Reconstructing the Jordan structure of $C^*$-algebras from the metric on positive unit spheres

Abstract

Let $A$ and $B$ be unital $C^*$-algebras, and let $\mathrm{S}_{A^+}$ and $\mathrm{S}_{B^+}$ denote their positive unit spheres. We first show that the diametral relation on the positive unit sphere already determines Jordan structure. More precisely, suppose that $Λ:\mathrm{S}_{A^+}\to\mathrm{S}_{B^+}$ is a bijection satisfying \[ \|Λ(u)-Λ(v)\|=1 \quad\Longleftrightarrow\quad \|u-v\|=1 \qquad (u,v\in\mathrm{S}_{A^+}). \] If $A$ has no two-dimensional irreducible $^*$-representation, then $A$ and $B$ are Jordan $^*$-isomorphic. Moreover, the Jordan $^*$-isomorphism can be chosen to be compatible with $Λ$ on the peak and zero projections of positive norm-one elements. The proof is based on recovering the order and orthogonality structure of Akemann closed projections in the bidual. As a main application, we give an affirmative answer to the positive-sphere extension problem for unital $C^*$-algebras of real rank zero. Namely, if $A$ has real rank zero (but could has two-dimensional irreducible $^*$-representations) and $Λ:\mathrm{S}_{A^+}\to\mathrm{S}_{B^+}$ is a surjective isometry, then $Λ$ extends to a Jordan $^*$-isomorphism from $A$ onto $B$.