Absolutely compatible pairs in a von Neumann algebra

Abstract

Let $a,b$ be elements in a unital C${}^{\ast }$-algebra with $0\le a,b\le 1$. The element $a$ is absolutely compatible with $b$ if $$|a-b|+|1-a-b|=1.$$ 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 ${\mathbb{M}}_n$.

20 pages