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 J B ∗ -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: M and N are isometrically isomorphic as (complex) Banach spaces; M and N are isometrically isomorphic as real Banach spaces; 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 e i M s a . If we assume that M and N are J B W ∗ -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 J B ∗ -algebras.