{"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. 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\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02707","kind":"arxiv","version":1},"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"}