Counting spaces of functions on separable compact lines

TL;DR

This paper explores the classification of Banach spaces of continuous functions on compact lines, revealing that the number of isomorphism types depends on set-theoretic assumptions and varies with the size of the underlying space.

Contribution

It establishes the exact number of isomorphism types of $C(K)$ spaces for uncountable regular cardinals and shows the classification depends on additional set-theoretic axioms.

Findings

For any uncountable regular cardinal $\kappa$, there are $2^\kappa$ isomorphism types of $C(K)$.

Under the continuum hypothesis, there are $2^{\omega_1}$ types for spaces of weight $\omega_1$.

Assuming Baumgartner's axiom, there is only one isomorphism type for these spaces.

Abstract

We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the supremum norm: Let $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ are there, for $K\in \mathcal{K}$? We prove that for any uncountable regular cardinal number $\kappa$, there exist exactly $2^\kappa$ isomorphism types of spaces $C(K)$ for compact spaces of weight $\kappa$. We show that, for the class $\mathcal{L}_{\omega_1}$ of separable compact linearly ordered spaces of weight $\omega_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{\omega_1}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{\omega_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type.