{"paper":{"title":"Berkovich spectra of elements in Banach Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.NT","math.RA","math.SP"],"primary_cat":"math.FA","authors_text":"Chi-Keung Ng, Chi-Wai Leung","submitted_at":"2014-10-22T01:31:00Z","abstract_excerpt":"Adapting the notion of the spectrum $\\Sigma_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum $\\sigma^{Ber}_R(u)$ of an element $u$ in a Banach ring $R$. This spectrum is a compact subset of the affine analytic space $A_Z^1$ over $Z$, and the later can be identified with the \"equivalence classes\" of all elements in all complete valuation fields.\n  If $R$ is generated by $u$ as a unital Banach ring, then $\\sigma^{Ber}_R(u)$ coincides with the spectrum of $R$ (as defined by Berkovich). If $R$ is a unital comple"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5893","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"}