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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1209.2966","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-13T17:18:28Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"59a4427b01269390819a7b8b29883b3d02e81268f03b47cf0e2719dcfbf7d5c8","abstract_canon_sha256":"de16df7713d04df2efaecdfac1c9b39da478d802c1a8fc3d4e893e4029c6b74b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:49.897519Z","signature_b64":"FQRD6BzDho2e6BMwWv9OCwZTxCo6P+GcWmMsIIGA4Z7cCyl9dDB/JdN0wpDfd6AVAaoElZYWq/9iG+tlWn7HCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64c5835a7d20ae87c90fdf26c03ef29982c4c03bf2c62c1678889eabd6b2540d","last_reissued_at":"2026-05-18T03:04:49.896797Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:49.896797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\rho(ab)\\leq\\rho(a)\\rho(b)$. 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