Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Counting spaces of functions on separable compact lines

Maciej Korpalski, Piotr Koszmider, Witold Marciszewski

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 standard supremum norm:Let $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ of real-valued continuous functions on $K$ are there, for $K\in \mathcal{K}$? We prove that for any uncountable regular cardinal number $κ$, there exist exactly $2^κ$ isomorphism types of spaces $C(K)$ for compact spaces of weight $κ$. We show that, for the class $\mathcal{L}_{ω_1}$ of separable compact linearly ordered spaces of weight $ω_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{2^ω}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{ω_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type.

Counting spaces of functions on separable compact lines

Abstract

We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces of continuous real-valued functions on a compact space , equipped with the standard supremum norm:Let be a class of compact spaces. How many isomorphism types of Banach spaces of real-valued continuous functions on are there, for ? We prove that for any uncountable regular cardinal number , there exist exactly isomorphism types of spaces for compact spaces of weight . We show that, for the class of separable compact linearly ordered spaces of weight , the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are isomorphism types of , for , and assuming a certain axiom proposed by Baumgartner, there is only one type.
Paper Structure (15 sections, 33 theorems, 33 equations)

This paper contains 15 sections, 33 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set-theoretical terminology
  4. Banach spaces of continuous functions and their duals
  5. Scattered spaces
  6. Separable compact linearly ordered spaces
  7. Counting isomorphism types of C(K) of given weight
  8. Topological properties of separable compact lines
  9. General properties of spaces C(K A)
  10. Counting spaces C(K A) of weight kappa when pow(2, omega1) > pow(2, omega)
  11. Counting spaces C(K A) under Baumgartner's Axiom
  12. Function spaces on finite products of separable compact lines
  13. Final remarks
  14. Compact spaces of weight κ
  15. Separable compact lines

Key Result

Theorem 1.1

Suppose that $\kappa$ is an uncountable regular cardinal. There is a family of cardinality $2^\kappa$ of compact spaces $K$ of weight $\kappa$, such that the corresponding Banach spaces $C(K)$ are pairwise nonisomorphic.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: BA
  • Corollary 1.4: BA
  • Theorem 2.2: Ostaszewski, Os74
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 46 more