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We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,\\pi_\\varphi)$ and reduced 1-cohomology $\\bar{H}^1(G,\\pi_\\varphi)$. For example, $\\bar{H}^1(G,\\pi_\\varphi)=0$ if and only if either $Hom(G,\\mathbb{C})=0$ or $\\mu_\\varphi(1_G)=0$, where"},"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":"1303.4185","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-03-18T08:52:07Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"407fa677c91f5f696f473769f4a3228c5bf7f49a43c51b24f7c92ab259015dc5","abstract_canon_sha256":"a9c6945dcc63ca848a3eeaafca92e83fc3bf3fc22ab05de778a682d1ef1c4ff9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:39.462850Z","signature_b64":"rbkiirqbtInsB/6lRAJ8bvu0hBbcUcmfTqRMgWfsftStHcZMfjWHf4uSU+KvC9xOHUd0VBmZFNyayvKYPIu8Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a3bd5fe2fa8d4873677315e5346a5396dc1d3c5b133cb1bb58708dcf2f35e6b","last_reissued_at":"2026-05-18T03:30:39.461873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:39.461873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On 1-cocycles induced by a positive definite function on a locally compact abelian group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.RT","authors_text":"Alain Valette, Jordan Franks","submitted_at":"2013-03-18T08:52:07Z","abstract_excerpt":"For $\\varphi$ a normalized positive definite function on a locally compact abelian group $G$, we consider on the one hand the unitary representation $\\pi_\\varphi$ associated to $\\varphi$ by the GNS construction, on the other hand the probability measure $\\mu_\\varphi$ on the Pontryagin dual $\\hat{G}$ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,\\pi_\\varphi)$ and reduced 1-cohomology $\\bar{H}^1(G,\\pi_\\varphi)$. For example, $\\bar{H}^1(G,\\pi_\\varphi)=0$ if and only if either $Hom(G,\\mathbb{C})=0$ or $\\mu_\\varphi(1_G)=0$, where"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4185","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1303.4185","created_at":"2026-05-18T03:30:39.462040+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.4185v1","created_at":"2026-05-18T03:30:39.462040+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.4185","created_at":"2026-05-18T03:30:39.462040+00:00"},{"alias_kind":"pith_short_12","alias_value":"NI55L7RPVDKI","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"NI55L7RPVDKIONTX","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"NI55L7RP","created_at":"2026-05-18T12:27:52.871228+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/NI55L7RPVDKIONTXGFPFGRVFHF","json":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF.json","graph_json":"https://pith.science/api/pith-number/NI55L7RPVDKIONTXGFPFGRVFHF/graph.json","events_json":"https://pith.science/api/pith-number/NI55L7RPVDKIONTXGFPFGRVFHF/events.json","paper":"https://pith.science/paper/NI55L7RP"},"agent_actions":{"view_html":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF","download_json":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF.json","view_paper":"https://pith.science/paper/NI55L7RP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.4185&json=true","fetch_graph":"https://pith.science/api/pith-number/NI55L7RPVDKIONTXGFPFGRVFHF/graph.json","fetch_events":"https://pith.science/api/pith-number/NI55L7RPVDKIONTXGFPFGRVFHF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF/action/storage_attestation","attest_author":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF/action/author_attestation","sign_citation":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF/action/citation_signature","submit_replication":"https://pith.science/pith/NI55L7RPVDKIONTXGFPFGRVFHF/action/replication_record"}},"created_at":"2026-05-18T03:30:39.462040+00:00","updated_at":"2026-05-18T03:30:39.462040+00:00"}