{"paper":{"title":"On finite sums of projections and Dixmier's averaging theorem for type ${\\rm II}_1$ factors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Junsheng Fang, Xinyan Cao, Zhaolin Yao","submitted_at":"2020-12-01T12:21:58Z","abstract_excerpt":"Let $\\mathcal{M}$ be a type ${\\rm II_1}$ factor and let $\\tau$ be the faithful normal tracial state on $\\mathcal{M}$. In this paper, we prove that given an $X \\in \\mathcal{M}$, $X=X^*$, then there is a decomposition of the identity into $N \\in \\mathbb{N}$ mutually orthogonal nonzero projections $E_j\\in\\mathcal{M}$, $I=\\sum_{j=1}^NE_j$, such that $E_jXE_j=\\tau(X) E_j$ for all $j=1,\\cdots,N$. Equivalently, there is a unitary operator $U \\in \\mathcal{M}$ with $U^N=I$ and $\\frac{1}{N}\\sum_{j=0}^{N-1}{U^*}^jXU^j=\\tau(X)I.$ As the first application, we prove that a positive operator $A\\in \\mathcal{M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.00440","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2012.00440/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"}