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Any W$^*$-subalgebra $Q$ in a II$_1$ factor $M$ which admits a diffuse W$^*$-algebra $Q_0\\subset M$ that's free independent to $Q$, is compressible in $M$. We prove that if $Q\\subset M$ is compressible, then $_NL^2M_Q$ contains a copy of "},"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":"2510.17076","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-10-20T01:06:58Z","cross_cats_sorted":[],"title_canon_sha256":"9dab433212ed06c5ab83adf8360adfcff5f8a0895f74ef0a40c35c100d83e2de","abstract_canon_sha256":"4ae83cf330d206334bbe48758fa890f703bd8d45ad1075b08d18138933ab6b9e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T01:17:11.479755Z","signature_b64":"sw0te2zU6ekT7mcbBJSRB06YS4ap3SzICl2YzUUNs67Sflro/szgp39n9hNdfJY41F1I4xekWMZpVCaodyaKBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee66b879cac5fff4dd32ece90f087459fb5c8045d836eb0b5d9fbf162158d676","last_reissued_at":"2026-07-01T01:17:11.479223Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T01:17:11.479223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Compressible 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":"2025-10-20T01:06:58Z","abstract_excerpt":"Given a II$_1$ factor $M$, a W$^*$-subalgebra $Q\\subset M$ is {\\it compressible} if for any $\\varepsilon>0$ there exists a finite set of unitary elements $\\Cal U_0\\subset \\Cal U(M)$ such that $\\| \\frac{1}{|\\Cal U_0|}\\sum_{u\\in \\Cal U_0} uxu^* -E_{1\\otimes \\Bbb M_K(\\Bbb C)}(x)\\|\\leq \\varepsilon$, $\\forall K\\geq 1$, $\\forall x\\in (Q\\otimes \\Bbb M_K(\\Bbb C))_1$. Any W$^*$-subalgebra $Q$ in a II$_1$ factor $M$ which admits a diffuse W$^*$-algebra $Q_0\\subset M$ that's free independent to $Q$, is compressible in $M$. We prove that if $Q\\subset M$ is compressible, then $_NL^2M_Q$ contains a copy of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.17076","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.17076/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2510.17076","created_at":"2026-07-01T01:17:11.479290+00:00"},{"alias_kind":"arxiv_version","alias_value":"2510.17076v2","created_at":"2026-07-01T01:17:11.479290+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.17076","created_at":"2026-07-01T01:17:11.479290+00:00"},{"alias_kind":"pith_short_12","alias_value":"5ZTLQ6OKYX77","created_at":"2026-07-01T01:17:11.479290+00:00"},{"alias_kind":"pith_short_16","alias_value":"5ZTLQ6OKYX77JXJS","created_at":"2026-07-01T01:17:11.479290+00:00"},{"alias_kind":"pith_short_8","alias_value":"5ZTLQ6OK","created_at":"2026-07-01T01:17:11.479290+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/5ZTLQ6OKYX77JXJS5TUQ6CDULH","json":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH.json","graph_json":"https://pith.science/api/pith-number/5ZTLQ6OKYX77JXJS5TUQ6CDULH/graph.json","events_json":"https://pith.science/api/pith-number/5ZTLQ6OKYX77JXJS5TUQ6CDULH/events.json","paper":"https://pith.science/paper/5ZTLQ6OK"},"agent_actions":{"view_html":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH","download_json":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH.json","view_paper":"https://pith.science/paper/5ZTLQ6OK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2510.17076&json=true","fetch_graph":"https://pith.science/api/pith-number/5ZTLQ6OKYX77JXJS5TUQ6CDULH/graph.json","fetch_events":"https://pith.science/api/pith-number/5ZTLQ6OKYX77JXJS5TUQ6CDULH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH/action/storage_attestation","attest_author":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH/action/author_attestation","sign_citation":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH/action/citation_signature","submit_replication":"https://pith.science/pith/5ZTLQ6OKYX77JXJS5TUQ6CDULH/action/replication_record"}},"created_at":"2026-07-01T01:17:11.479290+00:00","updated_at":"2026-07-01T01:17:11.479290+00:00"}