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Let $\\E_n$ denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for $1 < p < 2$ and $x \\in L_p(\\M,\\tau)$ one can find $a, b \\in L_p(\\M,\\tau)$ and contractions $u_n, v_n \\in \\M$ such that $$\\E_n(x) = a u_n + v_n b \\quad \\mbox{and} \\quad \\max \\big\\{ \\|a\\|_p, \\|b\\|_p \\big\\"},"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":"1507.02707","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-07-09T20:49:14Z","cross_cats_sorted":["math.CA","math.PR"],"title_canon_sha256":"c66878a7319ed807f5a335a7c7f4a10daec242d93d584aa97faca711b7841595","abstract_canon_sha256":"5235f88e6a7bcf30a399acd85f37a54d8ad92a365e60d8ae6deb27f6d9ca0bc4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:52.766392Z","signature_b64":"LnoKIgtTHUTS1Pol36kfWUEnDdFEhKApurvtiYKBjBcV9ZaJSEkex1+QJTN7aEy2ZECFjq0GzsJjlRNlyRm1BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92027d2fba86b3381631e1d96206c257579a457f436c22da03fe7edfef0a4801","last_reissued_at":"2026-05-18T01:15:52.765826Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:52.765826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic Davis decomposition and asymmetric Doob inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.PR"],"primary_cat":"math.OA","authors_text":"Guixiang Hong, Javier Parcet, Marius Junge","submitted_at":"2015-07-09T20:49:14Z","abstract_excerpt":"In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\\M,\\tau)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann subalgebras $(\\M_n)_{n \\ge 1}$. Let $\\E_n$ denote the corresponding family of conditional expectations. 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