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

Abstract

Let $M$ and $N$ be two unital JB$^*$-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 $\Delta:\mathcal{U} (M) \to \mathcal{U} (N)$ we can find a surjective real linear isometry $\Psi:M\to N$ which coincides with $\Delta$ on the subset $e^{i M_{sa}}$. If we assume that $M$ and $N$ are JBW$^*$-algebras, then every surjective isometry $\Delta:\mathcal{U} (M) \to \mathcal{U} (N)$ 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$^*$-algebras.