{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:DZB5SNIIC6JDE2QWJLGDVWHVJV","short_pith_number":"pith:DZB5SNII","schema_version":"1.0","canonical_sha256":"1e43d935081792326a164acc3ad8f54d5a98a925aba4d607f4c8b9db9b039020","source":{"kind":"arxiv","id":"1101.4320","version":2},"attestation_state":"computed","paper":{"title":"Multi-norms and the injectivity of $L^p(G)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"H. Garth Dales, Hung Le Pham, Matthew Daws, Paul Ramsden","submitted_at":"2011-01-22T20:32:59Z","abstract_excerpt":"Let $G$ be a locally compact group, and take $p\\in(1,\\infty)$. We prove that the Banach left $L^1(G)$-module $L^p(G)$ is injective (if and) only if the group $G$ is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1101.4320","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-22T20:32:59Z","cross_cats_sorted":[],"title_canon_sha256":"9a9cee1c79264b3e311f534ecb8051b8f1705f1142e510576fca7a61dd73d5e2","abstract_canon_sha256":"7f9e2e3c0725efc4f45135ad3f56d8d17f50dd574a223f1c236115f4f30c6767"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:26.398517Z","signature_b64":"2xiPlMbqnixa+Uj8WRSLu4uKbGcMyxXKMSwfYkHS7xxY7lGarYSmIpSpqKQJlVJ4J5LAIzpsUT3D4TECFFbjAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e43d935081792326a164acc3ad8f54d5a98a925aba4d607f4c8b9db9b039020","last_reissued_at":"2026-05-18T04:01:26.397902Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:26.397902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multi-norms and the injectivity of $L^p(G)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"H. Garth Dales, Hung Le Pham, Matthew Daws, Paul Ramsden","submitted_at":"2011-01-22T20:32:59Z","abstract_excerpt":"Let $G$ be a locally compact group, and take $p\\in(1,\\infty)$. We prove that the Banach left $L^1(G)$-module $L^p(G)$ is injective (if and) only if the group $G$ is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4320","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1101.4320","created_at":"2026-05-18T04:01:26.398013+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.4320v2","created_at":"2026-05-18T04:01:26.398013+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.4320","created_at":"2026-05-18T04:01:26.398013+00:00"},{"alias_kind":"pith_short_12","alias_value":"DZB5SNIIC6JD","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"DZB5SNIIC6JDE2QW","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"DZB5SNII","created_at":"2026-05-18T12:26:26.731475+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV","json":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV.json","graph_json":"https://pith.science/api/pith-number/DZB5SNIIC6JDE2QWJLGDVWHVJV/graph.json","events_json":"https://pith.science/api/pith-number/DZB5SNIIC6JDE2QWJLGDVWHVJV/events.json","paper":"https://pith.science/paper/DZB5SNII"},"agent_actions":{"view_html":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV","download_json":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV.json","view_paper":"https://pith.science/paper/DZB5SNII","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.4320&json=true","fetch_graph":"https://pith.science/api/pith-number/DZB5SNIIC6JDE2QWJLGDVWHVJV/graph.json","fetch_events":"https://pith.science/api/pith-number/DZB5SNIIC6JDE2QWJLGDVWHVJV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV/action/storage_attestation","attest_author":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV/action/author_attestation","sign_citation":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV/action/citation_signature","submit_replication":"https://pith.science/pith/DZB5SNIIC6JDE2QWJLGDVWHVJV/action/replication_record"}},"created_at":"2026-05-18T04:01:26.398013+00:00","updated_at":"2026-05-18T04:01:26.398013+00:00"}