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We prove that an element $u\\in \\partial_e(\\mathcal{B}_M)$ is a unitary if and only if the set $$\\mathcal{M}_{u} = \\{e\\in \\partial_e(\\mathcal{B}_M) : \\|u\\pm e\\|\\leq \\sqrt{2} \\}$$ contains an isolated point. This is a new geometric characterisation of unitaries in $M$ in terms of the set of extreme points of $\\mathcal{B}_M$."},"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":"1907.04738","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2019-07-10T14:09:05Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"70bf1fd4bc40d4724f2a8115e5060025024b3e5b1a62b12974ea7475ee256120","abstract_canon_sha256":"b4a332cbb1e7c5eada7f1087718f8210e67d786f65ec8ec5e58f2dd311bf08f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:56.939870Z","signature_b64":"e20miqacU3GUBA/6HTH4A/R70mExoDG/OVTsaBKygguirHCf7TtrMZME40PcuEfU7arW1e24cCzaZhqCKtq5Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95279e6e2fc35ec4e947c838bf5d6522d533f6dd53e7ac33cedb182dff5e9c04","last_reissued_at":"2026-05-17T23:40:56.939056Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:56.939056Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Metric characterisation of unitaries in JB$^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Antonio M. Peralta, Mar\\'ia Cueto-Avellaneda","submitted_at":"2019-07-10T14:09:05Z","abstract_excerpt":"Let $M$ be a unital JB$^*$-algebra whose closed unit ball is denoted by $\\mathcal{B}_M$. Let $\\partial_e(\\mathcal{B}_M)$ denote the set of all extreme points of $\\mathcal{B}_M$. We prove that an element $u\\in \\partial_e(\\mathcal{B}_M)$ is a unitary if and only if the set $$\\mathcal{M}_{u} = \\{e\\in \\partial_e(\\mathcal{B}_M) : \\|u\\pm e\\|\\leq \\sqrt{2} \\}$$ contains an isolated point. 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