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

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 $E\times E^{\ast }$, based on new descriptions of the Banach spaces $\mathcal{C}([1,\ddot{I}{‰}^{\hat{I}}\pm ])$ for countable ordinal numbers $\hat{I}\pm$ and $\mathcal{C}(2^{\ddot{I}}‰)$. 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.