{"paper":{"title":"Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.MP","quant-ph"],"primary_cat":"math-ph","authors_text":"Davide Pastorello (Math. Dept. - Trento University), Valter Moretti","submitted_at":"2012-05-21T07:32:55Z","abstract_excerpt":"Consider a finite dimensional complex Hilbert space $\\cH$, with $dim(\\cH) \\geq 3$, define $\\bS(\\cH):= \\{x\\in \\cH \\:|\\: ||x||=1\\}$, and let $\\nu_\\cH$ be the unique regular Borel positive measure invariant under the action of the unitary operators in $\\cH$, with $\\nu_\\cH(\\bS(\\cH))=1$. We prove that if a complex frame function $f : \\bS(\\cH)\\to \\bC$ satisfies $f \\in \\cL^2(\\bS(\\cH), \\nu_\\cH)$, then it verifies Gleason's statement: There is a unique linear operator $A: \\cH \\to \\cH$ such that $f(u) = < u| A u>$ for every $u \\in \\bS(\\cH)$. $A$ is Hermitean when $f$ is real. No boundedness requirement "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4504","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"}