We investigate isomorphic embeddings T: C(K) → 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 π-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) → C(L) can be, in a sense, reduced to positive embeddings.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm216-2-5,
author = {Grzegorz Plebanek},
title = {On positive embeddings of C(K) spaces},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {179-192},
zbl = {1286.46026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm216-2-5}
}
Grzegorz Plebanek. On positive embeddings of C(K) spaces. Studia Mathematica, Tome 215 (2013) pp. 179-192. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm216-2-5/