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Peralta","submitted_at":"2018-04-11T06:15:37Z","abstract_excerpt":"Let $E$ and $P$ be subsets of a Banach space $X$, and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P) :=\\left\\{ x\\in P : \\|x-b\\|=1 \\hbox{ for all } b\\in E \\right\\}.$$ Given a C$^*$-algebra $A$, and a subset $E\\subset A,$ we shall write $Sph^+ (E)$ or $Sph_A^+ (E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ stands for the set of all positive operators in the unit sphere of $A$. We prove that, for an arbitrary complex Hilbert space $H$, then a positive element $a$ in the unit sphere of $B(H)$ is a projection if and only if $Sph^+_{B(H)} \\left( Sph^+_{B(H)}(\\{a\\}) \\right) ="},"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":"1804.04507","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-04-11T06:15:37Z","cross_cats_sorted":[],"title_canon_sha256":"3bd8bf06afe7f74e299ad20636ade209c36b979c436a5aa7fda96f9ae7b08b18","abstract_canon_sha256":"dbca1d3f916dd32ea8fec46877a8614ebff0e41e021fd26d00a1716a9be93045"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:37.349545Z","signature_b64":"bW2U427MC2EQkKjNEn6XLbSAN58OGefdQ3UUDgUfkiIsQDj9S4fcLzsPMDZr3LuV2amDJi2/V7InNwsi3huBAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d86173faf072d170512ab472040fe545f12414deb23f3527b9882b8bc6e6169","last_reissued_at":"2026-05-18T00:18:37.349020Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:37.349020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterizing projections among positive operators in the unit sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Antonio M. Peralta","submitted_at":"2018-04-11T06:15:37Z","abstract_excerpt":"Let $E$ and $P$ be subsets of a Banach space $X$, and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P) :=\\left\\{ x\\in P : \\|x-b\\|=1 \\hbox{ for all } b\\in E \\right\\}.$$ Given a C$^*$-algebra $A$, and a subset $E\\subset A,$ we shall write $Sph^+ (E)$ or $Sph_A^+ (E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ stands for the set of all positive operators in the unit sphere of $A$. We prove that, for an arbitrary complex Hilbert space $H$, then a positive element $a$ in the unit sphere of $B(H)$ is a projection if and only if $Sph^+_{B(H)} \\left( Sph^+_{B(H)}(\\{a\\}) \\right) ="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.04507","kind":"arxiv","version":1},"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":"1804.04507","created_at":"2026-05-18T00:18:37.349091+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.04507v1","created_at":"2026-05-18T00:18:37.349091+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.04507","created_at":"2026-05-18T00:18:37.349091+00:00"},{"alias_kind":"pith_short_12","alias_value":"TWDBOP5PA4WR","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"TWDBOP5PA4WROBIS","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"TWDBOP5P","created_at":"2026-05-18T12:32:56.356000+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/TWDBOP5PA4WROBISVNDSAQH6KR","json":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR.json","graph_json":"https://pith.science/api/pith-number/TWDBOP5PA4WROBISVNDSAQH6KR/graph.json","events_json":"https://pith.science/api/pith-number/TWDBOP5PA4WROBISVNDSAQH6KR/events.json","paper":"https://pith.science/paper/TWDBOP5P"},"agent_actions":{"view_html":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR","download_json":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR.json","view_paper":"https://pith.science/paper/TWDBOP5P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.04507&json=true","fetch_graph":"https://pith.science/api/pith-number/TWDBOP5PA4WROBISVNDSAQH6KR/graph.json","fetch_events":"https://pith.science/api/pith-number/TWDBOP5PA4WROBISVNDSAQH6KR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR/action/storage_attestation","attest_author":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR/action/author_attestation","sign_citation":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR/action/citation_signature","submit_replication":"https://pith.science/pith/TWDBOP5PA4WROBISVNDSAQH6KR/action/replication_record"}},"created_at":"2026-05-18T00:18:37.349091+00:00","updated_at":"2026-05-18T00:18:37.349091+00:00"}