Involution-preserving ring isomorphisms in norm between unital C^*-algebras

Abstract

Let A and B be nonzero unital C^*-algebras with units 1_A and 1_B, respectively, and let T: A→ B be a bijection satisfying |T(a+b)|=|T(a)+T(b)|, |T(ab)|=|T(a)T(b)|, T(a^*)=T(a)^* for all a,b∈ A. We prove that there exist a central symmetry u in B and a real *-isomorphism Φ: A→ B such that T(a)=uΦ(a) (a∈ A). Moreover, u and Φ are uniquely determined by T. Conversely, if u∈ B is a central symmetry and Φ: A→ B is a real $ * $-isomorphism, then T=uΦ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map uT to preserve the full product, not merely the Jordan product.

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BibTeX RIS

BibTeX

@misc{matsuzaki2026involutionpreserving,
  title = {{Involution-preserving ring isomorphisms in norm between unital $C^*$-algebras}},
  author = {Izuho Matsuzaki},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18121},
  eprint = {2609.18121},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18121},
}

RIS

TY  - UNPB
TI  - Involution-preserving ring isomorphisms in norm between unital C^*-algebras
AU  - Izuho Matsuzaki
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18121
UR  - https://arxiv.org/abs/2609.18121
AB  - Let A and B be nonzero unital C^*-algebras with units 1_A and 1_B, respectively, and let T: A→ B be a bijection satisfying |T(a+b)|=|T(a)+T(b)|, |T(ab)|=|T(a)T(b)|, T(a^*)=T(a)^* for all a,b∈ A. We prove that there exist a central symmetry u in B and a real *-isomorphism Φ: A→ B such that T(a)=uΦ(a) (a∈ A). Moreover, u and Φ are uniquely determined by T. Conversely, if u∈ B is a central symmetry and Φ: A→ B is a real $ * $-isomorphism, then T=uΦ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map uT to preserve the full product, not merely the Jordan product.
ER  -

Abstract

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