{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:FRKF3XZV4WQBILXNMYGTTTMAJ3","short_pith_number":"pith:FRKF3XZV","canonical_record":{"source":{"id":"1411.4301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-11-16T20:31:57Z","cross_cats_sorted":[],"title_canon_sha256":"8edc7115231b702724ae1a3c5745a16da42d0c4d0eb25b8a0c15f8fcb3c7973d","abstract_canon_sha256":"26c5aadd94f965e8941d341a6f76af33daef2655f5078dfe6e1148dc56b5ff68"},"schema_version":"1.0"},"canonical_sha256":"2c545ddf35e5a0142eed660d39cd804eff62c305275634a5c53de98e551b0e3f","source":{"kind":"arxiv","id":"1411.4301","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.4301","created_at":"2026-05-18T02:35:45Z"},{"alias_kind":"arxiv_version","alias_value":"1411.4301v1","created_at":"2026-05-18T02:35:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.4301","created_at":"2026-05-18T02:35:45Z"},{"alias_kind":"pith_short_12","alias_value":"FRKF3XZV4WQB","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FRKF3XZV4WQBILXN","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FRKF3XZV","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:FRKF3XZV4WQBILXNMYGTTTMAJ3","target":"record","payload":{"canonical_record":{"source":{"id":"1411.4301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-11-16T20:31:57Z","cross_cats_sorted":[],"title_canon_sha256":"8edc7115231b702724ae1a3c5745a16da42d0c4d0eb25b8a0c15f8fcb3c7973d","abstract_canon_sha256":"26c5aadd94f965e8941d341a6f76af33daef2655f5078dfe6e1148dc56b5ff68"},"schema_version":"1.0"},"canonical_sha256":"2c545ddf35e5a0142eed660d39cd804eff62c305275634a5c53de98e551b0e3f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:35:45.969911Z","signature_b64":"ZnIzgxlzkBlrRbPOV6Zf+4p6X1DsHdTagRfEjkUATnPuqUK9v6FWXERzeFikRtdF4jQXWJhqqnY7A/t3Xk0sBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c545ddf35e5a0142eed660d39cd804eff62c305275634a5c53de98e551b0e3f","last_reissued_at":"2026-05-18T02:35:45.969469Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:35:45.969469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.4301","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":"2026-05-18T02:35:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"61Kl1yiqFRWk+7Zw/AfhFx7dSy/Obs7X/eVZRcIR4cQuNAjs7OnCmwNOE5AnNeh7bSLlm4dffROiRU2CI+fVCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T01:14:00.629422Z"},"content_sha256":"1b449c5cc951c377da1d0fedb106103162b3d0c03b93691bef5da0f360eb6ff4","schema_version":"1.0","event_id":"sha256:1b449c5cc951c377da1d0fedb106103162b3d0c03b93691bef5da0f360eb6ff4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:FRKF3XZV4WQBILXNMYGTTTMAJ3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dilations for Systems of Imprimitivity acting on Banach Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Bei Liu, David R. Larson, Deguang Han, Rui Liu","submitted_at":"2014-11-16T20:31:57Z","abstract_excerpt":"Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.4301","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":"2026-05-18T02:35:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3dOH2WU5L4VRxFxnWZcO5W65sx1l9VbUyUfGmsk75aJ4kVaxF0E98FUzgqh13v8yUh9cQjqHEACTRoRWZ+65Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T01:14:00.630101Z"},"content_sha256":"7e79f3c64b1b34ccdd33e0b48c4bdb32e4621225c8260e95b83856e292d33b9d","schema_version":"1.0","event_id":"sha256:7e79f3c64b1b34ccdd33e0b48c4bdb32e4621225c8260e95b83856e292d33b9d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FRKF3XZV4WQBILXNMYGTTTMAJ3/bundle.json","state_url":"https://pith.science/pith/FRKF3XZV4WQBILXNMYGTTTMAJ3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FRKF3XZV4WQBILXNMYGTTTMAJ3/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":"2026-05-26T01:14:00Z","links":{"resolver":"https://pith.science/pith/FRKF3XZV4WQBILXNMYGTTTMAJ3","bundle":"https://pith.science/pith/FRKF3XZV4WQBILXNMYGTTTMAJ3/bundle.json","state":"https://pith.science/pith/FRKF3XZV4WQBILXNMYGTTTMAJ3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FRKF3XZV4WQBILXNMYGTTTMAJ3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:FRKF3XZV4WQBILXNMYGTTTMAJ3","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":"26c5aadd94f965e8941d341a6f76af33daef2655f5078dfe6e1148dc56b5ff68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-11-16T20:31:57Z","title_canon_sha256":"8edc7115231b702724ae1a3c5745a16da42d0c4d0eb25b8a0c15f8fcb3c7973d"},"schema_version":"1.0","source":{"id":"1411.4301","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.4301","created_at":"2026-05-18T02:35:45Z"},{"alias_kind":"arxiv_version","alias_value":"1411.4301v1","created_at":"2026-05-18T02:35:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.4301","created_at":"2026-05-18T02:35:45Z"},{"alias_kind":"pith_short_12","alias_value":"FRKF3XZV4WQB","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FRKF3XZV4WQBILXN","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FRKF3XZV","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:7e79f3c64b1b34ccdd33e0b48c4bdb32e4621225c8260e95b83856e292d33b9d","target":"graph","created_at":"2026-05-18T02:35:45Z","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":"Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in","authors_text":"Bei Liu, David R. Larson, Deguang Han, Rui Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-11-16T20:31:57Z","title":"Dilations for Systems of Imprimitivity acting on Banach Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.4301","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":"sha256:1b449c5cc951c377da1d0fedb106103162b3d0c03b93691bef5da0f360eb6ff4","target":"record","created_at":"2026-05-18T02:35:45Z","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":"26c5aadd94f965e8941d341a6f76af33daef2655f5078dfe6e1148dc56b5ff68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-11-16T20:31:57Z","title_canon_sha256":"8edc7115231b702724ae1a3c5745a16da42d0c4d0eb25b8a0c15f8fcb3c7973d"},"schema_version":"1.0","source":{"id":"1411.4301","kind":"arxiv","version":1}},"canonical_sha256":"2c545ddf35e5a0142eed660d39cd804eff62c305275634a5c53de98e551b0e3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c545ddf35e5a0142eed660d39cd804eff62c305275634a5c53de98e551b0e3f","first_computed_at":"2026-05-18T02:35:45.969469Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:35:45.969469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZnIzgxlzkBlrRbPOV6Zf+4p6X1DsHdTagRfEjkUATnPuqUK9v6FWXERzeFikRtdF4jQXWJhqqnY7A/t3Xk0sBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:35:45.969911Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.4301","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b449c5cc951c377da1d0fedb106103162b3d0c03b93691bef5da0f360eb6ff4","sha256:7e79f3c64b1b34ccdd33e0b48c4bdb32e4621225c8260e95b83856e292d33b9d"],"state_sha256":"317b8b999fc3d250d3ac9c1f463f8cf447f5fd6ff32a2134dd78a9b471c2768b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JSjEe6tdB83NjzgRGZdJuM7GkvFsxCtIx+i5CoKMoP0CQI2wd/Zwii8PncWBEaJF8Vn//fmjCvgnwQoEMozDAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T01:14:00.633744Z","bundle_sha256":"765b40f9259a963112a326628c531abe3ea68237c518cfec2e190c91ae13b06c"}}