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

TL;DR

This paper characterizes when the space of continuous functions on a compact space can be complemented inside a Banach space, using tree structures and topological properties, with applications to isometric embeddings and measure spaces.

Contribution

It provides a new characterization of the complementability of $ ext{C}(L)$ in Banach spaces based on tree structures and topology, extending classical theorems.

Findings

Characterization of complementability via trees in $E imes E^*$

Description of $ ext{C}([1, ext{omega}^ ext{alpha}])$ spaces

A variant of Holsztyński's theorem for isometric embeddings

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 $\mathcal{C}([1, \omega^{\alpha}])$ for countable ordinal numbers $\alpha$ and $\mathcal{C}(2^{\omega})$. 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 Holszty\'{n}ski theorem for isometric embeddings onto complemented subspaces.