On some geometric properties of Banach spaces of continuous functions on separable compact lines

We study properties of Banach spaces $C(L)$ of all continuous scalar (real or complex) functions on compact lines $L$. First we show that if $L$ is a separable compact line, then for every closed linear subspace $X$ of $C(L)$ with separable dual the quotient space $C(L)/X$ possesses a sequence of continuous linear functionals separating its points. Next we show that for any compact line $L$ the space $C(L)$ contains no subspace isomorphic to a $C(K)$ space where $K$ is a separable nonmetrizable scattered compact Hausdorff space with countable height.