{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MCRXJ5XZ6SB67PVGXOOGYQORMP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"927dec3b70108d0e1a24fd3e76f0f21c7a6f3e33d8c5bdf4026454588f7eccdc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-01T04:41:23Z","title_canon_sha256":"a1bf9e083af085dbc2b229d1ee7e1fd5397d5276865892003e9b4d2c3b22dbda"},"schema_version":"1.0","source":{"id":"1209.0055","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.0055","created_at":"2026-05-18T03:46:30Z"},{"alias_kind":"arxiv_version","alias_value":"1209.0055v1","created_at":"2026-05-18T03:46:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0055","created_at":"2026-05-18T03:46:30Z"},{"alias_kind":"pith_short_12","alias_value":"MCRXJ5XZ6SB6","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MCRXJ5XZ6SB67PVG","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MCRXJ5XZ","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:8f7af102cc12ab3820de315dc73a04cc2efd49a4d3a053b18f82eb774fb3c8fe","target":"graph","created_at":"2026-05-18T03:46:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce the (T)-property, and prove that every Banach space with the (T)-property has the Mazur-Ulam property (briefly MUP). As its immediate applications, we obtain that almost-CL-spaces admitting a smooth point(specially, separable almost-CL-spaces) and a two-dimensional space whose unit sphere is a hexagon has the MUP. Furthermore, we discuss the stability of the spaces having the MUP by the $c_0$- and $\\ell_1$-sums, and show that the space $C(K,X)$ of the vector-valued continuous functions has the the MUP, where $X$ is a separable almost-CL-space and $K$ is a compact metric space.","authors_text":"Dongni Tan, Rui Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-01T04:41:23Z","title":"A Note on The Mazur-Ulam Property of Almost-CL-spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0055","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fb23a5e195c589693c0104688abf04fb91263b97470708a2dac81a1cc6b52ed7","target":"record","created_at":"2026-05-18T03:46:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"927dec3b70108d0e1a24fd3e76f0f21c7a6f3e33d8c5bdf4026454588f7eccdc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-01T04:41:23Z","title_canon_sha256":"a1bf9e083af085dbc2b229d1ee7e1fd5397d5276865892003e9b4d2c3b22dbda"},"schema_version":"1.0","source":{"id":"1209.0055","kind":"arxiv","version":1}},"canonical_sha256":"60a374f6f9f483efbea6bb9c6c41d163df6c6cea994d9426ca9fc560b867795b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"60a374f6f9f483efbea6bb9c6c41d163df6c6cea994d9426ca9fc560b867795b","first_computed_at":"2026-05-18T03:46:30.143729Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:30.143729Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vol2B3rCAAX2oSpg34O5gJrqHk1EuSwaE1ZPpmIZpl4cT24yDx30/X411Jnab1/U7+mvp1GgcSk6uUgPeNuxDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:30.144488Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.0055","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb23a5e195c589693c0104688abf04fb91263b97470708a2dac81a1cc6b52ed7","sha256:8f7af102cc12ab3820de315dc73a04cc2efd49a4d3a053b18f82eb774fb3c8fe"],"state_sha256":"749ae9eebd525554e6190b09886a87e47e08b59ab7d005be54bb467e1a73823a"}