On embeddings of $C_0(K)$ spaces into $C_0(L,X)$ spaces

TL;DR

This paper investigates conditions under which the space of continuous functions vanishing at infinity, $C_0(K)$, can be embedded into $C_0(L,X)$ spaces, revealing restrictions on the topological spaces involved based on properties of the Banach space $X$.

Contribution

It establishes new embedding restrictions for $C_0(K)$ into $C_0(L,X)$ spaces when $X$ lacks a copy of $c_0$, linking the structure of $K$ and $L$ with properties of $X$.

Findings

If $X$ contains no $c_0$ and admits an embedding of $C_0(K)$ into $C_0(L,X)$ with $X$ separable or $X^*$ having Radon-Nikodym property, then $K$ is finite or $|K| extless=|L|.

Embedding $C_0(K)$ into $C_0(L,X)$ with $X$ not containing $c_0$ and $L$ scattered implies $K$ is scattered.

Abstract

Let $C_0(K, X)$ denote the space of all continuous $X$-valued functions defined on the locally compact Hausdorff space $K$ which vanish at infinity, provided with the supremum norm. If $X$ is the scalar field, we denote $C_0(K, X)$ by simply $C_0(K)$. In this paper we prove that for locally compact Hausdorff spaces $K$ and $L$ and for Banach space $X$ containing no copy of $c_0$, if there is a isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where either $X$ is separable or $X^*$ has the Radon-Nikod\'ym property, then either $K$ is finite or $|K|\leq |L|$. As a consequence of this result, if there is a isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where $X$ contains no copy of $c_0$ and $L$ is scattered, then $K$ must be scattered.