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

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

Content of this paper has been merged with 1801.01216