2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112633We study refinements between spectral resolutions in an arbitrary II$_1$ factor $\M$ and obtain diffuse (maximal) refinements of spectral resolutions. We construct models of operators with respect to diffuse spectral resolutions. As an application we obtain new characterizations of sub-majorization and spectral preorder between positive operators in $\M$ and new versions of some known inequalities involving these preorders.Functional AnalysisOperator Algebras46L51; 47A63text