Continuous maps induced by embeddings of $C_{0}(K)$ spaces into $C_{0}(S, X)$ spaces

Abstract

Let K and S be locally compact Hausdorff spaces and X a Banach space. Suppose that T is a linear operator from $C_{0}(K)$ into $C_{0}(S, X)$ with

$$\begin{aligned} \Vert T\Vert \ \Vert T^{-1}\Vert < \lambda (X), \end{aligned}$$

where $\lambda (X)$ is a parameter introduced by Jarosz in 1989. We prove that there exist a subset $S_0$ of S and a continuous function from $S_0$ onto K. This vector-valued version of the 1966 classical Holsztyński’s theorem is optimal in the case where $X=l_{p}$, $2 \le p< \infty$. Moreover, if T satisfies the following stronger condition

$$\begin{aligned} \Vert T\Vert \ \Vert T^{-1}\Vert < \frac{3\lambda (X)}{\lambda (X)+2}, \end{aligned}$$

then for each ordinal $\alpha$ there exist a subset $S_{\alpha }$ of the $\alpha$th derivative of S and a continuous function from $S_{\alpha }$ onto the $\alpha$th derivative of K. When K is compact, the set $S_{\alpha }$ may be taken closed.