{"paper":{"title":"Divisibility and Real Rank Zero","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Xuanlong Fu","submitted_at":"2026-05-20T19:10:13Z","abstract_excerpt":"Let $A$ be a simple separable exact $C^*$-algebra that has traces. We show the following existed regularity properties are equivalent:\n  \\quad(1) $l^\\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace kernel ideal.\n  \\quad(2) $A$ is tracially almost divisible.\n  \\quad(3) $A$ is tracially $m$-almost divisible for some $m\\in\\N\\cup\\{0\\}.$\n  \\quad(4) $A$ has tracial approximate oscillation zero.\n  \\quad(5) $A$ has Property (TM).\n  We also show that for an algebraically simple separable stable rank one \\CA\\ $B$ with non-empty compact ${\\rm T}(B)$ and locally finite nuclear dimension, its un"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21655","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.21655/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}