Metric characterisation of unitaries in JB$^*$-algebras

Abstract: Let $M$ be a unital 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 $$\mathcal{M}{u} = {e\in \partial_e(\mathcal{B}_M) : |u\pm e|\leq \sqrt{2} }$$ contains an isolated point. This is a new geometric characterisation of unitaries in $M$ in terms of the set of extreme points of $\mathcal{B}_M$.