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

Abstract

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 \(e\leq \exp(ih)\), 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 result of M. Mori and N. Ozawa who have proved the same for unital C\(^*\)-algebras.