{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:OI64HUNO66MUUGYTR6Q3IYDR6R","short_pith_number":"pith:OI64HUNO","canonical_record":{"source":{"id":"1812.07257","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.FA","submitted_at":"2018-12-18T09:33:54Z","cross_cats_sorted":[],"title_canon_sha256":"497ee8360f7d17f946aefe96784b3c3d74023fa3616a59d7347a68f5a6b24229","abstract_canon_sha256":"eeadee8944a6df2fc3bda45c387fdb5c4ac10226dbc50e5a490397504e09e86d"},"schema_version":"1.0"},"canonical_sha256":"723dc3d1aef7994a1b138fa1b46071f45da93b334b5fea7d1c931a27ec1267f4","source":{"kind":"arxiv","id":"1812.07257","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.07257","created_at":"2026-05-17T23:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"1812.07257v1","created_at":"2026-05-17T23:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.07257","created_at":"2026-05-17T23:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"OI64HUNO66MU","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OI64HUNO66MUUGYT","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OI64HUNO","created_at":"2026-05-18T12:32:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:OI64HUNO66MUUGYTR6Q3IYDR6R","target":"record","payload":{"canonical_record":{"source":{"id":"1812.07257","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.FA","submitted_at":"2018-12-18T09:33:54Z","cross_cats_sorted":[],"title_canon_sha256":"497ee8360f7d17f946aefe96784b3c3d74023fa3616a59d7347a68f5a6b24229","abstract_canon_sha256":"eeadee8944a6df2fc3bda45c387fdb5c4ac10226dbc50e5a490397504e09e86d"},"schema_version":"1.0"},"canonical_sha256":"723dc3d1aef7994a1b138fa1b46071f45da93b334b5fea7d1c931a27ec1267f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:01.050397Z","signature_b64":"LYeTQ5sv4dOhXP1FbafTGVvAkfolB+o484M/uhNiWhsCQJxZbS2sWta2fFNhQAOIKfM3XuFC3Fe1ZjaVHOkpCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"723dc3d1aef7994a1b138fa1b46071f45da93b334b5fea7d1c931a27ec1267f4","last_reissued_at":"2026-05-17T23:58:01.049745Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:01.049745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.07257","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":"2026-05-17T23:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"np8Vgngc+SMb8F/jOr+ToYYJYSwd6SgRWmU1J0dzcFyRMCD5DjcNxez99A9Loe8L4Fbtg7Bg/hZMQCUacA3sCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:26:12.429931Z"},"content_sha256":"223ecc42c816c71e447e61cb3eaba858728a3bc04a58dfb28644e9c5731dc27f","schema_version":"1.0","event_id":"sha256:223ecc42c816c71e447e61cb3eaba858728a3bc04a58dfb28644e9c5731dc27f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:OI64HUNO66MUUGYTR6Q3IYDR6R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a question related to bounded approximate identities of ideals in Banach algebras","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohammad Fozouni","submitted_at":"2018-12-18T09:33:54Z","abstract_excerpt":"In this paper we give an example of a Banach algebra $A$ and a closed ideal $I$ of $A$ such that the multiplier algebra of $I$ is equal to $A$ but $I$ does not have any bounded approximate identity. In the case that $I$ has an approximate identity, we give a necessary condition on $I$ for which $A=\\mathcal{M}(I)$, where $\\mathcal{M}(I)$ denotes the multiplier algebra of $I$. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier-Stieltjes algebra."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.07257","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":"2026-05-17T23:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2NS44DvKBhm5I2ZM92xb9zJ/O1kbgjF01DjHHOUZhJWgxCNU2IZ6RMZ70bojqFDp28a+yM3Rhc+0zKWvpinVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:26:12.431830Z"},"content_sha256":"431eaa90a6fcd963508a070dc7f42979a89b0cafa57538e595f3f30fc307620c","schema_version":"1.0","event_id":"sha256:431eaa90a6fcd963508a070dc7f42979a89b0cafa57538e595f3f30fc307620c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OI64HUNO66MUUGYTR6Q3IYDR6R/bundle.json","state_url":"https://pith.science/pith/OI64HUNO66MUUGYTR6Q3IYDR6R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OI64HUNO66MUUGYTR6Q3IYDR6R/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":"2026-06-06T19:26:12Z","links":{"resolver":"https://pith.science/pith/OI64HUNO66MUUGYTR6Q3IYDR6R","bundle":"https://pith.science/pith/OI64HUNO66MUUGYTR6Q3IYDR6R/bundle.json","state":"https://pith.science/pith/OI64HUNO66MUUGYTR6Q3IYDR6R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OI64HUNO66MUUGYTR6Q3IYDR6R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:OI64HUNO66MUUGYTR6Q3IYDR6R","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":"eeadee8944a6df2fc3bda45c387fdb5c4ac10226dbc50e5a490397504e09e86d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.FA","submitted_at":"2018-12-18T09:33:54Z","title_canon_sha256":"497ee8360f7d17f946aefe96784b3c3d74023fa3616a59d7347a68f5a6b24229"},"schema_version":"1.0","source":{"id":"1812.07257","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.07257","created_at":"2026-05-17T23:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"1812.07257v1","created_at":"2026-05-17T23:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.07257","created_at":"2026-05-17T23:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"OI64HUNO66MU","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OI64HUNO66MUUGYT","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OI64HUNO","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:431eaa90a6fcd963508a070dc7f42979a89b0cafa57538e595f3f30fc307620c","target":"graph","created_at":"2026-05-17T23:58:01Z","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":"In this paper we give an example of a Banach algebra $A$ and a closed ideal $I$ of $A$ such that the multiplier algebra of $I$ is equal to $A$ but $I$ does not have any bounded approximate identity. In the case that $I$ has an approximate identity, we give a necessary condition on $I$ for which $A=\\mathcal{M}(I)$, where $\\mathcal{M}(I)$ denotes the multiplier algebra of $I$. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier-Stieltjes algebra.","authors_text":"Mohammad Fozouni","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.FA","submitted_at":"2018-12-18T09:33:54Z","title":"On a question related to bounded approximate identities of ideals in Banach algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.07257","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":"sha256:223ecc42c816c71e447e61cb3eaba858728a3bc04a58dfb28644e9c5731dc27f","target":"record","created_at":"2026-05-17T23:58:01Z","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":"eeadee8944a6df2fc3bda45c387fdb5c4ac10226dbc50e5a490397504e09e86d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.FA","submitted_at":"2018-12-18T09:33:54Z","title_canon_sha256":"497ee8360f7d17f946aefe96784b3c3d74023fa3616a59d7347a68f5a6b24229"},"schema_version":"1.0","source":{"id":"1812.07257","kind":"arxiv","version":1}},"canonical_sha256":"723dc3d1aef7994a1b138fa1b46071f45da93b334b5fea7d1c931a27ec1267f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"723dc3d1aef7994a1b138fa1b46071f45da93b334b5fea7d1c931a27ec1267f4","first_computed_at":"2026-05-17T23:58:01.049745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:01.049745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LYeTQ5sv4dOhXP1FbafTGVvAkfolB+o484M/uhNiWhsCQJxZbS2sWta2fFNhQAOIKfM3XuFC3Fe1ZjaVHOkpCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:01.050397Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.07257","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:223ecc42c816c71e447e61cb3eaba858728a3bc04a58dfb28644e9c5731dc27f","sha256:431eaa90a6fcd963508a070dc7f42979a89b0cafa57538e595f3f30fc307620c"],"state_sha256":"7509150bd09c7bd829fcab547d73a69db165482ac60acc66a4bda8efbb7854be"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hsj17wdDoWJAK3aDNkS4P+Tk1pLki0XoyTkKzW8vjxTAe2I9Dofv49+uP154C5hXkVeJOFcpWhV59Qi5RxnaAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T19:26:12.463213Z","bundle_sha256":"2ea10662bfa896781470feb59b6c381aa38e32118069491ce82632dc62bc40d0"}}