{"paper":{"title":"Poisson geometrical aspects of the Tomita-Takesaki modular theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.OA","authors_text":"Anatol Odzijewicz, Daniel Beltita","submitted_at":"2019-10-31T13:46:50Z","abstract_excerpt":"We investigate some genuine Poisson geometric objects in the modular theory of an arbitrary von Neumann algebra $\\mathfrak{M}$. Specifically, for any standard form realization $(\\mathfrak{M},\\mathcal{H},J,\\mathcal{P})$, we find a canonical foliation of the Hilbert space $\\mathcal{H}$, whose leaves are Banach manifolds that are weakly immersed into~$\\mathcal{H}$, thereby endowing $\\mathcal{H}$ with a richer Banach manifold structure to be denoted by~$\\widetilde{\\mathcal{H}}$. We also find that $\\widetilde{\\mathcal{H}}$ has the structure of a Banach-Lie groupoid $\\widetilde{\\mathcal{H}}\\rightrig"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.14466","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/1910.14466/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"}