{"paper":{"title":"Universality of free random variables: atoms for non-commutative rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Guillaume C\\'ebron, Octavio Arizmendi, Roland Speicher, Sheng Yin","submitted_at":"2021-07-24T01:43:56Z","abstract_excerpt":"Consider a tuple $(Y_1,\\dots,Y_d)$ of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each $Y_i$. We address the question what one can say about the sizes of the eigenspaces for any non-commutative polynomial $P(Y_1,\\dots,Y_d)$ in those operators? We show that for each polynomial $P$ there are unavoidable eigenspaces, which occur in $P(Y_1,\\dots,Y_d)$ for any $(Y_1,\\dots,Y_d)$ with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms - where it is given by realizations via matrices over"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.11507","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/2107.11507/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"}