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 On positive embeddings of CK spaces

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

Author: Grzegorz Plebanek

Source: https://archive.org/


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