{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:2RP5EMCE6UXICMPBO3NHNN5O3F","short_pith_number":"pith:2RP5EMCE","canonical_record":{"source":{"id":"1908.05671","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-15T01:30:08Z","cross_cats_sorted":["math.OA","math.RA"],"title_canon_sha256":"2c2fd14052a04575e30c3952471c0397b11cc147d5e7056d3d8c36b4caa1de00","abstract_canon_sha256":"74c21b55779c9af9510e919763c8bd4b21015bd5713879c0befbdf109321eb0e"},"schema_version":"1.0"},"canonical_sha256":"d45fd23044f52e8131e176da76b7aed972f426d21161d96ff2ef184c9fe69799","source":{"kind":"arxiv","id":"1908.05671","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05671","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05671v1","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05671","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"pith_short_12","alias_value":"2RP5EMCE6UXI","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"pith_short_16","alias_value":"2RP5EMCE6UXICMPB","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"pith_short_8","alias_value":"2RP5EMCE","created_at":"2026-07-04T23:57:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:2RP5EMCE6UXICMPBO3NHNN5O3F","target":"record","payload":{"canonical_record":{"source":{"id":"1908.05671","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-15T01:30:08Z","cross_cats_sorted":["math.OA","math.RA"],"title_canon_sha256":"2c2fd14052a04575e30c3952471c0397b11cc147d5e7056d3d8c36b4caa1de00","abstract_canon_sha256":"74c21b55779c9af9510e919763c8bd4b21015bd5713879c0befbdf109321eb0e"},"schema_version":"1.0"},"canonical_sha256":"d45fd23044f52e8131e176da76b7aed972f426d21161d96ff2ef184c9fe69799","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:57:52.318418Z","signature_b64":"fBi5g+Aa4dnHbjRvY6gxd7f5iI9OZGyTL4CEjxuMaesGjeKh4XONpRoLXWYWn7ie4TnVtjTpjY9KAJ7SWL7TCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d45fd23044f52e8131e176da76b7aed972f426d21161d96ff2ef184c9fe69799","last_reissued_at":"2026-07-04T23:57:52.317966Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:57:52.317966Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.05671","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":"2026-07-04T23:57:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fP/DTDD3TRZGDZLdhmV4uZnRmsh2Uh/VPIQSCHyhJaI9Vwc0yTibtp6602HHliCnOZFvmh17AFakR0IL5LRfBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T19:25:41.391737Z"},"content_sha256":"7e7b45aa359ee4c868c658b699336cfcd056c65439a88b3d4d31306d1c60ef1f","schema_version":"1.0","event_id":"sha256:7e7b45aa359ee4c868c658b699336cfcd056c65439a88b3d4d31306d1c60ef1f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:2RP5EMCE6UXICMPBO3NHNN5O3F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Additive Local Multiplications and zero-preserving maps on $C(X)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA","math.RA"],"primary_cat":"math.FA","authors_text":"Qian Hu","submitted_at":"2019-08-15T01:30:08Z","abstract_excerpt":"Suppose $X$ is a compact Hausdorff space. In terms of topolocical properties of $X$, we find topological conditions on $X$ that are equivalent to each of the following: 1. every additive local multiplication on $C\\left( X\\right) $ is a multiplication, 2. every additive local multiplication on $C_{R}\\left( X\\right) $ is a multiplication, and 3. every additive map on $C\\left( X\\right) $ that is zero-preserving (i.e., $f\\left( x\\right) =0$ implies $\\left( Tf\\right) \\left( x\\right) =0$) has the form $T\\left( f\\right) =T\\left( 1\\right) \\operatorname{Re}f+T\\left( i\\right) \\operatorname{Im}f$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05671","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.05671/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T23:57:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c+ArfMP6c7XGIaOgGa+0DbEDUb0EmnrPQxD4uQq9tN07S10cELZ8Fb55EYMlC5lMxiVlX7nG1NBOAZHZb5KUCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T19:25:41.392674Z"},"content_sha256":"e854dba7c8eb1386bc2850e7768e1277432f18079b6c2d3bd7d04f38716c0fdf","schema_version":"1.0","event_id":"sha256:e854dba7c8eb1386bc2850e7768e1277432f18079b6c2d3bd7d04f38716c0fdf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2RP5EMCE6UXICMPBO3NHNN5O3F/bundle.json","state_url":"https://pith.science/pith/2RP5EMCE6UXICMPBO3NHNN5O3F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2RP5EMCE6UXICMPBO3NHNN5O3F/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":"2026-08-16T19:25:41Z","links":{"resolver":"https://pith.science/pith/2RP5EMCE6UXICMPBO3NHNN5O3F","bundle":"https://pith.science/pith/2RP5EMCE6UXICMPBO3NHNN5O3F/bundle.json","state":"https://pith.science/pith/2RP5EMCE6UXICMPBO3NHNN5O3F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2RP5EMCE6UXICMPBO3NHNN5O3F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:2RP5EMCE6UXICMPBO3NHNN5O3F","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":"74c21b55779c9af9510e919763c8bd4b21015bd5713879c0befbdf109321eb0e","cross_cats_sorted":["math.OA","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-15T01:30:08Z","title_canon_sha256":"2c2fd14052a04575e30c3952471c0397b11cc147d5e7056d3d8c36b4caa1de00"},"schema_version":"1.0","source":{"id":"1908.05671","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05671","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05671v1","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05671","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"pith_short_12","alias_value":"2RP5EMCE6UXI","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"pith_short_16","alias_value":"2RP5EMCE6UXICMPB","created_at":"2026-07-04T23:57:52Z"},{"alias_kind":"pith_short_8","alias_value":"2RP5EMCE","created_at":"2026-07-04T23:57:52Z"}],"graph_snapshots":[{"event_id":"sha256:e854dba7c8eb1386bc2850e7768e1277432f18079b6c2d3bd7d04f38716c0fdf","target":"graph","created_at":"2026-07-04T23:57:52Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.05671/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Suppose $X$ is a compact Hausdorff space. In terms of topolocical properties of $X$, we find topological conditions on $X$ that are equivalent to each of the following: 1. every additive local multiplication on $C\\left( X\\right) $ is a multiplication, 2. every additive local multiplication on $C_{R}\\left( X\\right) $ is a multiplication, and 3. every additive map on $C\\left( X\\right) $ that is zero-preserving (i.e., $f\\left( x\\right) =0$ implies $\\left( Tf\\right) \\left( x\\right) =0$) has the form $T\\left( f\\right) =T\\left( 1\\right) \\operatorname{Re}f+T\\left( i\\right) \\operatorname{Im}f$.","authors_text":"Qian Hu","cross_cats":["math.OA","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-15T01:30:08Z","title":"Additive Local Multiplications and zero-preserving maps on $C(X)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05671","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":"sha256:7e7b45aa359ee4c868c658b699336cfcd056c65439a88b3d4d31306d1c60ef1f","target":"record","created_at":"2026-07-04T23:57:52Z","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":"74c21b55779c9af9510e919763c8bd4b21015bd5713879c0befbdf109321eb0e","cross_cats_sorted":["math.OA","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-15T01:30:08Z","title_canon_sha256":"2c2fd14052a04575e30c3952471c0397b11cc147d5e7056d3d8c36b4caa1de00"},"schema_version":"1.0","source":{"id":"1908.05671","kind":"arxiv","version":1}},"canonical_sha256":"d45fd23044f52e8131e176da76b7aed972f426d21161d96ff2ef184c9fe69799","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d45fd23044f52e8131e176da76b7aed972f426d21161d96ff2ef184c9fe69799","first_computed_at":"2026-07-04T23:57:52.317966Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:57:52.317966Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fBi5g+Aa4dnHbjRvY6gxd7f5iI9OZGyTL4CEjxuMaesGjeKh4XONpRoLXWYWn7ie4TnVtjTpjY9KAJ7SWL7TCw==","signature_status":"signed_v1","signed_at":"2026-07-04T23:57:52.318418Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.05671","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e7b45aa359ee4c868c658b699336cfcd056c65439a88b3d4d31306d1c60ef1f","sha256:e854dba7c8eb1386bc2850e7768e1277432f18079b6c2d3bd7d04f38716c0fdf"],"state_sha256":"d0bebc599d4fc0d455a04709e5107efff81a54dccb81814d1ada14a55927f7ec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kUexd+pBpB9W3inJu2nMgBNTLtzv7L8zI8tq6EPOBvsYgKbUIT91xRNnc4ZzvIHBDU1wsx0V8m7J3F71S+f5Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T19:25:41.398772Z","bundle_sha256":"b0f68fc94ad32baa99c2a2aecf4274f532c5e163dda51630318ab5b305a57cc9"}}