Absolute compatibility and poincaré sphere

Abstract

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