The classification of $C(K)$ spaces for countable compacta by positive isomorphisms

TL;DR

This paper classifies $C(K)$ spaces for countable compacta using positive isomorphisms, establishing conditions for isomorphisms and embeddings, and introduces a positive Banach-Mazur distance with explicit formulas.

Contribution

It provides new classification results for $C(K)$ spaces via positive maps, relates positive embeddings to Cantor-Bendixson heights, and introduces a positive Banach-Mazur distance with explicit computations.

Findings

Isomorphisms between $C(K)$ and $C(L)$ can be realized by positive maps or their inverses.

Positive embeddings imply bounds on Cantor-Bendixson heights of compact spaces.

Exact formulas for the positive Banach-Mazur distance between specific $C(K)$ spaces.

Abstract

We study the classification of spaces of continuous functions $C(K)$ under positive linear maps. For infinite countable compacta, we show that whenever $C(K)$ and $C(L)$ are isomorphic, there exists an isomorphism $T:C(K)\to C(L)$ satisfying either $T\geq 0$ or $T^{-1}\geq 0$. We also prove that for any compact spaces $K$ and $L$, the existence of a positive embedding $T: C(K) \to C(L)$ implies that the Cantor-Bendixson height of $K$ does not exceed the height of $L$. Furthermore, we introduce a one-sided positive Banach-Mazur distance and obtain new estimates for both the classical and positive distances. Notably, we prove the exact formula $d_{BM}(C(\omega^{\omega^\alpha}), C(\omega^{\omega^\alpha n})) = n+\sqrt{(n-1)(n+3)}$.