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The notion of a WCG space is a common roof for separable Banach spaces and reflexive ones. Nevertheless, the class is stable under $\ell^p(\Gamma)$-sums for any set $\Gamma$ when $p>1$ and countable $\Gamma$ when $p=1$, it is not closed under projective tensor products. More-less folklore fact states that $\ell^p(\Gamma) \hat{\otimes} \ell^p(\Gamma)$ ($\Gamma$ uncountable, $1

Is the same true for the injective tensor product? Or maybe injective tensor product of two WCG spaces is again WCG? (I think we should look for reflexive conunterexamples but I do not know any).

EDIT: Note that the natural guess $\ell^1 \otimes_\varepsilon \ell^1$ is isometrically isomorphic to $\ell^1(\ell^1)$ which is WCG.

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    When speaking of stability under $\ell^p(\Gamma)$-sums, don't you need to restrict to $1 \lt p \lt \infty$ since $\ell^{\infty}$ is not weakly compactly generated? By the way: for those unfamiliar with weakly compactly generated spaces, here's Dirk Werner's [Springer encyclopaedia entry on them](http://eom.springer.de/w/w120030.htm).2011-08-29
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    Of course, I mean $p\in (1,\infty)$ when writing $p>1$. Sorry if this makes anyone confused.2011-08-29

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