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We obtain this result by associating to abelian, but not maximal abelian, subalgebras of a II_1 factor, an equivalence relation that can be of type III. In particular, we associate to L(BS(n,m)) a canonical equivalence relation of type III_|n/m|."},"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":"1301.0510","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-01-03T17:10:32Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d6941a588a2c9b95fe842bfe199f6558b5cc39e1c96ee4bd494a24a6e97c250e","abstract_canon_sha256":"b2ba639aeaaa4b7c4494cb05a1b3116249dc96f1a8dbb4000aac102063f58bd1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:04.745086Z","signature_b64":"jRUBZbMokww8g/Y+PyybxLUEegqkQGizhQ7ZVG2ObAxnop1KlrWOcgyojdeOtbG+ATY3lakH5p4n/0sIKmEyAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5784fff453d9dad2699a64ff898d62ea74842dcbbc8bb202ef7ef08a60cf130c","last_reissued_at":"2026-05-18T02:46:04.744588Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:04.744588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partial classification of the Baumslag-Solitar group von Neumann algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.OA","authors_text":"Niels Meesschaert, Stefaan Vaes","submitted_at":"2013-01-03T17:10:32Z","abstract_excerpt":"We prove that the rational number |n/m| is an invariant of the group von Neumann algebra of the Baumslag-Solitar group BS(n,m). More precisely, if L(BS(n,m)) is isomorphic with L(\\BS(n',m')), then |n'/m'| = |n/m| or |m/n|. We obtain this result by associating to abelian, but not maximal abelian, subalgebras of a II_1 factor, an equivalence relation that can be of type III. 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