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A complement to the main result, which may be new even if h = 1, is that if a and b are self-adjoint in A, a \\le h, and b \\le h - delta 1 for delta > 0, then there is a self"},"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":"1404.1383","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-04-04T20:04:52Z","cross_cats_sorted":[],"title_canon_sha256":"0132dbd25bf677a3fde3474f4b004f3d38e9035a4515fac637a6d16fdcdf7b13","abstract_canon_sha256":"cfaf91548760bd5b3de4b04f8163a9d58f30e9a570f3ed06568c5cfdc95c6116"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:49.229124Z","signature_b64":"LFISoOOeG5oCLg3xmwLTbQeiBR/EFMUO6rX+GQ01iv/JkpEqIHsPaUTCBTraH+MEMf3UUUs8nyxtkpxjkjKnCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4aec757b4b509be3bca91c9671a19053daaddb96b75e72398275daf5486046a2","last_reissued_at":"2026-05-18T00:42:49.228550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:49.228550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some directed subsets of C*-algebras and semicontinuity theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Lawrence G. 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