On linear extendability of isometrical embeddings (Q511309)

It is an interesting problem to find some conditions under which an isometry between two normed linear spaces is affine. This question is related to the Mazur-Ulam theorem; see [\textit{T. Figiel}, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 16, 185--188 (1968; Zbl 0155.18301)] and [\textit{M. S. Moslehian} and \textit{G. Sadeghi}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 10, 3405--3408 (2008; Zbl 1160.46049)]. Motivated by \textit{R. Villa} [Stud. Math. 129, No. 3, 197--205 (1998; Zbl 0915.46007)], the authors investigate linear extendability of an isometric embedding \(T : U \to Y\) from an open subset \(U\) of a real Banach space \(X\) into a real Banach space \(Y\) in the case when \(Y\) is either the space \(C_{\mathbb{R}}(K)\) of continuous real-valued functions on a compact space \(K\), or is a strictly convex Banach space. In the first case, they show that there exists a closed subset \(L \subseteq K\) and a real-linear isometry \(\widetilde{T} : X \to C_{\mathbb{R}}(L)\) which extends \(Q \circ T\) (up to a translation), where \(Q : C_{\mathbb{R}}(K) \to C_{\mathbb{R}}(L)\) is the restriction map, and in the second case, when \(Y\) is strictly convex, it turns out that such an isometry \(T: U\to Y\) is extended to a real-linear isometry from \(X\) into \(Y\) up to a translation.