Absolute compatibility and poincaré 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 $\left((p-x_0)\otimes I_2\right)P_0+(x_0\otimes I_2)P$, $\left((p-x_0)\otimes I_2\right)P_0+(x_0\otimes I_2)P^{\prime }$ of $M_2({\mathcal{M}}_0)$ where ${\mathcal{M}}_0$ is an abelian von Neumann algebra, $x_0$ is a strict element of ${\mathcal{M}}_0^{+}$, $P_0=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\in M_2({\mathcal{M}}_0)$ and $P$ is a strict projection in $M_2({\mathcal{M}}_0)$. We also discuss the geometric form of this representation when $\mathcal{M}={\mathbb{M}}_2$.

12 pages