{"paper":{"title":"Obstructions to lifting cocycles on groupoids and the associated $C^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Alex Kumjian, Marius Ionescu","submitted_at":"2016-12-21T18:08:33Z","abstract_excerpt":"Given a short exact sequence of locally compact abelian groups $0 \\to A \\to B \\to C \\to 0$ and a continuous $C$-valued $1$-cocycle $\\phi$ on a locally compact Hausdorff groupoid $\\Gamma$ we construct a twist of $\\Gamma$ by $A$ that is trivial if and only if $\\phi$ lifts. The cocycle determines a strongly continuous action of $\\widehat{C}$ into $\\operatorname{Aut} C^*(\\Gamma)$ and we prove that the $C^*$-algebra of the twist is isomorphic to the induced algebra of this action if $\\Gamma$ is amenable. We apply our results to a groupoid determined by a locally finite cover of a space $X$ and a co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07257","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"}