Can one identify two unital JB${}^{\ast }$-algebras by the metric spaces determined by their sets of unitaries?

Let $M$ and $N$ be two unital JB${}^{\ast }$-algebras and let $\mathcal{U}(M)$ and $\mathcal{U}(N)$ 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 $\Delta: $\mathcal{U}$(M)\to $\mathcal{U}$(N).$ We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry $\Delta:$\mathcal{U}$ (M) \to $\mathcal{U}$ (N)$wecanfindasurjectivereallinearisometry$\Psi:M\to N$whichcoincideswith$\Delta$onthesubset$e^{i M_{sa}}$\mathrm{.}Ifweassumethat$M$and$N$areJBW$^*$-algebras,theneverysurjectiveisometry$\Delta:$\mathcal{U}$ (M) \to $\mathcal{U}$ (N)$ admits a (unique) extension to a surjective real linear isometry from $M$ onto $N$. This is an extension of the Hatori–Moln{'a}r theorem to the setting of JB${}^{\ast }$-algebras.