Surjective isometries between unitary sets of unital JB⁎-algebras1

This paper is, in a first stage, devoted to establishing a topological-algebraic characterization of the principal component, U (M), of the set of unitary elements, U0(M), in a unital JB*-algebra M. We arrive to the conclusion that, as in the case of unital C*-algebras, ... is analytically arcwise connected. Actually, U0 (M) is the smallest quadratic subset of U(M) containing the set eiMsa. Our second goal is to provide a complete description of the surjective isometries between the principal components of two unital JB*-algebras M and N. Contrary to the case of unital C*-algebras, we shall deduce the existence of connected components in %{M) which are not isometric as metric spaces. We shall also establish necessary and sufficient conditions to guarantee that a surjective isometry A : U{M) → U(N) admits an extension to a surjective linear isometry between M and N, a conclusion which is not always true. Among the consequences it is proved that M and N are Jordan *-isomorphic i£ and only i£ their principal components are isometric as metric spaces if and only if, there exists a surjective isometry A : U(M) → U(N) mapping the unit of M to an element in U0 (N). These results provide an extension to the setting of unital JB*-algebras of the results obtained by O. Hatori for unital C*-algebras.(ProQuest: … denotes formulae omitted.)