{"paper":{"title":"B-Valued Free Convolution for Unbounded Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.OA","authors_text":"John D. Williams","submitted_at":"2015-07-09T16:23:58Z","abstract_excerpt":"Consider the $\\mathcal{B}$-valued probability space $(\\mathcal{A}, E, \\mathcal{B})$, where $\\mathcal{A}$ is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators $\\tilde{\\mathcal{A}}$. For a random variable $X \\in \\tilde{\\mathcal{A}}^{sa}$ we study the Cauchy transform $G_{X}$ and show that the operator algebra $(\\mathcal{B} \\cup \\{X\\})\"$ can be recovered from this function. In the case where $\\mathcal{B}$ is finite dimensional, we show that, when $X, Y \\in \\tilde{\\mathcal{A}}^{sa}$ are assumed to be $\\mathcal{B}$-free, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02580","kind":"arxiv","version":3},"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"}