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Let $E$ be the projective tensor product of $c_{0}$ by $c_{0}$. Does it follow that $E$ is isomorphic to no subspace of $C(K)$, where $K$ is countable compact metric space?

When $C(K)$ is isomorphic to $c_{0}$ the answer to this question is positive.

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    $\begingroup$ May be this can help you $\endgroup$
    – Norbert
    Commented Jun 5, 2014 at 21:40
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    $\begingroup$ Norbert, I know this article (and most likely so does the OP) and the results they obtain in that paper do not imply in an obvious way (at least to me) anything definite. If one could prove that $c_0\hat{\otimes}c_0$ has the Szlenk index ω, it would have solved the problem (that spaces wouldn't embed). It looks to me that it is really so; I'll try to make the calculations tomorrow. $\endgroup$
    – Tomasz Kania
    Commented Jun 5, 2014 at 23:14

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