Refinements of spectral resolutions and modelling of operators in II$_1$ factors

We 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 n