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Problem 1. Characterize compact Hausdorff spaces $K$ for which the Banach space $C(K)$ of continuous real-valued functions embeds into the Banach space $c_0$.

Since $c_0$ has separable dual, such $K$ must me countable. So, we can make Problem 1 more precise:

Problem 2. Is it true that for every compact countable space $K$ the Banach space $C(K)$ is isomorphic to a subspace of $c_0$?

Another possible option:

Problem 3. Let $K$ be a compact Hausdorff space. Is it true that the Banach space $C(K)$ is isomorphic to $c_0$ if $C(K)$ is isomorphic to a subspace of $c_0$?

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  • $\begingroup$ Problem 3: $K$ finite makes trivial counterexamples. Surprisingly (?) the answer is yes for $K$ infinite according to Tomek Kania's answer. $\endgroup$
    – YCor
    Commented Jun 3, 2019 at 6:32

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The Szlenk index is the answer.

A space $C(K)$, where $K$ is infinite compact Hausdorff space, is embeddable into $c_0$ if and only if $K$ is homeomorphic to an ordinal below $\omega^\omega$ and if this is the case (and $K$ is infinite) the space itself is isomorphic to $c_0$.

So the answer to problem 2 is no however the answer to problem 3 is yes.

For details see Rosenthal's chapter in the Handbook of Banach spaces.

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    $\begingroup$ Thank you very much for the answer. This is exactly what I need! $\endgroup$
    – Taras Banakh
    Commented Jun 3, 2019 at 5:27
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    $\begingroup$ Rosenthal, Haskell P. The Banach spaces C(K). Handbook of the geometry of Banach spaces, Vol. 2, 1547-1602, North-Holland, Amsterdam, 2003. $\endgroup$
    – YCor
    Commented Jun 3, 2019 at 6:23

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