Absolutely compatible pairs in a von Neumann algebra

TL;DR

This paper characterizes absolutely compatible pairs of elements in von Neumann algebras, providing geometric and algebraic insights, especially for 2x2 matrices, and explores their proximity to orthogonality and commutativity.

Contribution

It offers new technical characterizations of absolutely compatible pairs in von Neumann algebras and applies these to geometric interpretations in matrix algebras.

Findings

Characterizations of absolutely compatible pairs in von Neumann algebras

Geometric interpretation for 2x2 matrices involving ellipsoids

Descriptions of compatible pairs in matrix algebras

Abstract

Let $a,b$ be elements in a unital C$^*$-algebra with $0\leq a,b\leq 1$. The element $a$ is absolutely compatible with $b$ if $$\vert a - b \vert + \vert 1 - a - b \vert = 1.$$ In this note we find some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra. These characterizations are applied to measure how close is a pair of absolute compatible positive elements in the closed unit ball from being orthogonal or commutative. In the case of 2 by 2 matrices the results offer a geometric interpretation in terms of an ellipsoid determined by one of the points. The conclusions for 2 by 2 matrices are also applied to describe absolutely compatible pairs of positive elements in the closed unit ball of $\mathbb{M}_n$.