Subspaces isomorphic to C[0,omega_1]

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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 followin 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.

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1 Answer

Have you searched the literature? There are a large number of papers about uncomplemented and complemented embeddings of $C(K)$ into $C(K)$. In particular, you should look at papers of Bessaga and Pelczynski from the 1960s (especially their fourth one on spaces of continuous functions), papers of Dan Amir from the late 1960s and 1970s, and Alspach and Benyamini from the 1970s. For an overview of part of the material, read Rosenthal's article in volume 2 of the Handbook of the Geometry of Banach Spaces. In particular, in Section 3C of that article you will find that there is an isomorphic copy of $C[0,\omega^\omega]$ in itself that is not complemented; from that it is very easy to prove that there is an isomorphic copy of $C[0,\omega_1]$ in itself that is not complemented.

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Please read what I wrote. From the uncomplementation result for $C[0,\omega^\omega]$ you immediately get the answer to your first question, so I do not understand why you asked the first question. (Just observe that $C[0,\omega_1]$ is isomorphic to $C[0,\omega_1]\oplus C[0,\omega^\omega]$.) – Bill Johnson May 17 2011 at 18:08
Indeed, sometimes I write faster than I think but I'm trying to fight with that, sorry. By the way, it is not hard to prove that every closed non-separable subspace of $C[0,\omega_1]$ contains a complemented copy of $c_0(\omega_1)$... – Tomek Kania May 17 2011 at 19:54
BTW: For deep results on non separable $C(K)$ spaces, some of which rely on special set theoretic axioms, look at recent papers by Koszmider. (Probably you know this, but I mention it just in case.) – Bill Johnson May 17 2011 at 20:04
Now, we know that each subspace of $C[0,\omega_1]$ isomorphic to $C[0,\omega_1]$ contains a further subspace isomorphic to itself and complemented in $C[0,\omega_1]$. The preprint is about to be uploaded. – Tomek Kania Jun 18 at 12:11
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