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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1103.3404","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2011-03-17T13:23:34Z","cross_cats_sorted":[],"title_canon_sha256":"971d678ad574fa381bf51faa07a84ecf5d3a4af08bb1601bb5d83d69b40cace4","abstract_canon_sha256":"dee9d08bb403802b684f6bed4b3b99aed210b08e4fe143ebd57c166bed4ff45b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:20.352072Z","signature_b64":"nPa96CI8DrQ3W9pmE2qh/zfpMhmK7CVPU5RJqqEkad/yWALHtUrI8ct+hJqg8tXm1ZSKr30Q6d0VY0zQDSZODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be7c8b6977c6d23524689947a35a7f461be962d54fe78e9ad06dafa039d2692d","last_reissued_at":"2026-05-18T03:12:20.351276Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:20.351276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1103.3404","created_at":"2026-05-18T03:12:20.351428+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.3404v2","created_at":"2026-05-18T03:12:20.351428+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3404","created_at":"2026-05-18T03:12:20.351428+00:00"},{"alias_kind":"pith_short_12","alias_value":"XZ6IW2LXY3JD","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"XZ6IW2LXY3JDKJDI","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"XZ6IW2LX","created_at":"2026-05-18T12:26:47.523578+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY","json":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY.json","graph_json":"https://pith.science/api/pith-number/XZ6IW2LXY3JDKJDITFD2GWT7IY/graph.json","events_json":"https://pith.science/api/pith-number/XZ6IW2LXY3JDKJDITFD2GWT7IY/events.json","paper":"https://pith.science/paper/XZ6IW2LX"},"agent_actions":{"view_html":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY","download_json":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY.json","view_paper":"https://pith.science/paper/XZ6IW2LX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.3404&json=true","fetch_graph":"https://pith.science/api/pith-number/XZ6IW2LXY3JDKJDITFD2GWT7IY/graph.json","fetch_events":"https://pith.science/api/pith-number/XZ6IW2LXY3JDKJDITFD2GWT7IY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY/action/storage_attestation","attest_author":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY/action/author_attestation","sign_citation":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY/action/citation_signature","submit_replication":"https://pith.science/pith/XZ6IW2LXY3JDKJDITFD2GWT7IY/action/replication_record"}},"created_at":"2026-05-18T03:12:20.351428+00:00","updated_at":"2026-05-18T03:12:20.351428+00:00"}