{"paper":{"title":"Simultaneous power factorization in modules over Banach algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Marcel de Jeu, Xingni Jiang","submitted_at":"2016-10-06T14:24:05Z","abstract_excerpt":"Let $A$ be a Banach algebra with a bounded left approximate identity $\\{e_\\lambda\\}_{\\lambda\\in\\Lambda}$, let $\\pi$ be a continuous representation of $A$ on a Banach space $X$, and let $S$ be a non-empty subset of $X$ such that $\\lim_{\\lambda}\\pi(e_\\lambda)s=s$ uniformly on $S$. If $S$ is bounded, or if $\\{e_\\lambda\\}_{\\lambda\\in\\Lambda}$ is commutative, then we show that there exist $a\\in A$ and maps $x_n: S\\to X$ for $n\\geq 1$ such that $s=\\pi(a^n)x_n(s)$ for all $n\\geq 1$ and $s\\in S$. The properties of $a\\in A$ and the maps $x_n$, as produced by the constructive proof, are studied in some "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.01885","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"}