Let $E$ and $F$ be Banach spaces, and let $\mathfrak L\_{co}(E,F)$ denote the space of bounded linear operators $E \to F$ equipped with the topology of uniform convergence on the absolutely convex compact subsets of $E$.


Defant and Floret [DF93, §5.5] point out that there is a natural linear map 
$$ D\_F : F' \mathop{\tilde\otimes\_\pi} E \to (\mathfrak L\_{co}(E,F))' $$
which is surjective. (No continuity claims are made.) Furthermore, they prove [DF93, §5.7] that $D\_F$ is injective if $F$ is reflexive or if $F'$ or $E$ has the approximation property, so in that case one has an isomorphism $F' \mathop{\tilde\otimes\_\pi} E \cong (\mathfrak L\_{co}(E,F))'$ of vector spaces. They conclude with:



> 
> *We do not know whether or not $D\_F$ is always injective.*
> 
> 
> 


**Question.** Has this since been settled? Can someone provide (a reference to) a proof or a counterexample?




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Some background: $D\_F$ is defined by taking the natural map
$$ \Phi\_F : \mathfrak L(E,F) \stackrel{1}{\hookrightarrow} \mathfrak L(E,F'') \stackrel{1}{\cong} (E \mathop{\tilde\otimes\_\pi} F')' \stackrel{1}{\cong} (F' \mathop{\tilde\otimes\_\pi} E)'. $$
It is shown that $\Phi\_F$ is continuous as a map $\mathfrak L\_{co}(E,F) \to [(F' \mathop{\tilde\otimes\_\pi} E)',\text{weak-$\*$}]$, and then $D\_F$ is defined as the adjoint of this map. (I guess one could deduce some continuity property of $D\_F$ from this, but the authors steer clear of that, presumably to focus on questions related to Banach spaces.)


Another way to interpret the question is this: the map $D\_F$ (or $\Phi\_F$) gives rise to a bilinear map $(F' \mathop{\tilde\otimes\_\pi} E) \times \mathfrak L(E,F) \to \mathbb{F}$. One readily verifies that this is simply the natural map
$$ \left(\sum\_{n=1}^\infty y\_n' \mathop{\otimes} x\_n \, , \, T\right) \mapsto \sum\_{n=1}^\infty y\_n'(Tx\_n). $$
Since we have $\mathfrak L(E,F) \stackrel{1}{\hookrightarrow} (F' \mathop{\tilde\otimes\_\pi} E)'$, it is clear that $F' \mathop{\tilde\otimes\_\pi} E$ separates points on $\mathfrak L(E,F)$. The question whether $D\_F$ is injective is equivalent to the question whether $\mathfrak L(E,F)$ separates points on $F' \mathop{\tilde\otimes\_\pi} E$ (so that the bilinear map becomes a dual pairing).


Yet another equivalent formulation: is the image of $\Phi\_F$ weak-$\*$ dense in $(F' \mathop{\tilde\otimes\_\pi} E)'$?




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**References.**


[DF93] A. Defant, K. Floret, *Tensor Norms and Operator Ideals* (1993), Mathematics Studies 176, North-Holland.



Do the operators in $B(E,F)$ separate points on the projective tensor product $F' \mathop{\tilde\otimes_\pi} E$?
This question was settled in 2012 by Petr Hájek and Richard J. Smith [HS12]. They prove the following beautiful result (reformulated here to match the notation from the question).



> 
> **Theorem** (cf. [HS12, Theorem 2.5])**.** Let $F$ be a Banach space with the AP. Then the following conditions are equivalent:
> 
> 
> * $F'$ has the AP.
> * For every Banach space $E$, the map $F' \mathbin{\tilde\otimes\_\pi} E \to (\mathfrak L\_{co}(E,F))'$ is injective.
> * The map $F' \mathbin{\tilde\otimes\_\pi} F'' \to (\mathfrak L\_{co}(F'',F))'$ is injective.
> 
> 
> 


As there are known examples of Banach spaces with the AP whose dual fails the AP, the question is settled in the negative.




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**References.**


[HS12]: Petr Hájek, Richard J. Smith, *Some duality relations in the theory of tensor products*, Expositiones Mathematicae, volume 30 (2012), issue 3, pages 239–249. <https://doi.org/10.1016/j.exmath.2012.08.004>


