{"paper":{"title":"Infinite order decompositions of C$^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"F.N. Arzikulov","submitted_at":"2011-03-17T13:23:34Z","abstract_excerpt":"In the given article infinite order decompositions of C$^*$-algebras are investigated. We give complete proofs of the following statements:\n  1) If the order unit space $\\sum_{\\xi,\\eta}^\\oplus p_\\xi Ap_\\eta$ is monotone complete in $B(H)$ (i.e. ultraweakly closed), then $\\sum_{\\xi,\\eta}^\\oplus p_\\xi Ap_\\eta$ is a C$^*$-algebra.\n  2) If $A$ is monotone complete in $B(H)$ (i.e. a von Neumann algebra), then $A=\\sum_{\\xi,\\eta}^\\oplus p_\\xi Ap_\\eta$.\n  3) If $\\sum_{\\xi,\\eta}^\\oplus p_\\xi Ap_\\eta$ is a C$^*$-algebra then this algebra is a von Neumann algebra."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3404","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":""},"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"}