On embedding separable spaces $\mathcal{C}(L)$ in arbitrary spaces $\mathcal{C}(K)$

Abstract

Supplementing and expanding classical results, for compact spaces $K$ and $L$, $L$ metric, and their Banach spaces $\mathcal{C}(L)$ and $\mathcal{C}(K)$ of continuous real-valued functions, we provide several characterizations of the existence of isometric, resp. isomorphic, embeddings of $\mathcal{C}(L)$ into $\mathcal{C}(K)$. In particular, we show that if the embedded space $\mathcal{C}(L)$ is separable, then the classical theorems of Holszty\'{n}ski and Gordon become equivalences. We also obtain new results describing the relative cellularities of the perfect kernel of a given compact space $K$ and of the Cantor--Bendixson derived sets of $K$ of countable order in terms of the presence of isometric copies of specific spaces $\mathcal{C}(L)$ inside $\mathcal{C}(K)$.