Absolutely compatibility in $\mathbb{M}_2$ and ellipticity

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 describe a complete list of absolutely compatible pairs in $\mathbb{M}_2$. We note that non-commutative pairs exhibit an elliptic behaviour.