I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective tensor product of the same pair of Banach spaces can fail to be injective; for obscure reasons, I'd like to see a specific example of this behaviour.
I seem to remember---forgive me, this was years ago, and my memory is not what it once was---that this failure of injectivity was said to be somehow tied to the failure of the approximation property (in general). I.e., that one or both of the Banach spaces involved had to not have the approximation property in order for the map to not be injective.
The Wikipedia article on the approximation property claims that $c_0$ and $\ell^p$ for $p\neq2$ have closed subspaces that do not have the approximation property, but there is no citation given. (The only cited counter-example is the original one of Enflo.) Can anyone provide citations for these (presumably easier) counter-examples?
Also, if a citation (or refutation) of the link between the approximation property and the (non-)injectivity of the map $x \overline\otimes_\pi y \to x \overline\otimes_\varepsilon y$ can also be provided, that'd be great!