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

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 $L$. Further, we introduce a one-sided positive Banach-Mazur distance and compute it in several families of the corresponding spaces of continuous functions. Our estimates also yield new exact values for the classical Banach-Mazur distance between such spaces.