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

TL;DR

The paper addresses the problem of classifying spaces $\mathcal{C}(K)$ up to isomorphism under positive linear maps when $K$ is a compact space, with a focus on countable scattered $K$. It proves that for infinite countable $K,L$, a positive isomorphism exists if and only if $\mathcal{C}(K)\simeq\mathcal{C}(L)$ and $ht(K)\le ht(L)$, and that a positive embedding forces a height inequality. The authors construct explicit positive isomorphisms via a weighted, block-interval decomposition yielding sharp quantitative bounds; they also derive exact values for both classical and one-sided Banach–Mazur distances in a broad ordinal regime, notably $d_{BM}(\mathcal{C}(\omega^{\omega^{\alpha}}), \mathcal{C}(\omega^{\omega^{\alpha}n}))=n+\sqrt{(n-1)(n+3)}$. These results advance Kaplansky-type understanding for $C(K)$ spaces, provide tight distance estimates, and raise open questions about extending one-sided positive classification to uncountable compacta and related nonlinear phenomena.

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 $d_{BM}(C(ω^{ω^α}), C(ω^{ω^αn})) = n+\sqrt{(n-1)(n+3)}$.

Theorems & Definitions (60)

• Theorem A
• Corollary A
• Theorem B
• Theorem 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• proof : Proof of Theorem\ref{['thm:embedMain']}
• Theorem 2.4