On positive embeddings of C(K) spaces

TL;DR

This paper explores conditions under which positive embeddings between spaces of continuous functions imply specific topological and set-theoretic properties of the underlying spaces, revealing how such embeddings influence the structure of these spaces.

Contribution

It establishes that positive embeddings of C(K) into C(L) impose topological constraints on K derived from L, and shows how general isomorphic embeddings can be related to positive ones.

Findings

Positive embeddings imply K is an image of L via an upper semicontinuous set-function.

K has a π-base of sets whose closures are continuous images of compact subspaces of L.

Topological properties like countable tightness and Fréchet property pass from L to K.

Abstract

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 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 $C(K)$ can be positively embedded into $C(L)$ then some topological properties of $L$, such as countable tightness of Frechetness, pass to the space $K$. We show that some arbitrary isomorphic embeddings $C(K)\to C(L)$ can be, in a sense, reduced to positive embeddings.