Metric Characterisation of Unitaries in $\hbox {JB}^*$-Algebras

Abstract

Let M be a unital $\hbox {JB}^*$-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

$$\begin{aligned} {\mathcal {M}}_{u} = \{e\in \partial _e({\mathcal {B}}_M) : \Vert u\pm e\Vert \le \sqrt{2} \} \end{aligned}$$

contains an isolated point. This is a new geometric characterisation of unitaries in M in terms of the set of extreme points of ${\mathcal {B}}_M$.