{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:I672R6QUAVEX4LIZX5FVR4GIRW","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":"adbca88b55f020f85ef44bf66aa33ab0e49afd0d579edf154f45b036e4532131","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-12-09T04:10:13Z","title_canon_sha256":"03902c4bab379705795bb836433c0231e1d9710ec2f8777dc57137bb523a43d4"},"schema_version":"1.0","source":{"id":"1312.2304","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.2304","created_at":"2026-05-18T01:08:36Z"},{"alias_kind":"arxiv_version","alias_value":"1312.2304v1","created_at":"2026-05-18T01:08:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.2304","created_at":"2026-05-18T01:08:36Z"},{"alias_kind":"pith_short_12","alias_value":"I672R6QUAVEX","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"I672R6QUAVEX4LIZ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"I672R6QU","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:91467d0e2684b8b8b9683ab05d76bd9fbcf6fface8f18a46925e8352b8d0cc28","target":"graph","created_at":"2026-05-18T01:08:36Z","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":"Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if $AC(\\sigma_1)$ is algebra isomorphic to $AC(\\sigma_2)$ then $\\sigma_1$ is homeomorphic to $\\sigma_2$. The converse however is false. In a positive direction we show that the converse implication does hold if the sets $\\sigma_1$ and $\\sigma_2$ are confined to a restricted collection of compact sets, such as the set of all simple polygons.","authors_text":"Ian Doust, Michael Leinert","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-12-09T04:10:13Z","title":"Isomorphisms of $AC(\\sigma)$ spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2304","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":"sha256:94978c053eb3f9d78f68d6c11b7240fa7a94a94fb8836d4a06b9c731b3cfc36e","target":"record","created_at":"2026-05-18T01:08:36Z","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":"adbca88b55f020f85ef44bf66aa33ab0e49afd0d579edf154f45b036e4532131","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-12-09T04:10:13Z","title_canon_sha256":"03902c4bab379705795bb836433c0231e1d9710ec2f8777dc57137bb523a43d4"},"schema_version":"1.0","source":{"id":"1312.2304","kind":"arxiv","version":1}},"canonical_sha256":"47bfa8fa1405497e2d19bf4b58f0c88da56cb4f365adf5f3931f721bad42812e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47bfa8fa1405497e2d19bf4b58f0c88da56cb4f365adf5f3931f721bad42812e","first_computed_at":"2026-05-18T01:08:36.074500Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:36.074500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nM2uL4fGQgjTEXTOKwFRLmAkBez/ltFDUfRsMwyUdsmrip8ahizwFnr6Gf9A/5zuyEl2j1B1FOXuVM6R6bLOAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:36.075148Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.2304","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94978c053eb3f9d78f68d6c11b7240fa7a94a94fb8836d4a06b9c731b3cfc36e","sha256:91467d0e2684b8b8b9683ab05d76bd9fbcf6fface8f18a46925e8352b8d0cc28"],"state_sha256":"d36eb01dbae6f3a5628e59e9da98050d62ce92b24466dab6af636b0f74ae6c9e"}