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

Leandro Candido¹

¹ Instituto de Ciência e Tecnologia Universidade Federal de São Paulo Campus São José dos Campos – Parque Tecnológico Avenida Cesare Monsueto Giulio Lattes, 1211 12231-280 São José dos Campos – SP, Brazil

Studia Mathematica, Tome 232 (2016) no. 1, pp. 1-6

Résumé

For a locally compact Hausdorff space $K$ and a Banach space $X$ let $C_0(K, X)$ denote the space of all continuous functions $f:K\to X$ which vanish at infinity, equipped with the supremum norm. If $X$ is the scalar field, we denote $C_0(K, X)$ simply by $C_0(K)$. We prove that for locally compact Hausdorff spaces $K$ and $L$ and for a Banach space $X$ containing no copy of $c_0$, if there is an isomorphic embedding of $C_0(K)$ into $C_0(L,X)$, then either $K$ is finite or $|K|\leq |L|$. As a consequence, if there is an 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.

Détail

DOI : 10.4064/sm7857-3-2016

Keywords: locally compact hausdorff space banach space denote space continuous functions which vanish infinity equipped supremum norm scalar field denote simply prove locally compact hausdorff spaces and banach space containing copy there isomorphic embedding either finite leq consequence there isomorphic embedding where contains copy scattered scattered