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

@misc{leung2026reconstructing,
  title = {{Reconstructing the Jordan structure of $C^*$-algebras from the metric on positive unit spheres}},
  author = {Chi-Wai Leung and Chi-Keung Ng and Ngai-Ching Wong},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32029},
  eprint = {2610.32029},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32029},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - Reconstructing the Jordan structure of C^*-algebras from the metric on positive unit spheres
AU  - Chi-Wai Leung
AU  - Chi-Keung Ng
AU  - Ngai-Ching Wong
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32029
UR  - https://arxiv.org/abs/2610.32029
AB  - Let A and B be unital C^*-algebras, and let S_A^+ and 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:S_A^+→S_B^+ is a bijection satisfying |Λ(u)−Λ(v)|=1 Longleftrightarrow |u−v|=1 (u,v∈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:S_A^+→S_B^+ is a surjective isometry, then Λ extends to a Jordan ^*-isomorphism from A onto B.
ER  -