{"paper":{"title":"Noncommutative Riesz transforms -- a probabilistic approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Marius Junge, Tao Mei","submitted_at":"2008-01-12T01:37:17Z","abstract_excerpt":"For $2\\le p<\\infty$ we show the lower estimates\n  \\[ \\|A^{\\frac 12}x\\|_p \\kl c(p)\\max\\{\\pl \\|\\Gamma(x,x)^{{1/2}}\\|_p,\\pl \\|\\Gamma(x^*,x^*)^{{1/2}}\\|_p\\} \\] for the Riesz transform associated to a semigroup $(T_t)$ of completely positive maps on a von Neumann algebra with negative generator $T_t=e^{-tA}$, and gradient form \\[ 2\\Gamma(x,y)\\lel Ax^*y+x^*Ay-A(x^*y)\\pl .\\] As additional hypothesis we assume that $\\Gamma^2\\gl 0$ and the existence of a Markov dilation for $(T_t)$. We give applications to quantum metric spaces and show the equivalence of semigroup Hardy norms and martingale Hardy norm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0801.1873","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0801.1873/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}