On embeddings of C₀(K) spaces into C₀(L,X) spaces

2013/08/29 by Candido, Leandro
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 46E40 #Secondary 46B25

paper · doi:10.48550/arxiv.1308.6555

Abstract

Let C₀(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 C₀(K, X) by simply C₀(K). In this paper we prove that for locally compact Hausdorff spaces K and L and for Banach space X containing no copy of c₀, if there is a isomorphic embedding of C₀(K) into C₀(L,X) where either X is separable or X^* has the Radon-Nikodým property, then either K is finite or |K|≤ |L|. As a consequence of this result, if there is a isomorphic embedding of C₀(K) into C₀(L,X) where X contains no copy of c₀ and L is scattered, then K must be scattered.

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