On positive embeddings of $C(K)$ spaces
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.