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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","authors_text":"Junsheng Fang, Xinyan Cao, Zhaolin Yao","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2020-12-01T12:21:58Z","title":"On finite sums of projections and Dixmier's averaging theorem for type ${\\rm II}_1$ factors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.00440","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:35d523ad2cb1a2b9d0312d28686e3785a0daeb1a38a789c88810ce96b3aae492","target":"record","created_at":"2026-07-05T02:40:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ea1fbad8cc18aba3537497d014748ab1aaf144142a6b1ebb396ee07df3dc84c9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2020-12-01T12:21:58Z","title_canon_sha256":"58a732355b06d8dba2e6c83d741314fd194aca1d262a5b407211f6ef5154f8fc"},"schema_version":"1.0","source":{"id":"2012.00440","kind":"arxiv","version":2}},"canonical_sha256":"50f8cf38b20f6e5a173a4d600ac9c5ea6cc909efb6bdd77e511c55ef2b53f896","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50f8cf38b20f6e5a173a4d600ac9c5ea6cc909efb6bdd77e511c55ef2b53f896","first_computed_at":"2026-07-05T02:40:30.366068Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:40:30.366068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SKBJVvYCFvpGroRzqysdMNC4y5NKUT/dVYDn3bjxY8M7f76WN9hzgomAnsL3Gj2X+r6qM3BnTp9yKKXnTEtPDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:40:30.366463Z","signed_message":"canonical_sha256_bytes"},"source_id":"2012.00440","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35d523ad2cb1a2b9d0312d28686e3785a0daeb1a38a789c88810ce96b3aae492","sha256:f34f2cb6b8a97b676b59c49a840eea9694593c3baf5ce4522de0e6907f5e9370"],"state_sha256":"9c919ec48e5be49c64f0968cefdba86f7f1de96b0a2c95e62e6837087cec4f93"}