{"paper":{"title":"Noncommutative Fractional integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OA","authors_text":"Lian Wu, Narcisse Randrianantoanina","submitted_at":"2015-01-24T09:36:33Z","abstract_excerpt":"Let $\\M$ be a hyperfinite finite von Nemann algebra and $(\\M_k)_{k\\geq 1}$ be an increasing filtration of finite dimensional von Neumann subalgebras of $\\M$. We investigate abstract fractional integrals associated to the filtration $(\\M_k)_{k\\geq 1}$. For a finite noncommutative martingale $x=(x_k)_{1\\leq k\\leq n} \\subseteq L_1(\\M)$ adapted to $(\\M_k)_{k\\geq 1}$ and $0<\\alpha<1$, the fractional integral of $x$ of order $\\alpha$ is defined by setting: $$I^\\alpha x = \\sum_{k=1}^n \\zeta_k^{\\alpha} dx_k$$ for an appropriate sequence of scalars $(\\zeta_k)_{k\\geq 1}$. For the case of noncommutative "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06016","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"}