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A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated and also every quotient of $C(\alpha)$ contains $c_0$. Since a closed subspace of a quotient is a quotient of a closed subspace one may ask the following.

Question I. Let $X$ be a closed subspace of $C(\alpha)$. Is every quotient map $Q:X\to X/Y$ not strictly singular ? In particular what is the answer when $X/Y\cong c_0$ ?

This is formally a stronger question than the initial problem and has a positive answer when $X=C(\alpha)$. A possible negative answer indicates that the problem probably also has a negative answer. Moreover, the above question has a negative answer if the following has a positive one.

Question II. Does there exist a universal constant $C$ such that for every equivalent norm $|\|\cdot\||$ on $c_0$, there exists a countable ordinal $\alpha$ such that $(c_0,|\|\cdot\||)$ is $C$-isomoprhic to a subspace of $C(\alpha)$ ?

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    $\begingroup$ Do you know the answer to Question 1 when $X$ is the Schreier space (or, more generally, when $X$ has an unconditional basis)? $\endgroup$
    – Bill Johnson
    Commented May 11, 2020 at 20:21
  • $\begingroup$ No I do not know. But I really concern for Question II. Is it possible Zippin's work to be related to that question? $\endgroup$
    – S Argyros
    Commented May 11, 2020 at 21:01
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    $\begingroup$ Nice question. My initial thoughts. The answer to Q2 should be negative. The collection of renormings of $c_0$ is analytic, and so you can cook up a universal space for the property in the question that doesn't contain $C[0,1]$. In some sense (this isn't precise, since the universal space need not be $C(\alpha)$) this implies there is a countable $\alpha$ for all renormings, which shouldn't be true. $\endgroup$
    – Bunyamin Sari
    Commented May 11, 2020 at 21:14
  • $\begingroup$ Is it possible to show that if for every specific equivalent norm there exists an $\alpha$ such that $C_0$ with this norm embeds into $C(\alpha)$ with a universal constant $C$ then there exists a countable ordinal $\beta$ such that $C(\beta)$ is universal for all these embeddings? $\endgroup$
    – S Argyros
    Commented May 11, 2020 at 21:41
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    $\begingroup$ @ Bill Johnson I. Gasparis informed me that Ted Odell ( arXiv:math/9201219) has proved that the quotients of the Schreier space are $c_0$ saturated and also the quotient map is not strictly singular. $\endgroup$
    – S Argyros
    Commented May 12, 2020 at 15:14

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