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We first show that any intermediate subalgebra $M_1 \\subset Q \\subset M$ which has nontrivial central sequences in $M$ is necessarily equal to $M_1$. Then we obtain a general structural result for all the intermediate subalgebras $M_1 \\subset Q \\subset M$ with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra wit"},"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":"1403.4098","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-03-17T14:00:43Z","cross_cats_sorted":[],"title_canon_sha256":"e6d3411c6f5463b85213a0a725da25b2e5c33dc6ab350eb5ba3e66b1fb352fce","abstract_canon_sha256":"ee999524edabfe258fcc5a51890c1f5858dd8f222484d6c3bb22cae43acd0873"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:44:00.485407Z","signature_b64":"kWX9HOUxqbFKFLWNihJOjxWhrDJ8PzeRX28T/qMSf7UxpLitjdmbeFa6EiTedtw/G0JLXGiPWsp5p7EgPOiHCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd51bcd53a3a20bf0c16d2e0e2d6987cfae000e755e639039339fcb9bd453098","last_reissued_at":"2026-05-18T01:44:00.484853Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:44:00.484853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gamma stability in free product von Neumann algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Cyril Houdayer","submitted_at":"2014-03-17T14:00:43Z","abstract_excerpt":"Let $(M, \\varphi) = (M_1, \\varphi_1) \\ast (M_2, \\varphi_2)$ be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer $M_1^{\\varphi_1}$ is diffuse. We first show that any intermediate subalgebra $M_1 \\subset Q \\subset M$ which has nontrivial central sequences in $M$ is necessarily equal to $M_1$. Then we obtain a general structural result for all the intermediate subalgebras $M_1 \\subset Q \\subset M$ with expectation. 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