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Structure of Jordan *-isomorphisms between general C*-algebras

Characterize the precise structural form of Jordan *-isomorphisms between arbitrary C*-algebras, equivalently determining the exact structure of surjective unital linear isometries between general C*-algebras.

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Background

Kadison proved that surjective unital isometries between a von Neumann algebra and a unital C*-algebra are exactly Jordan *-isomorphisms decomposing into homomorphic and antihomomorphic parts via a central projection. However, beyond the von Neumann case, the exact structural description is not known.

The paper notes that the same structural conclusion does not extend verbatim to all C*-algebras, although weaker, related statements exist. A complete characterization would settle the structure of isometries and Jordan -isomorphisms in full generality for C-algebras.

References

To the best of our knowledge, the precise structure of a Jordan -isomorphism (and hence of an isometry) from a general C-algebra A onto another C*-algebra B is still unknown (what is known is that exactly the same result as in the case where A is a von Neumann algebra does not hold [Example 5.1]{B2}, although a similar but not as definitive does [St]).

— Jordan homomorphisms: A survey (2510.16876 - Brešar et al., 19 Oct 2025) in Section 5 (Jordan homomorphisms on A^{(+)}: The early development)