Quasi‐isometric embeddings of C0(K,X)$C_{0}(K, X)$ spaces which induce isometries whenever X$X$ is a Hilbert space

• Published in: Mathematische Nachrichten Vol. 298; no. 9; pp. 2975 - 2985
• Main Author: Galego, Elói Medina
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.09.2025
• Subjects: C0(K,X) spaces; Continuity (mathematics); Hilbert space; Hilbert spaces; Isomorphism; locally compact spaces; quasi‐isometric embeddings; Real numbers
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.12033

Suppose that K$K$ and S$S$ are locally compact Hausdorff spaces and X$X$ is a Hilbert space. It is proven that if there exist real numbers M≥1$M \ge 1$, L≥0$L \ge 0$ and a map T$T$ from C0(K,X)$C_{0}(K,X)$ to C0(S,X)$C_{0}(S,X)$ satisfying 1M∥f−g∥−L≤∥T(f)−T(g)∥≤M∥f−g∥+L,$$\begin{equation*} \frac{1}{M}\Vert f-g\Vert -L\le \Vert T(f)-T(g)\Vert \le M\Vert f-g\Vert +L,\ \end{equation*}$$for every f$f$ and g$g$ in C0(K,X)${C}_{0}(K,X)$, then there are a locally compact subset S0$S_0$ of S$S$ and a proper continuous function φ$\varphi$ of S0$S_0$ onto K$K$, on the assumption that M2<27−23.$$\begin{equation*} M^{2}< \frac{2\sqrt {7}-2}{3}. \end{equation*}$$ In this case, as an immediate consequence, φ$\varphi$ generates a linear isometry of C0(K)$C_{0}(K)$ into C0(S0)$C_{0}(S_0)$. Even in the Lipschitz case (L=0$L=0$), this result is the first nonlinear vector generalization of a classical Jarosz theorem (1984) concerning the into linear isomorphisms of spaces of continuous functions on locally compact Hausdorff spaces.