A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact:
Suppose that $X$ is a compact, non-metrizable space satisfying the countable chain condition. Then the dual space of $C(X)$ contains a non-separable Hilbert space.
Any hints where to find this theorem appreciated.
EDIT: This is not true. There exist separable scattered compact spaces $X$, in which case $C(X)^* = \ell_1(|X|)$.
Let $\mathcal{A}$ be an uncountable family of infinite subsets of $\mathbb{N}$ which have finite intersection. Consider the Boolean subalgebra of the power-set algebra $\wp(\mathbb{N})$ generated by $\mathcal{A}$ and all finite subsets. The Stone space $K$ of this algebra is scattered (actually of Cantor-Bandixon height 3), which is separable as $\mathbb{N}$ is dense in $K$. Since $K$ is scattered, $C(K)^* \cong \ell_1(|K|)$. Consequently, $C(K)^*$ contains no reflexive infinite-dimensional subspace.
The spaces $K$ of the above form have googlable names: Mrówka space, Mrówka-Isbell space.