{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:S3SH5R6CZ63MGDDUJV3H6R7EYB","short_pith_number":"pith:S3SH5R6C","canonical_record":{"source":{"id":"1110.2970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-10-13T14:58:09Z","cross_cats_sorted":["math.FA","math.LO"],"title_canon_sha256":"07e233bab6f2311187a5803ddccb5bf244805123d1bde353b9e87c987a9ab5d5","abstract_canon_sha256":"63ed773389d3b119917d3c5725a19fba8e16f5eb14d403297aca328fd750542e"},"schema_version":"1.0"},"canonical_sha256":"96e47ec7c2cfb6c30c744d767f47e4c059b05866f321a59cef5a52955684bec9","source":{"kind":"arxiv","id":"1110.2970","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.2970","created_at":"2026-05-18T04:11:06Z"},{"alias_kind":"arxiv_version","alias_value":"1110.2970v1","created_at":"2026-05-18T04:11:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2970","created_at":"2026-05-18T04:11:06Z"},{"alias_kind":"pith_short_12","alias_value":"S3SH5R6CZ63M","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"S3SH5R6CZ63MGDDU","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"S3SH5R6C","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:S3SH5R6CZ63MGDDUJV3H6R7EYB","target":"record","payload":{"canonical_record":{"source":{"id":"1110.2970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-10-13T14:58:09Z","cross_cats_sorted":["math.FA","math.LO"],"title_canon_sha256":"07e233bab6f2311187a5803ddccb5bf244805123d1bde353b9e87c987a9ab5d5","abstract_canon_sha256":"63ed773389d3b119917d3c5725a19fba8e16f5eb14d403297aca328fd750542e"},"schema_version":"1.0"},"canonical_sha256":"96e47ec7c2cfb6c30c744d767f47e4c059b05866f321a59cef5a52955684bec9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:06.536154Z","signature_b64":"QP9KCyJ+zXl2dBc80af7v6dtv+d0kA+miziPqkjoIQ/G3KJ7iSSegaeZkIIbEQF8+MYPCiegvCO7lspWWzQsCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"96e47ec7c2cfb6c30c744d767f47e4c059b05866f321a59cef5a52955684bec9","last_reissued_at":"2026-05-18T04:11:06.535686Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:06.535686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.2970","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:11:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bStTLMtZlyjUhZVMEWIRs39Pq3wbm3yL8sHkZVJjm/KnX4OwFcwabGbOOOWIlOQTYdTbLKBwmEIFkEhfArraDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:31:36.527587Z"},"content_sha256":"b536911477a96868c8143568340a124f2dfe5331b4d8f6c66b992c42e609575e","schema_version":"1.0","event_id":"sha256:b536911477a96868c8143568340a124f2dfe5331b4d8f6c66b992c42e609575e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:S3SH5R6CZ63MGDDUJV3H6R7EYB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Displaying Polish groups on separable Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.LO"],"primary_cat":"math.GR","authors_text":"Christian Rosendal, Valentin Ferenczi","submitted_at":"2011-10-13T14:58:09Z","abstract_excerpt":"A display of a topological group G on a Banach space X is a topological isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm ||.|| on X, where the latter group is equipped with the strong operator topology.\n  Displays of Polish groups on separable real spaces are studied. It is proved that any closed subgroup of the infinite symmetric group S_\\infty containing a non-trivial central involution admits a display on any of the classical spaces c0, C([0,1]), lp and Lp for 1 <=p <\\infty. Also, for any Polsih group G, there exists a separable space X on which {-1,1} x G has"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2970","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:11:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CLxQ5PTKclgmTxDeu4Lqcz6uYbNrASGJNGHMTWPPQ5fGrO9DyADs9fWn/RJOZ58w7yZxfDRCVe61g6TJvb2KBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:31:36.527943Z"},"content_sha256":"53a63bbdfbec30200b1cd22b63ae03734b4e243840ce9976ee790576076ce178","schema_version":"1.0","event_id":"sha256:53a63bbdfbec30200b1cd22b63ae03734b4e243840ce9976ee790576076ce178"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S3SH5R6CZ63MGDDUJV3H6R7EYB/bundle.json","state_url":"https://pith.science/pith/S3SH5R6CZ63MGDDUJV3H6R7EYB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S3SH5R6CZ63MGDDUJV3H6R7EYB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T21:31:36Z","links":{"resolver":"https://pith.science/pith/S3SH5R6CZ63MGDDUJV3H6R7EYB","bundle":"https://pith.science/pith/S3SH5R6CZ63MGDDUJV3H6R7EYB/bundle.json","state":"https://pith.science/pith/S3SH5R6CZ63MGDDUJV3H6R7EYB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S3SH5R6CZ63MGDDUJV3H6R7EYB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:S3SH5R6CZ63MGDDUJV3H6R7EYB","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":"63ed773389d3b119917d3c5725a19fba8e16f5eb14d403297aca328fd750542e","cross_cats_sorted":["math.FA","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-10-13T14:58:09Z","title_canon_sha256":"07e233bab6f2311187a5803ddccb5bf244805123d1bde353b9e87c987a9ab5d5"},"schema_version":"1.0","source":{"id":"1110.2970","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.2970","created_at":"2026-05-18T04:11:06Z"},{"alias_kind":"arxiv_version","alias_value":"1110.2970v1","created_at":"2026-05-18T04:11:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2970","created_at":"2026-05-18T04:11:06Z"},{"alias_kind":"pith_short_12","alias_value":"S3SH5R6CZ63M","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"S3SH5R6CZ63MGDDU","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"S3SH5R6C","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:53a63bbdfbec30200b1cd22b63ae03734b4e243840ce9976ee790576076ce178","target":"graph","created_at":"2026-05-18T04:11:06Z","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":"A display of a topological group G on a Banach space X is a topological isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm ||.|| on X, where the latter group is equipped with the strong operator topology.\n  Displays of Polish groups on separable real spaces are studied. It is proved that any closed subgroup of the infinite symmetric group S_\\infty containing a non-trivial central involution admits a display on any of the classical spaces c0, C([0,1]), lp and Lp for 1 <=p <\\infty. Also, for any Polsih group G, there exists a separable space X on which {-1,1} x G has","authors_text":"Christian Rosendal, Valentin Ferenczi","cross_cats":["math.FA","math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-10-13T14:58:09Z","title":"Displaying Polish groups on separable Banach spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2970","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:b536911477a96868c8143568340a124f2dfe5331b4d8f6c66b992c42e609575e","target":"record","created_at":"2026-05-18T04:11:06Z","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":"63ed773389d3b119917d3c5725a19fba8e16f5eb14d403297aca328fd750542e","cross_cats_sorted":["math.FA","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-10-13T14:58:09Z","title_canon_sha256":"07e233bab6f2311187a5803ddccb5bf244805123d1bde353b9e87c987a9ab5d5"},"schema_version":"1.0","source":{"id":"1110.2970","kind":"arxiv","version":1}},"canonical_sha256":"96e47ec7c2cfb6c30c744d767f47e4c059b05866f321a59cef5a52955684bec9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"96e47ec7c2cfb6c30c744d767f47e4c059b05866f321a59cef5a52955684bec9","first_computed_at":"2026-05-18T04:11:06.535686Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:06.535686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QP9KCyJ+zXl2dBc80af7v6dtv+d0kA+miziPqkjoIQ/G3KJ7iSSegaeZkIIbEQF8+MYPCiegvCO7lspWWzQsCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:06.536154Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.2970","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b536911477a96868c8143568340a124f2dfe5331b4d8f6c66b992c42e609575e","sha256:53a63bbdfbec30200b1cd22b63ae03734b4e243840ce9976ee790576076ce178"],"state_sha256":"12b2e0cc6bafd770824edb4e5ecd54d8f5e3799a0899cb1e9007d3ccdd4fb956"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H1mJNSzSKnSyrvS4QjDzcSQsGzF+16/g/oAtnz+Mcc2+MC8PZCwPz9TbeVBIkj8uZAscbEd9g3pCctFBlc5EAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T21:31:36.530933Z","bundle_sha256":"d20a6d8eae9d9a4e1d1250de33050bbf7a9c1dee90a838a31d231de004588157"}}