On positive embeddings of $C(K)$ spaces

Grzegorz Plebanek¹

¹ Instytut Matematyczny Uniwersytet Wrocławski Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Studia Mathematica, Tome 216 (2013) no. 2, pp. 179-192

Résumé

We investigate isomorphic embeddings $T: C(K)\to C(L)$ between Banach spaces of continuous functions. We show that if such an embedding $T$ is a positive operator then $K$ is the image of $L$ under an upper semicontinuous set-function having finite values. Moreover we show that $K$ has a $\pi $-base of sets whose closures are continuous images of compact subspaces of $L$. Our results imply in particular that if $C(K)$ can be positively embedded into $C(L)$ then some topological properties of $L$, such as countable tightness or Fréchetness, are inherited by $K$. We show that some isomorphic embeddings $C(K)\to C(L)$ can be, in a sense, reduced to positive embeddings.

Détail

DOI : 10.4064/sm216-2-5

Keywords: investigate isomorphic embeddings between banach spaces continuous functions embedding positive operator image under upper semicontinuous set function having finite values moreover has base sets whose closures continuous images compact subspaces results imply particular positively embedded topological properties countable tightness chetness inherited nbsp isomorphic embeddings sense reduced positive embeddings