Do the operators in $B(E,F)$ separate points on the projective tensor product $F' \mathop{\tilde\otimes_\pi} E$?

Question

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?

---

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 $(F' \mathop{\tilde\otimes_\pi} E)'$?

---

References.

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

Answer 1

It follows from this answer of Bill Johnson that it can already fail if $E = F$, so the question has been settled in the negative.

Answer 2

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.

---

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