A metric characterization of projections among positive norm-One elements in unital C${}^{\ast }$-algebras

Abstract

We characterize projections among positive norm-one elements in unital C${}^{\ast }$-algebras in pure geometric terms determined by the norm of the underlying Banach space. Concretely, let $A$ be a C${}^{\ast }$-algebra (or a JB${}^{\ast }$-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 $\mathcal{S}$ in ${\mathrm{S}}_{A^{+}}$ is the set $${Sph}_{{}_{{\mathrm{S}}_{A^{+}}}}(\mathcal{S}):=\{x\in {\mathrm{S}}_{A^{+}}:\Vert x-s\Vert =1\; for\; alls\in \mathcal{S}\}.$$ 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, ${Sph}_{{}_{{\mathrm{S}}_{A^{+}}}}\left({Sph}_{{}_{{\mathrm{S}}_{A^{+}}}}\left(\{a\}\right)\right)=\{a\}.$