Reconstructing the Jordan structure of $C^{\ast }$-algebras from the metric on positive unit spheres

Abstract

Let $A$ and $B$ be unital $C^{\ast }$-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 $\Lambda :{\mathrm{S}}_{A^{+}}\to {\mathrm{S}}_{B^{+}}$ 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 ${}^{\ast }$-representation, then $A$ and $B$ are Jordan ${}^{\ast }$-isomorphic. Moreover, the Jordan ${}^{\ast }$-isomorphism can be chosen to be compatible with $\Lambda$ 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 $C^{\ast }$-algebras of real rank zero. Namely, if $A$ has real rank zero (but could has two-dimensional irreducible ${}^{\ast }$-representations) and $\Lambda :{\mathrm{S}}_{A^{+}}\to {\mathrm{S}}_{B^{+}}$ is a surjective isometry, then $\Lambda$ extends to a Jordan ${}^{\ast }$-isomorphism from $A$ onto $B$.

arXiv comment32 pages