Nonlinear embeddings of spaces of continuous functions

Galego, Elói; da Silva, André

doi:10.1090/proc/14798

Nonlinear embeddings of spaces of continuous functions
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by Elói Medina Galego and André Luis Porto da Silva PDF

Proc. Amer. Math. Soc. 148 (2020), 1555-1566 Request permission

Abstract:

We provide a nonlinear version of an extension of the classical Holsztyński theorem due to Jarosz (1984) concerning the into isomorphisms of spaces of continuous functions. More precisely, supposing that $K$ and $S$ are locally compact Hausdorff spaces, we prove that if there exists a map $T$ from an extremely regular subspace $A$ of $C_{0}(K)$ to $C_{0}(S)$ satisfying \begin{equation*} \frac {1}{M} \|f-g\|-L \leq \|T(f)-T(g)\|\leq M \|f-g\|+L\ \forall f, g \in A, \end{equation*} with $1\leq M^{2}<2$ and $L\geq 0$, then there exist a subset $S_{0}$ of $S$ and a proper mapping $\varphi$ of $S_{0}$ onto $K$.

We show that $\varphi$ is not only continuous in the obvious case when K is compact, but also in the case when $M^{2}< 4/3$, where $S_0$ can be taken locally compact.

In the Lipschitz case, that is, when $L=0$, and $K$ and $S$ are intervals of ordinal numbers, our result improves some others by Procházka and Sánchez-González (2017) concerning the Lipschitz embeddings between $C(K)$ spaces and also solves a problem raised by them.

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