{"paper":{"title":"$\\mathrm{W}^*$-algebraic Integration Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jan G{\\l}owacki, Yui Kuramochi","submitted_at":"2026-06-25T17:59:24Z","abstract_excerpt":"Given a pair of $\\mathrm{W}^*$-algebras $(\\mathcal{M}_\\mathcal{S},\\mathcal{M}_\\mathcal{R})$ with $(\\mathcal{M}_\\mathcal{S})_*$ separable, a measurable space $(\\Sigma, \\mathcal{F})$ and a POVM $\\mathsf{E}: \\mathcal{F} \\to \\mathcal{E}(\\mathcal{M}_\\mathcal{R})$, the integral of a function $f: \\Sigma \\to \\mathcal{M}_\\mathcal{S}$ is defined as an element of the spatial tensor product $\\int f \\otimes d\\mathsf{E} \\in \\mathcal{M}_\\mathcal{S} \\bar{\\otimes} \\mathcal{M}_\\mathcal{R}$. The space $B_b(\\Sigma,\\mathcal{F},\\mathcal{M}_\\mathcal{S})$ of uniformly bounded ultraweakly measurable functions is the u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27366","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.27366/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"}