# Injective tensor products of WCG spaces

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, $1<p<\infty$, projective tensor product) is not WCG.

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.

• 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. – t.b. Aug 29 '11 at 2:44
• Of course, I mean $p\in (1,\infty)$ when writing $p>1$. Sorry if this makes anyone confused. – F.R. Rogers Aug 29 '11 at 10:32