Absolute compatibility and poincaré sphere

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 absolutely compatible pair strict elements a {M}$$ unitarily equivalent to $$\left( (1 - x_0) \otimes I_2 \right) P_0 + (x_0 I_2) P$$ P'$$ algebra, $$x_0$$ element {M}_0^+$$ , $$P_0 = \begin{bmatrix} 0 &{} \\ 1 \end{bmatrix} \in M_2(\mathcal P projection . We also discuss geometric form representation when {M} \mathbb