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 let U(M) U(N) denote the sets of all unitaries in N, respectively. We prove that following statements are equivalent: isometrically isomorphic as (complex) Banach spaces;M real spaces;there exists a surjective isometry Δ:U(M)→U(N).We actually establish more general statement asserting that, under some mild extra conditions, for each Δ:U(M)→U(N), we can find linear Ψ:M→N which coincides with Δ on subset eiMsa. If assume JBW∗-algebras, then every Δ:U(M)→U(N) admits (unique) extension to from onto N. This is an Hatori–Molnár theorem setting JB∗-algebras.