{"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"}