{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:7D6QWRVTHBVI35J46PEZSP5XTV","short_pith_number":"pith:7D6QWRVT","canonical_record":{"source":{"id":"1502.04506","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.FA","submitted_at":"2015-02-16T12:03:28Z","cross_cats_sorted":[],"title_canon_sha256":"a83eedf8a4fc0f537a4ffa6d4edf46143e172ed160a7df6c38c7f24e8be177e5","abstract_canon_sha256":"48d2a9f2f771953e3cb01c9a304ba9393e33b0f884994f793c8872971be18ba6"},"schema_version":"1.0"},"canonical_sha256":"f8fd0b46b3386a8df53cf3c9993fb79d692f30019f5b23c304f439381f68622f","source":{"kind":"arxiv","id":"1502.04506","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.04506","created_at":"2026-05-18T01:19:20Z"},{"alias_kind":"arxiv_version","alias_value":"1502.04506v2","created_at":"2026-05-18T01:19:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.04506","created_at":"2026-05-18T01:19:20Z"},{"alias_kind":"pith_short_12","alias_value":"7D6QWRVTHBVI","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"7D6QWRVTHBVI35J4","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"7D6QWRVT","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:7D6QWRVTHBVI35J46PEZSP5XTV","target":"record","payload":{"canonical_record":{"source":{"id":"1502.04506","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.FA","submitted_at":"2015-02-16T12:03:28Z","cross_cats_sorted":[],"title_canon_sha256":"a83eedf8a4fc0f537a4ffa6d4edf46143e172ed160a7df6c38c7f24e8be177e5","abstract_canon_sha256":"48d2a9f2f771953e3cb01c9a304ba9393e33b0f884994f793c8872971be18ba6"},"schema_version":"1.0"},"canonical_sha256":"f8fd0b46b3386a8df53cf3c9993fb79d692f30019f5b23c304f439381f68622f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:20.666877Z","signature_b64":"wFGf+KYcgF7OCPUbMORHdZOu27ElmCniaqZRggZYec7pA6h1zOCgZ+qjjYCYMTs12yCZgwUzSy2KOVpfdaLVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8fd0b46b3386a8df53cf3c9993fb79d692f30019f5b23c304f439381f68622f","last_reissued_at":"2026-05-18T01:19:20.666399Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:20.666399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.04506","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":"2026-05-18T01:19:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9BKEPm6P0XXMMfDS5PR1Sa7rzebPHLs7dohQ29swMItIJQwRxD5yNn0/OdtbJmFHLvna9InkKT4u69vgYY8EDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:56:17.793646Z"},"content_sha256":"aa547d2cce8a8fe645a1bfc3613b9af2adab0557ae2584f97de675e81749f70f","schema_version":"1.0","event_id":"sha256:aa547d2cce8a8fe645a1bfc3613b9af2adab0557ae2584f97de675e81749f70f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:7D6QWRVTHBVI35J46PEZSP5XTV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized injectivity of Banach modules","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohammad Fozouni","submitted_at":"2015-02-16T12:03:28Z","abstract_excerpt":"In this paper, we study the notion of $\\phi$-injectivity in the special case that $\\phi=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\\infty$ are 0-injective Banach $L^{1}(G)$ modules."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04506","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":"2026-05-18T01:19:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u7GGyLi7txotU0+NFpYejSkW10pJ5PAi5xQMveF+LKTARYOy8FyaEcIr9uPl8J+RA4Gy/7p6t2XKVBgSy/z2BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:56:17.794356Z"},"content_sha256":"9d7c919b8a366c065f300c3aed8ecd20f521c2509776ed7b1c1e8e5946928193","schema_version":"1.0","event_id":"sha256:9d7c919b8a366c065f300c3aed8ecd20f521c2509776ed7b1c1e8e5946928193"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7D6QWRVTHBVI35J46PEZSP5XTV/bundle.json","state_url":"https://pith.science/pith/7D6QWRVTHBVI35J46PEZSP5XTV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7D6QWRVTHBVI35J46PEZSP5XTV/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":"2026-06-06T19:56:17Z","links":{"resolver":"https://pith.science/pith/7D6QWRVTHBVI35J46PEZSP5XTV","bundle":"https://pith.science/pith/7D6QWRVTHBVI35J46PEZSP5XTV/bundle.json","state":"https://pith.science/pith/7D6QWRVTHBVI35J46PEZSP5XTV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7D6QWRVTHBVI35J46PEZSP5XTV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7D6QWRVTHBVI35J46PEZSP5XTV","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":"48d2a9f2f771953e3cb01c9a304ba9393e33b0f884994f793c8872971be18ba6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.FA","submitted_at":"2015-02-16T12:03:28Z","title_canon_sha256":"a83eedf8a4fc0f537a4ffa6d4edf46143e172ed160a7df6c38c7f24e8be177e5"},"schema_version":"1.0","source":{"id":"1502.04506","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.04506","created_at":"2026-05-18T01:19:20Z"},{"alias_kind":"arxiv_version","alias_value":"1502.04506v2","created_at":"2026-05-18T01:19:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.04506","created_at":"2026-05-18T01:19:20Z"},{"alias_kind":"pith_short_12","alias_value":"7D6QWRVTHBVI","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"7D6QWRVTHBVI35J4","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"7D6QWRVT","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:9d7c919b8a366c065f300c3aed8ecd20f521c2509776ed7b1c1e8e5946928193","target":"graph","created_at":"2026-05-18T01:19:20Z","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 study the notion of $\\phi$-injectivity in the special case that $\\phi=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\\infty$ are 0-injective Banach $L^{1}(G)$ modules.","authors_text":"Mohammad Fozouni","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.FA","submitted_at":"2015-02-16T12:03:28Z","title":"Generalized injectivity of Banach modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04506","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:aa547d2cce8a8fe645a1bfc3613b9af2adab0557ae2584f97de675e81749f70f","target":"record","created_at":"2026-05-18T01:19:20Z","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":"48d2a9f2f771953e3cb01c9a304ba9393e33b0f884994f793c8872971be18ba6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.FA","submitted_at":"2015-02-16T12:03:28Z","title_canon_sha256":"a83eedf8a4fc0f537a4ffa6d4edf46143e172ed160a7df6c38c7f24e8be177e5"},"schema_version":"1.0","source":{"id":"1502.04506","kind":"arxiv","version":2}},"canonical_sha256":"f8fd0b46b3386a8df53cf3c9993fb79d692f30019f5b23c304f439381f68622f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8fd0b46b3386a8df53cf3c9993fb79d692f30019f5b23c304f439381f68622f","first_computed_at":"2026-05-18T01:19:20.666399Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:20.666399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wFGf+KYcgF7OCPUbMORHdZOu27ElmCniaqZRggZYec7pA6h1zOCgZ+qjjYCYMTs12yCZgwUzSy2KOVpfdaLVCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:20.666877Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.04506","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa547d2cce8a8fe645a1bfc3613b9af2adab0557ae2584f97de675e81749f70f","sha256:9d7c919b8a366c065f300c3aed8ecd20f521c2509776ed7b1c1e8e5946928193"],"state_sha256":"8e613e1dd04dd632e2d0e39543ac1d01f9531305389bb82e7e331951ff99972f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"85E+q/LiqcCxGFnZKGEk0TAUMHyl6COsvgS7U1l1S1A8u2J9r4yJBb7ZvZ6zGeWo6dJIRKKZc3T46/uWTlAIAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T19:56:17.798701Z","bundle_sha256":"9288932bb5484d135eb264e09ccddf1d93f2bce3ad99c5926be2e1ba468ae1ea"}}