{"paper":{"title":"The Selfless Dichotomy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Miles Gould","submitted_at":"2026-06-08T15:42:45Z","abstract_excerpt":"The purpose of this note is to address the gap in the stably finite/purely infinite dichotomy of selfless $C^*$-probability spaces. In particular, we show that nonfaithful selfless $C^*$-probability spaces are purely infinite, simple. This completes the dichotomy: Every selfless $C^*$-algebra is either purely infinite or stably finite. Notably, this shows that every selfless $C^*$-algebra is pure. To accomplish this, we show that infinite reduced free products of $C^*$-probability spaces with nonfaithful states inducing faithful GNS representations are often purely infinite, simple. Having res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09654","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/2606.09654/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"}