On the extension of isometries between the unit spheres of a JBW$^*$-triple and a Banach space

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

“…, 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} ′,E * = {u(a)} ′,E * , and {a} ′,E * ′,E * * = ({u(a)} ′,E * )…”

“…Lemma 5.1). We first observe that if K = {t 0 }, then C(K, H) is isometrically isomorphic to H, and thus the desired conclusion follows, for example, from [6,Proposition 4.15].…”

“…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 3.4.…”

“…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. This space is not, in general, a C * -algebra nor a JBW * -triple because it is neither a dual Banach space.…”

“…M. Mori and N. Ozawa prove in [44, Theorem 2] that every 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].…”