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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!

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Taking your last question first: there is an accessible discussion of the link between the AP and the injectivity of the comparison map "projective tensor product to injective tensor product" in Section 4.1 of

R. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, 2002.

In particular, it seems that one can extract the following from Proposition 4.6 in that book:

Theorem. Let $X$ be a Banach space. TFAE:

1) $X$ has the AP;

2) for every Banach space $Y$, the canonical map $X\overline{\otimes}_\pi Y \to X\overline{\otimes}_\varepsilon Y $ is injective.

As for the first question: I think that the first construction of subspaces of $\ell_p$ for $2 < p <\infty$ which do not have AP is due to A. M. Davie:

A. M. Davie, The Approximation Problem for Banach Spaces, Bull. London Math. Soc. 51 (1973), no. 3, 261–266. DOI link

These subspaces are not completely explicit, in the sense that Davie relies on some probabilistic inequalities, and hence, his result has the form

for any sequence $(\epsilon_n)_{n\geq 1}$ with values $\pm 1$, we define certain spaces; then there exists some choice of $(\epsilon_n)_{n\geq 1}$ such that these spaces fail AP.

Looking at his paper, I don't see how one gets the cases $1\leq p < 2$, but perhaps this was done in follow-up work.

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  • $\begingroup$ I guess this may still be somewhat unsatisfactory in the sense that one doesn't obtain explicit $X$ and $Y$ for which the comparison map between tensor products is non-injective. IIRC, the (in)famous Banach space of Pisier, which I'll call $P$, has the property that $P \overline{\otimes}_\pi P^* \to P \overline{\otimes}_{\varepsilon} P^*$ is non-injective. But this should probably wait for a proper expert to come along $\endgroup$
    – Yemon Choi
    Jul 4, 2018 at 3:06
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    $\begingroup$ Figiel independently proved the same thing as Davie using a combinatorial rather than probabilistic argument. He did not publish it because Davie wrote up his proof quickly, but I wrote an outline of his proof as a problem set in an article in an MAA volume edited by Bartle. Szankowski is the person who proved that $\ell_p$, $1\le p < 2$, has a subspace that fails the approximation property. $\endgroup$ Jul 4, 2018 at 3:55
  • $\begingroup$ Thanks! This is quite helpful. I will see if I can lay my hands on a copy of Ryan's book. $\endgroup$
    – Jeff Egger
    Jul 4, 2018 at 11:19
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    $\begingroup$ Thank you @BillJohnson - to be honest, I hesitated about leaving an answer when I feel I should just have resurrected the "ask-johnson" tag... I wasn't aware of Figiel's argument, but I will see if I can hunt down the MAA volume you refer to $\endgroup$
    – Yemon Choi
    Jul 4, 2018 at 17:19
  • $\begingroup$ @YemonChoi I wasn't aware of the MAA volume either, but it would seem to be this one: books.google.com.au/books/about/… $\endgroup$ Jul 5, 2018 at 11:04

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