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We say that the path $t\\mapsto E_t$ is smooth if for every $a\\in A$ and $v \\in H$, the map $$ I\\ni t\\mapsto E_t(a)v\\in H $$ is continuously differentiable. This condition implies the existence of the derivative operator $$ dE_t(a):H\\to H, \\ dE_t(a)v=\\frac{d}{dt}E_t(a)v. $$ If this operator verifies the additional boundedness condition, $$ \\int_J \\|dE_t(a)\\|_2^2 d"},"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":"1010.1045","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-10-06T01:03:51Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"5876433130f529360b4e90cf1c98158bb665bb9063ed0582a0ab4c16ee13a41a","abstract_canon_sha256":"8cdfba237207d191fb0d9e087465d52f81c628411df58a579b9b0c1f093c616b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:49.926037Z","signature_b64":"0FxCRpthqqEYkeqkWDHYBTOgzaQKBik6cEXa2aUKkB6Xd5bSD0dsnEyjT0MybfGqq0/FVODd18PR4AWxlQHyCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffc66706a23acd9b922514277b6211648398f39d9428605d4a4d04c75ef15a8f","last_reissued_at":"2026-05-18T04:39:49.925526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:49.925526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Smooth paths of conditional expectations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Esteban Andruchow, Gabriel Larotonda","submitted_at":"2010-10-06T01:03:51Z","abstract_excerpt":"Let A be a von Neumann algebra with a finite trace $\\tau$, represented in $H=L^2(A,\\tau)$, and let $B_t\\subset A$ be sub-algebras, for $t$ in an interval $I$. Let $E_t:A\\to B_t$ be the unique $\\tau$-preserving conditional expectation. We say that the path $t\\mapsto E_t$ is smooth if for every $a\\in A$ and $v \\in H$, the map $$ I\\ni t\\mapsto E_t(a)v\\in H $$ is continuously differentiable. This condition implies the existence of the derivative operator $$ dE_t(a):H\\to H, \\ dE_t(a)v=\\frac{d}{dt}E_t(a)v. $$ If this operator verifies the additional boundedness condition, $$ \\int_J \\|dE_t(a)\\|_2^2 d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1045","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":"1010.1045","created_at":"2026-05-18T04:39:49.925604+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.1045v1","created_at":"2026-05-18T04:39:49.925604+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1045","created_at":"2026-05-18T04:39:49.925604+00:00"},{"alias_kind":"pith_short_12","alias_value":"77DGOBVCHLGZ","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"77DGOBVCHLGZXERF","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"77DGOBVC","created_at":"2026-05-18T12:26:04.259169+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/77DGOBVCHLGZXERFCQTXWYQRMS","json":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS.json","graph_json":"https://pith.science/api/pith-number/77DGOBVCHLGZXERFCQTXWYQRMS/graph.json","events_json":"https://pith.science/api/pith-number/77DGOBVCHLGZXERFCQTXWYQRMS/events.json","paper":"https://pith.science/paper/77DGOBVC"},"agent_actions":{"view_html":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS","download_json":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS.json","view_paper":"https://pith.science/paper/77DGOBVC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.1045&json=true","fetch_graph":"https://pith.science/api/pith-number/77DGOBVCHLGZXERFCQTXWYQRMS/graph.json","fetch_events":"https://pith.science/api/pith-number/77DGOBVCHLGZXERFCQTXWYQRMS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS/action/storage_attestation","attest_author":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS/action/author_attestation","sign_citation":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS/action/citation_signature","submit_replication":"https://pith.science/pith/77DGOBVCHLGZXERFCQTXWYQRMS/action/replication_record"}},"created_at":"2026-05-18T04:39:49.925604+00:00","updated_at":"2026-05-18T04:39:49.925604+00:00"}