Metric characterisation of unitaries in JB${}^{\ast }$-algebras

Abstract

Let $M$ be a unital JB${}^{\ast }$-algebra whose closed unit ball is denoted by ${\mathcal{B}}_M$. Let ${\partial }_e({\mathcal{B}}_M)$ denote the set of all extreme points of ${\mathcal{B}}_M$. We prove that an element $u\in {\partial }_e({\mathcal{B}}_M)$ is a unitary if and only if the set $${\mathcal{M}}_u=\{e\in {\partial }_e({\mathcal{B}}_M):\Vert u\pm e\Vert \le \sqrt{2}\}$$ contains an isolated point. This is a new geometric characterisation of unitaries in $M$ in terms of the set of extreme points of ${\mathcal{B}}_M$.