A metric characterization of projections among positive norm-One elements in unital C$^*$-algebras

Abstract: We characterize projections among positive norm-one elements in unital C$^*$-algebras in pure geometric terms determined by the norm of the underlying Banach space. Concretely, let $A$ be a C$^*$-algebra (or a JB$^*$-algebra) whose positive cone and unit sphere are denoted by ${A}^+$ and $\mathrm{S}{A}$, respectively. The positive portion of the unit sphere in $A$, denoted by $\mathrm{S}{{A}^+}$, is the set ${A}^+ \cap \mathrm{S}{A}$, while the unit sphere of positive norm-one elements around a subset $\mathscr{S}$ in $\mathrm{S}{A^+}$ is the set $$\hbox{Sph}{{\mathrm{S}{{A}^+}}} (\mathscr{S}) :=\Big{ x\in \mathrm{S}{{A}^+} : |x-s|=1 \hbox{ for all } s\in \mathscr{S} \Big}.$$ Assuming that $A$ is unital, we establish that an element $a\in \mathrm{S}{{A}^+}$ is a projection if, and only if, it satisfies the double sphere property, that is, $ \hbox{Sph}{{\mathrm{S}{{A}^+}}} \left(\hbox{Sph}{{\mathrm{S}_{{A}^+}}} \left({a}\right) \right) = {a}.$