{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:2BJQOX7QT6MKEQ6DUB7POCMT75","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":"cffe00690826426964b740bd27f7e635c326ff75193cfdd626f667729b3a41fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-01-25T09:49:39Z","title_canon_sha256":"7253e1654ad10954528891764209d3acd4ee4e77e46716898d943f0d2eea23b0"},"schema_version":"1.0","source":{"id":"1701.07229","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.07229","created_at":"2026-05-18T00:52:07Z"},{"alias_kind":"arxiv_version","alias_value":"1701.07229v1","created_at":"2026-05-18T00:52:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.07229","created_at":"2026-05-18T00:52:07Z"},{"alias_kind":"pith_short_12","alias_value":"2BJQOX7QT6MK","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"2BJQOX7QT6MKEQ6D","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"2BJQOX7Q","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:ad8a2067df8b127bf72dc6f13e64ff509609eeaa66aa8521043f6d596fb72280","target":"graph","created_at":"2026-05-18T00:52:07Z","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":"Let $(G,+)$ be a topological abelian group with a neutral element $e$ and let $\\mu : G\\longrightarrow\\mathbb{C}$ be a continuous character of $G$. Let $(\\mathcal{H}, \\langle \\cdot,\\cdot \\rangle)$ be a complex Hilbert space and let $\\mathbf{B}(\\mathcal{H})$ be the algebra of all linear continuous operators of $\\mathcal{H}$ into itself. A continuous mapping $ \\Phi: G\\longrightarrow \\mathbf{B}(\\mathcal{H})$ will be called an operator-valued $\\mu$-cosine function if it satisfies both the $\\mu$-cosine equation $$\\Phi(x+y)+\\mu(y)\\Phi(x-y)=2\\Phi(x)\\Phi(y),\\; x,y\\in G$$ and the condition $\\Phi(e)=I,$ ","authors_text":"Bouikhalene Belaid, Elqorachi Elhoucien","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-01-25T09:49:39Z","title":"On the Operator-valued $\\mu$-cosine functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07229","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:e5650bd5c349b38b2c5738c8b085d016a3525699cfcfbf06ed58dc876ac4f5a3","target":"record","created_at":"2026-05-18T00:52:07Z","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":"cffe00690826426964b740bd27f7e635c326ff75193cfdd626f667729b3a41fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-01-25T09:49:39Z","title_canon_sha256":"7253e1654ad10954528891764209d3acd4ee4e77e46716898d943f0d2eea23b0"},"schema_version":"1.0","source":{"id":"1701.07229","kind":"arxiv","version":1}},"canonical_sha256":"d053075ff09f98a243c3a07ef70993ff4da1d46ed45906ee1d82bb4f318826b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d053075ff09f98a243c3a07ef70993ff4da1d46ed45906ee1d82bb4f318826b1","first_computed_at":"2026-05-18T00:52:07.119891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:52:07.119891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3UkQMjFgN23FvTSktr3ZlKYoPtcwbBl5wm6mxe0y8W1yxLTr56pBvlguYqAd7GFw4CrX9IUWTjkzA4Zjp/D/CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:52:07.120547Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.07229","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e5650bd5c349b38b2c5738c8b085d016a3525699cfcfbf06ed58dc876ac4f5a3","sha256:ad8a2067df8b127bf72dc6f13e64ff509609eeaa66aa8521043f6d596fb72280"],"state_sha256":"266d748932f84be0851c04b77c07249ba30df5fdb1f572b356fbf66a65391dab"}