TITLE: Subspaces isomorphic to $C[0, \omega_1]$
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Yes, it can deduced from Lemma 1.2 combined with Proposition 2 of 

D. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals. Israel J. Math. 27 (1977), 64–92.

Another proof can be found here (Cor. 1.12).