On positive embeddings of $C(K)$ spaces

Volume 216 / 2013

Grzegorz Plebanek Studia Mathematica 216 (2013), 179-192 MSC: Primary 46B26, 46B03, 46E15. DOI: 10.4064/sm216-2-5


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.


  • Grzegorz PlebanekInstytut Matematyczny
    Uniwersytet Wrocławski
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland

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