2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154058We introduce the notions of approximate innerness and central triviality for endomorphisms on separable von Neumann factors, and we characterize them for hyperfinite factors by Connes-Takesaki modules of endomorphisms and modular endomorphisms which are introduced by Izumi. Our result is a generalization of the corresponding result obtained by Kawahigashi-Sutherland-Takesaki in automorphism case.57 pagesOperator Algebras46L10, 46L37text