Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Therefore I have the following question:

Does every subspace of $C[0,\omega_1]$ isomorphic to $C[0,\omega_1]$ contain a complemented copy isomorphic to itself? The only (complemented) examples that I can "construct by hand", excluding the finite-codimensional ones, are of the form

$\mbox{cl lin}(\mathbf{1}_{[0,\gamma_{\sigma}]}\colon \sigma\leq \omega_1)$

where $(\gamma_\sigma)_{\sigma<\omega_1}$ is increasing long sequence of limit ordinals and $\sigma_{\omega_1} = \omega_1$ (note that the family $(\{\mathbf{1}_{[0,\alpha]}\colon \alpha \leq \omega_1\})$ forms the long Schauder basis for $C[0,\omega_1]$).

Thank you, T.