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 $$^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> -algebras, showing that for each minimal tripotent e in the bidual, $${\mathfrak {A}}^{**}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow/> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:mmultiscripts> </mml:math> , of a unital JB $$^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> -algebra $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> , there exists a self-adjoint element h in $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> satisfying $$e\le \exp (ih)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>≤</mml:mo> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mi>h</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , that is, e is bounded by a unitary in the principal connected component of the unitary elements in $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> . 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 $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> asserting that each such face F of $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> coincides with the norm closed convex hull of the unitaries of $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> which lie in F . Another geometric property derived from our results proves that every surjective isometry from the unit sphere of a unital JB $$^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> -algebra $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> 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 $$^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> -algebra $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> onto the unit sphere of any other Banach space Y admits an extension to a surjective real linear isometry from $${\mathfrak {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> onto Y . This extends a contribution by M. Mori and N. Ozawa who have proved the same result for unital C $$^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> -algebras.