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

TL;DR

This paper investigates spectral resolution refinements in II$_1$ factors, constructing models and deriving new characterizations of operator preorders and inequalities, advancing the understanding of operator structure in this setting.

Contribution

It introduces diffuse refinements of spectral resolutions and models of operators, providing new characterizations of sub-majorization and spectral preorders in II$_1$ factors.

Findings

Constructed diffuse (maximal) spectral resolution refinements.

Developed models of operators relative to these refinements.

Derived new characterizations and inequalities involving operator preorders.

Abstract

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 new versions of some known inequalities involving these preorders.