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Complementability of separable spaces $\mathcal{C}(K)$ in Banach spaces

Jakub Rondoš, Damian Sobota

Abstract

For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the product $E \times E^*$, based on new descriptions of the Banach spaces $\mathcal{C}([1, ω^α])$ for countable ordinal numbers $α$ and $\mathcal{C}(2^ω)$. Applying this general result in the case where $E=\mathcal{C}(K)$ for some compact space $K$, we further obtain a characterization of the existence of a positively $1$-complemented positively isometric copy of $\mathcal{C}(L)$ inside $\mathcal{C}(K)$ in terms of the topology of $K$ and the space of probability Radon measures on $K$. In the process, we also prove a variant of the classical Holsztyński theorem for isometric embeddings onto complemented subspaces.

Complementability of separable spaces $\mathcal{C}(K)$ in Banach spaces

Abstract

For a metric compact space and a Banach space , we provide a characterization of the complementability of the Banach space of continuous functions on inside in terms of the existence of a certain tree in the product , based on new descriptions of the Banach spaces for countable ordinal numbers and . Applying this general result in the case where for some compact space , we further obtain a characterization of the existence of a positively -complemented positively isometric copy of inside in terms of the topology of and the space of probability Radon measures on . In the process, we also prove a variant of the classical Holsztyński theorem for isometric embeddings onto complemented subspaces.
Paper Structure (16 sections, 31 theorems, 219 equations)

This paper contains 16 sections, 31 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Real numbers
  4. Sets and ordinal numbers
  5. Trees on $\omega$
  6. Topological spaces
  7. Banach spaces and Banach lattices
  8. Spaces $\mathcal{C}(K)$, $\mathcal{M}(K)$, and $\mathcal{P}(K)$
  9. Isometric descriptions of the spaces $\mathcal{C}([1, \omega^{\alpha}])$ and $\mathcal{C}(2^{\omega})$
  10. Isometry between the spaces $\mathcal{A}_\Lambda$ with $\Lambda\in\mathfrak{T}_{\alpha+1}$ and $\mathcal{C}([1,\omega^\alpha])$
  11. Isometry between the spaces $\mathcal{A}_{\mathbb L}$ and $\mathcal{C}(2^\omega)$
  12. A Markushevich basis and the dual of a space $\mathcal{A}_\Lambda$
  13. Projectional trees in Banach spaces
  14. Applications to $\mathcal{C}(K)$-spaces
  15. A variant of the theorem of Holsztyński
  16. ...and 1 more sections

Key Result

Theorem A

Let $E$ be a Banach space. (1) For a countable ordinal $\alpha$ and $m\in\mathbb N$, the following assertions are equivalent: (2) The following assertions are equivalent:

Theorems & Definitions (75)

  • Theorem A
  • Theorem B
  • Theorem C
  • proof : Sketch of the proof
  • Theorem D
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 65 more