Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:49.830839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic orthogonalization of subalgebras in 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":"2017-07-23T15:57:26Z","abstract_excerpt":"Let $M$ be a II$_1$ factor with a von Neumann subalgebra $Q\\subset M$ that has infinite index under any projection in $Q'\\cap M$ (e.g., $Q$ abelian; or $Q$ an irreducible subfactor with infinite Jones index). We prove that given any separable subalgebra $B$ of the ultrapower II$_1$ factor $M^\\omega$, for a non-principal ultrafilter $\\omega$ on $\\Bbb N$, there exists a unitary element $u\\in M^\\omega$ such that $uBu^*$ is orthogonal to $Q^\\omega$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07317","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":"1707.07317","created_at":"2026-05-18T00:02:49.830933+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.07317v3","created_at":"2026-05-18T00:02:49.830933+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.07317","created_at":"2026-05-18T00:02:49.830933+00:00"},{"alias_kind":"pith_short_12","alias_value":"RH7HYHOFQVHH","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_16","alias_value":"RH7HYHOFQVHHG3YF","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_8","alias_value":"RH7HYHOF","created_at":"2026-05-18T12:31:39.905425+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/RH7HYHOFQVHHG3YFJ2LAD2QN2P","json":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P.json","graph_json":"https://pith.science/api/pith-number/RH7HYHOFQVHHG3YFJ2LAD2QN2P/graph.json","events_json":"https://pith.science/api/pith-number/RH7HYHOFQVHHG3YFJ2LAD2QN2P/events.json","paper":"https://pith.science/paper/RH7HYHOF"},"agent_actions":{"view_html":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P","download_json":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P.json","view_paper":"https://pith.science/paper/RH7HYHOF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.07317&json=true","fetch_graph":"https://pith.science/api/pith-number/RH7HYHOFQVHHG3YFJ2LAD2QN2P/graph.json","fetch_events":"https://pith.science/api/pith-number/RH7HYHOFQVHHG3YFJ2LAD2QN2P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P/action/storage_attestation","attest_author":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P/action/author_attestation","sign_citation":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P/action/citation_signature","submit_replication":"https://pith.science/pith/RH7HYHOFQVHHG3YFJ2LAD2QN2P/action/replication_record"}},"created_at":"2026-05-18T00:02:49.830933+00:00","updated_at":"2026-05-18T00:02:49.830933+00:00"}