A strengthened Kadison’s transitivity theorem for unital JB\(^*\)-algebras with applications to the Mazur–Ulam property

The principal result in this note is a strengthened version of Kadison’s transitivity theorem for unital JB\(^*\)-algebras, showing that for each minimal tripotent e in the bidual, \({\mathfrak {A}}^{**}\), of a unital JB\(^*\)-algebra \({\mathfrak {A}}\), there exists a self-adjoint element h in \({\mathfrak {A}}\) satisfying \(e\le \exp (ih)\), that is, e is bounded by a unitary in the principal connected component of the unitary elements in \({\mathfrak {A}}\). This new result opens the way to attack new geometric results, for example, a Russo–Dye type theorem for maximal norm closed proper faces of the closed unit ball of \({\mathfrak {A}}\) asserting that each such face F of \({\mathfrak {A}}\) coincides with the norm closed convex hull of the unitaries of \({\mathfrak {A}}\) which lie in F. Another geometric property derived from our results proves that every surjective isometry from the unit sphere of a unital JB\(^*\)-algebra \({\mathfrak {A}}\) onto the unit sphere of any other Banach space is affine on every maximal proper face. As a final application we show that every unital JB\(^*\)-algebra \({\mathfrak {A}}\) satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of \({\mathfrak {A}}\) onto the unit sphere of any other Banach space Y admits an extension to a surjective real linear isometry from \({\mathfrak {A}}\) onto Y. This extends a contribution by M. Mori and N. Ozawa who have proved the same result for unital C\(^*\)-algebras.