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Given an infinite set $\Gamma$, I would like to know if the class of Banach spaces containing no copies of $\ell_2(\Gamma)$ is stable under finite products.

When $\Gamma$ is countable the answer is positive. This is a well-known result that can be proved using the fact that a Banach space contains no copies of $\ell_2$ if and only if it is totally incomparable with $\ell_2$.

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  • $\begingroup$ Let $Q_p$ be the same question, but with $\ell_2(\Gamma)$ replaced by $\ell_p(\Gamma)$. It is not hard to show that $Q_1$ has an affirmative answer. For $1<p<\infty$, the question $Q_p$ has an affirmative answer for complemented embeddings (the question reduces to checking that the sum of two $\ell_p(\Gamma)$ strictly singular operators on $\ell_p(\Gamma)$ is again $\ell_p(\Gamma)$ strictly singular), but my guess is that $Q_p$ itself is open in the reflexive range. $\endgroup$
    – Bill Johnson
    Commented Apr 13, 2014 at 18:00

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