2023/01/02 by Peralta, Antonio M., Švarc, Radovan · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2301.00895
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, \mathfrakA**, of a unital JB^*-algebra \mathfrakA, there exists a self-adjoint element h in \mathfrakA satisfying e≤ exp(ih), that is, e is bounded by a unitary in the principal connected component of the unitary elements in \mathfrakA. 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 \mathfrakA asserting that each such face F of \mathfrakA coincides with the norm closed convex hull of the unitaries of \mathfrakA 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 \mathfrakA 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 \mathfrakA satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of \mathfrakA onto the unit sphere of any other Banach space Y admits an extension to a surjective real linear isometry from \mathfrakA onto Y. This extends a result of M. Mori and N. Ozawa who have proved the same for unital C^*-algebras.