Absolutely compatible pairs in a von Neumann algebra

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 $\mathbb{M}_n$.