{"paper":{"title":"On Diximier's averaging theorem for operators in type ${\\rm II}_1$ factors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Junsheng Fang, Shilin Wen, Zhaolin Yao","submitted_at":"2023-03-19T08:39:20Z","abstract_excerpt":"Let $\\M$ be a type ${\\rm II_1}$ factor and let $\\tau$ be the faithful normal tracial state on $\\M$. In this paper, we prove that given finite elements $X_1,\\cdots X_n \\in \\M$, there is a finite decomposition of the identity into $N \\in \\NNN$ mutually orthogonal nonzero projections $E_j\\in\\M$, $I=\\sum_{j=1}^NE_j$, such that $E_jX_iE_j=\\tau(X_i) E_j$ for all $j=1,\\cdots,N$ and $i=1,\\cdots,n$. Equivalently, there is a unitary operator $U \\in \\M$ such that $\\frac{1}{N}\\sum_{j=0}^{N-1}{U^*}^jX_iU^j=\\tau(X_i)I$ for $i=1,\\cdots,n$. This result is a stronger version of Dixmier's averaging theorem for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.10602","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/2303.10602/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"}