{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:QQXQGWQ5NW4G3WYJVC6DH3C3V7","short_pith_number":"pith:QQXQGWQ5","canonical_record":{"source":{"id":"1805.09892","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-24T20:46:53Z","cross_cats_sorted":[],"title_canon_sha256":"04e9234a2494a80a7cdf1eb2d24d56b585d470880e2947817022d0e5028548a1","abstract_canon_sha256":"f1855c0bca3bdfe0762fae5b7f3fa1d1d4388966dff1b33a8ae4c6506edcc9fa"},"schema_version":"1.0"},"canonical_sha256":"842f035a1d6db86ddb09a8bc33ec5baff8f5fce523c6c81c5cb0649e29548ac2","source":{"kind":"arxiv","id":"1805.09892","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.09892","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"arxiv_version","alias_value":"1805.09892v2","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09892","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"pith_short_12","alias_value":"QQXQGWQ5NW4G","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QQXQGWQ5NW4G3WYJ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QQXQGWQ5","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:QQXQGWQ5NW4G3WYJVC6DH3C3V7","target":"record","payload":{"canonical_record":{"source":{"id":"1805.09892","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-24T20:46:53Z","cross_cats_sorted":[],"title_canon_sha256":"04e9234a2494a80a7cdf1eb2d24d56b585d470880e2947817022d0e5028548a1","abstract_canon_sha256":"f1855c0bca3bdfe0762fae5b7f3fa1d1d4388966dff1b33a8ae4c6506edcc9fa"},"schema_version":"1.0"},"canonical_sha256":"842f035a1d6db86ddb09a8bc33ec5baff8f5fce523c6c81c5cb0649e29548ac2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:37.832631Z","signature_b64":"H0j/L/gRzRNVAxLrsdqary26Ca/EmqlMV1vs2XkyexReI9x/nyy4CJXkKBWGRnhEtjuzQzOEWzxtpDdYQhMlCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"842f035a1d6db86ddb09a8bc33ec5baff8f5fce523c6c81c5cb0649e29548ac2","last_reissued_at":"2026-05-18T00:12:37.831927Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:37.831927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.09892","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y1kZ3E3WfS/AE1PGGImx8XvrCeqEHnXjbfyuUu0xAn46rcLCUDHbp9ZJstg1v5SkXALQdzuDrvbawS+rJNikAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:44:07.778689Z"},"content_sha256":"2272766f56d7d7dd8c3c256270f3df57f8630cb87f1c0e1eeebcdf3fef2c1d8e","schema_version":"1.0","event_id":"sha256:2272766f56d7d7dd8c3c256270f3df57f8630cb87f1c0e1eeebcdf3fef2c1d8e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:QQXQGWQ5NW4G3WYJVC6DH3C3V7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weakly almost periodic topologies, idempotents and ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Nico Spronk","submitted_at":"2018-05-24T20:46:53Z","abstract_excerpt":"Let (G,tau_G) be a topological group. We establish relationships between weakly almost periodic topologies on G coarser than tau_G, central idempotents in the weakly almost periodic compactification G^W, and certain ideals in the algebra of weakly almost periodic functions W(G). We gain decompositions of weakly almost periodic representations, generalizing many from the literature. We look at the role of pre-locally compact topologies, unitarizable topologies, and extend or decompositions to Fourier-Stieltjes algebras B(G)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09892","kind":"arxiv","version":2},"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":"2026-05-18T00:12:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q94RPsh1k2xiwvgwmW1sxxpSgW/Zm/OG+2sHP+08UB4GcMQmdO1ccEnQxW50kqvijFydYKwKNmsfmOs24RPkDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:44:07.779117Z"},"content_sha256":"6fc2ca19ea26bc4023caf05ce2fdc0a299b0d70a8dc65181f419d853ed7ca3ac","schema_version":"1.0","event_id":"sha256:6fc2ca19ea26bc4023caf05ce2fdc0a299b0d70a8dc65181f419d853ed7ca3ac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QQXQGWQ5NW4G3WYJVC6DH3C3V7/bundle.json","state_url":"https://pith.science/pith/QQXQGWQ5NW4G3WYJVC6DH3C3V7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QQXQGWQ5NW4G3WYJVC6DH3C3V7/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":"2026-05-28T16:44:07Z","links":{"resolver":"https://pith.science/pith/QQXQGWQ5NW4G3WYJVC6DH3C3V7","bundle":"https://pith.science/pith/QQXQGWQ5NW4G3WYJVC6DH3C3V7/bundle.json","state":"https://pith.science/pith/QQXQGWQ5NW4G3WYJVC6DH3C3V7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QQXQGWQ5NW4G3WYJVC6DH3C3V7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QQXQGWQ5NW4G3WYJVC6DH3C3V7","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":"f1855c0bca3bdfe0762fae5b7f3fa1d1d4388966dff1b33a8ae4c6506edcc9fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-24T20:46:53Z","title_canon_sha256":"04e9234a2494a80a7cdf1eb2d24d56b585d470880e2947817022d0e5028548a1"},"schema_version":"1.0","source":{"id":"1805.09892","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.09892","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"arxiv_version","alias_value":"1805.09892v2","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09892","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"pith_short_12","alias_value":"QQXQGWQ5NW4G","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QQXQGWQ5NW4G3WYJ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QQXQGWQ5","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:6fc2ca19ea26bc4023caf05ce2fdc0a299b0d70a8dc65181f419d853ed7ca3ac","target":"graph","created_at":"2026-05-18T00:12:37Z","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":"Let (G,tau_G) be a topological group. We establish relationships between weakly almost periodic topologies on G coarser than tau_G, central idempotents in the weakly almost periodic compactification G^W, and certain ideals in the algebra of weakly almost periodic functions W(G). We gain decompositions of weakly almost periodic representations, generalizing many from the literature. We look at the role of pre-locally compact topologies, unitarizable topologies, and extend or decompositions to Fourier-Stieltjes algebras B(G).","authors_text":"Nico Spronk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-24T20:46:53Z","title":"Weakly almost periodic topologies, idempotents and ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09892","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":"sha256:2272766f56d7d7dd8c3c256270f3df57f8630cb87f1c0e1eeebcdf3fef2c1d8e","target":"record","created_at":"2026-05-18T00:12:37Z","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":"f1855c0bca3bdfe0762fae5b7f3fa1d1d4388966dff1b33a8ae4c6506edcc9fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-24T20:46:53Z","title_canon_sha256":"04e9234a2494a80a7cdf1eb2d24d56b585d470880e2947817022d0e5028548a1"},"schema_version":"1.0","source":{"id":"1805.09892","kind":"arxiv","version":2}},"canonical_sha256":"842f035a1d6db86ddb09a8bc33ec5baff8f5fce523c6c81c5cb0649e29548ac2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"842f035a1d6db86ddb09a8bc33ec5baff8f5fce523c6c81c5cb0649e29548ac2","first_computed_at":"2026-05-18T00:12:37.831927Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:37.831927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H0j/L/gRzRNVAxLrsdqary26Ca/EmqlMV1vs2XkyexReI9x/nyy4CJXkKBWGRnhEtjuzQzOEWzxtpDdYQhMlCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:37.832631Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.09892","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2272766f56d7d7dd8c3c256270f3df57f8630cb87f1c0e1eeebcdf3fef2c1d8e","sha256:6fc2ca19ea26bc4023caf05ce2fdc0a299b0d70a8dc65181f419d853ed7ca3ac"],"state_sha256":"b6c5327dc6042ea80f457dbb2df0adbc3f1c6ab480a44f5c2ef72fd94221aa2a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U3zwytZNh6ubRM2VAaEtja5Ieb6TI66It6nXmPTl6f9HCL5/lddePg4cgvGMqveZapMSGJcW7XCGhLbTlgwbDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:44:07.782237Z","bundle_sha256":"87973f14475fc8b0ac224fc2bcaee095fadca56ad98b07c89dee087883a178e8"}}