A note on the Radon-Nikodym type theorem for operators on self-dual cones

Abstract

Let $(\mathcal{M},\mathcal{H}, J,\mathcal{H}^+)$ be a standard form of a von Neumann algebra. We consider an order for operators preserving a self-dual cone $\mathcal{H}^+$. Let $A,B$ be positive semi-definite operators on $\mathcal{H}$ such that $A$ preserves $\mathcal{H}^+$ and $B$ belongs to a strong closure of the positive part of an order automorphism group on $\mathcal{H}^+$. We prove that if $A$ is majorized by $B$, then there exists a positive semi-definite operator $c$ in the center $Z(Q\mathcal{M}|_{Q\mathcal{H}})$ with $\| c \| \le 1$ such that $QA|_{Q\mathcal{H}} = cB|_{Q\mathcal{H}}$ where $Q$ is a support projection of $B$