Schur-Horn theorems in II$_\infty$-factors

TL;DR

This paper extends Schur-Horn theorems to selfadjoint operators in $ ext{II}_ ext{infty}$ factors, characterizing majorization and spectral relations within this infinite-dimensional setting.

Contribution

It provides a spectral majorization characterization for operators in $ ext{II}_ ext{infty}$ factors, generalizing classical Schur-Horn theorems to this context.

Findings

Characterization of the measure topology closure of unitary orbits via majorization.

Extension of Schur-Horn theorems to positive and $ au$-integrable operators.

Spectral relations expressed through simple spectral conditions.

Abstract

We describe majorization between selfadjoint operators in a $\sigma$-finite II$_\infty$ factor $(\mathcal{M},\tau)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$ with trace-preserving conditional expectation $E_{\mathcal{A}}$, we characterize the closure in the measure topology of the image through $E_{\mathcal{A}}$ of the unitary orbit of a selfadjoint operator in $\mathcal{M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in $\mathcal{M}$ and for the unitary and contractive orbits of $\tau$-integrable operators in $\mathcal{M}$.