{"paper":{"title":"Characterizing group $C^\\ast$-algebras through their unitary groups: the Abelian case","license":"","headline":"","cross_cats":["math.GN","math.GR","math.KT"],"primary_cat":"math.OA","authors_text":"Ana-Mar'ia R'odenas, Jorge Galindo","submitted_at":"2007-04-27T15:06:52Z","abstract_excerpt":"We study to what extent group $C^\\ast$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^\\ast$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of $C^\\ast(\\Gamma)$, such as the $K$-theoretic $K_1(C^\\ast(\\Gamma))$ and find that $C^\\ast$-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic $K_1$-groups, in sharp contrast with the well-known fact that $C^\\ast(\\Gamma)$ (even $\\Gamma$) is characterized by the topological group structure of its unitary group"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.3687","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"}