Absolute compatibility and poincar\'{e} sphere

Abstract

In this paper, we introduce the notion of strict projections and prove that an absolutely compatible pair of strict elements in a von Neumann algebra $\mathcal{M}$ unitarily equivalent to the elements $ \left((p - x_0) \otimes I_2 \right) P_0 + (x_0 \otimes I_2) P$, $\left((p - x_0) \otimes I_2 \right) P_0 + (x_0 \otimes I_2) P'$ of $M_2(\mathcal{M}_0)$ where $\mathcal{M}_0$ is an abelian von Neumann algebra, $x_0$ is a strict element of $\mathcal{M}_0^+$, $P_0 = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \in M_2(\mathcal{M}_0)$ and $P$ is a strict projection in $M_2(\mathcal{M}_0)$. We also discuss the geometric form of this representation when $\mathcal{M} = \mathbb{M}_2$.