{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:LPLZVOBMVF24CA3APTSIY2WDQD","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":"43a46007bc27ebca98d940f42297eed3237a01f48f5e234a2782486e2da1058d","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-07-09T17:51:41Z","title_canon_sha256":"0684deabd12c9c26a09e758229c6545181942c843bb36214fc42e98aedeae37c"},"schema_version":"1.0","source":{"id":"1307.2526","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.2526","created_at":"2026-05-18T01:20:18Z"},{"alias_kind":"arxiv_version","alias_value":"1307.2526v2","created_at":"2026-05-18T01:20:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2526","created_at":"2026-05-18T01:20:18Z"},{"alias_kind":"pith_short_12","alias_value":"LPLZVOBMVF24","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LPLZVOBMVF24CA3A","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LPLZVOBM","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:77cfef5ee6b64d389d72c8e7cec8f378cfb0ee325dab3bd4eb7d9aae508d3d98","target":"graph","created_at":"2026-05-18T01:20:18Z","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 prove that the universal covering group $\\widetilde{\\mathrm{Sp}}(2,\\mathbb{R})$ of $\\mathrm{Sp}(2,\\mathbb{R})$ does not have the Approximation Property (AP). Together with the fact that $\\mathrm{SL}(3,\\mathbb{R})$ does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that $\\mathrm{Sp}(2,\\mathbb{R})$ does not have the AP, which was proved by the authors of this article, this finishes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one. By an adaptatio","authors_text":"Tim de Laat, Uffe Haagerup","cross_cats":["math.FA","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-07-09T17:51:41Z","title":"Simple Lie groups without the Approximation Property II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2526","kind":"arxiv","version":2},"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":"sha256:709799032dd98d53b5e7aac846bf75996306d85d27908f7d1f979a5e052c168f","target":"record","created_at":"2026-05-18T01:20:18Z","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":"43a46007bc27ebca98d940f42297eed3237a01f48f5e234a2782486e2da1058d","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-07-09T17:51:41Z","title_canon_sha256":"0684deabd12c9c26a09e758229c6545181942c843bb36214fc42e98aedeae37c"},"schema_version":"1.0","source":{"id":"1307.2526","kind":"arxiv","version":2}},"canonical_sha256":"5bd79ab82ca975c103607ce48c6ac380da1707761ddf3d68611daa0fc9910303","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5bd79ab82ca975c103607ce48c6ac380da1707761ddf3d68611daa0fc9910303","first_computed_at":"2026-05-18T01:20:18.008048Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:18.008048Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y+k0dUof+fQBjhCa40i6NLIm6NCufKLZ1H90Iybt+yofjACRjisWvemQ5eMh0EkZgGoo68WkUCI85PpkL/1hDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:18.008607Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.2526","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:709799032dd98d53b5e7aac846bf75996306d85d27908f7d1f979a5e052c168f","sha256:77cfef5ee6b64d389d72c8e7cec8f378cfb0ee325dab3bd4eb7d9aae508d3d98"],"state_sha256":"6115c61921d84cd49c4407edb1cd53c536954aa7715e5da49d4a45c78054c225"}