On the Extension of Isometries Between the Unit Spheres of a -Triple and a Banach Space

Abstract

We prove that every JBW * -triple M with rank one or rank bigger than or equal to three satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of M onto the unit sphere of another Banach space Y extends to a surjective real linear isometry from M onto Y . We also show that the same conclusion holds if M is not a JBW * -triple factor, or more generally, if the atomic part of M * * is not a rank two Cartan factor.

On the Mazur–Ulam property for the space of Hilbert-space-valued continuous functions

“…. We therefore conclude that u(a) ≥ s(φ) in X * * , and thus φ(u(a)) = 1, witnessing that {u(a)} ′,X * = {a} ′,X * and consequently, (6) {a} ′,X * ′,X * * = ({u(a)} ′,X * )…”

“…, for all natural n, E * * is weak * -closed in X * * , and τ ♯♯ is weak * -continuous, we deduce that τ ♯♯ (r(a)) = r(a) and τ ♯♯ (u(a)) = u(a), that is, the range and support tripotents of a in X * * are τ ♯♯symmetric elements in X * * , and thus they both are tripotents in E * * , called range and support tripotents of a in E * * . Combining (6) with the previous conclusions we get (8) {a}…”

“…In the setting of (complex) JB * -triples a new characterization of compact tripotents in the second dual has been recently established in [6]. The concrete result reads as follows: Theorem 3.6] Let X be a JB * -triple.…”

“…Apart from a wide list of classical Banach spaces satisfying the Mazur-Ulam property (cf. [56,46]), new achievements prove that this property is satisfied by commutative von Neumann algebras [15], unital complex C * -algebras and real von Neumann algebras [44], and more recently, JBW * -triples with rank one or rank bigger than or equal to three [6]. The latest two mentioned references naturally lead us to consider the Mazur-Ulam property on the space C(K, H) of all continuous functions from a compact Hausdorff space K into a real or complex Hilbert space H.…”

“…Banach space X such that the closed convex hull of the extreme points of its closed unit ball has non-empty interior satisfies that every convex body K ⊂ X has the strong Mankiewicz property, that is, every surjective isometry ∆ from K onto an arbitrary convex subset L in a normed space Y is affine. This is a key ingredient to prove that unital C * -algebras, real von Neumann algebras and JBW * -triples of rank 1 or bigger or equal than three satisfy the Mazur-Ulam property [44,6].…”