Absolute compatibility and poincar\'{e} sphere

TL;DR

This paper explores the structure of absolutely compatible pairs in von Neumann algebras, introducing strict projections and analyzing their geometric representation within the Poincaré sphere framework.

Contribution

It introduces the concept of strict projections and characterizes absolutely compatible pairs in von Neumann algebras using unitary equivalence and geometric forms.

Findings

Characterization of absolutely compatible pairs via strict projections

Representation of these pairs in matrix algebras

Geometric interpretation on the 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$, $\left((p - x_0) \otimes I_2 \right) P_0 + (x_0 \otimes I_2) P'$ 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 $\mathcal{M} = \mathbb{M}_2$.