Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?

Let M and N be two unital JB∗-algebras and let U(M) and U(N) denote the sets of all unitaries in M and N, respectively. We prove that the following statements are equivalent:

A) M and N are isometrically isomorphic as (complex) Banach spaces;

B) M and N are isometrically isomorphic as real Banach spaces;

C) there exists a surjective isometry Δ:U(M)→U(N).

We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry Δ:U(M)→U(N), we can find a surjective real linear isometry Ψ:M→N which coincides with Δ on the subset eiMsa. If we assume that M and N are JBW∗-algebras, then every surjective isometry Δ:U(M)→U(N) admits a (unique) extension to a surjective real linear isometry from M onto N. This is an extension of the Hatori–Molnár theorem to the setting of JB∗-algebras.