{"paper":{"title":"Comparison and pure infiniteness of crossed products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.DS","authors_text":"Xin Ma","submitted_at":"2018-08-07T14:40:01Z","abstract_excerpt":"Let $\\alpha: G\\curvearrowright X$ be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action $\\alpha$ is topologically free and has no $G$-invariant regular Borel probability measure on $X$, dynamical comparison implies that the reduced crossed product of $\\alpha$ is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.02404","kind":"arxiv","version":3},"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"}