{"paper":{"title":"Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.FA","authors_text":"Mehdi Ghasemi, Murray Marshall, Salma Kuhlmann","submitted_at":"2012-09-13T17:18:28Z","abstract_excerpt":"Let $A$ be a commutative unital $\\mathbb{R}$-algebra and let $\\rho$ be a seminorm on $A$ which satisfies $\\rho(ab)\\leq\\rho(a)\\rho(b)$. We apply T. Jacobi's representation theorem to determine the closure of a $\\sum A^{2d}$-module $S$ of $A$ in the topology induced by $\\rho$, for any integer $d\\ge1$. We show that this closure is exactly the set of all elements $a\\in A$ such that $\\alpha(a)\\ge0$ for every $\\rho$-continuous $\\mathbb{R}$-algebra homomorphism $\\alpha : A \\rightarrow \\mathbb{R}$ with $\\alpha(S)\\subseteq[0,\\infty)$, and that this result continues to hold when $\\rho$ is replaced by an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2966","kind":"arxiv","version":1},"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"}