{"paper":{"title":"Reduction of free independence to tensor independence","license":"","headline":"","cross_cats":["math.OA","math.RA"],"primary_cat":"math.QA","authors_text":"Romuald Lenczewski","submitted_at":"2002-10-23T12:12:36Z","abstract_excerpt":"We show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators `affiliated' with a given unital *-algebra and call the associated closure `monotone'. Then we prove that monotone closed operators of the form $$ X'= \\sum_{k=1}^{\\infty}X(k)\\bar{\\otimes} p_{k}, X''=\\sum_{k=1}^{\\infty} p_{k}\\bar{\\otimes}X(k) $$ are free with respect to a tensor product state, where $X(k)$ are tensor independent copies of a random variable $X$ and $(p_{k})$ is a sequence of orthogonal projections. For unital free *-algebras, we constru"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0210358","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":""},"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"}