Refinements of spectral resolutions and modelling of operators in II<SUB>1</SUB> factors

Abstract

We study refinements between spectral resolutions in an arbitrary II<SUB>1</SUB> 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.