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

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.

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.

• (Disclaimer: I did search for related questions before asking, but only noticed the linked answer after meticulously crafting a list of all problems concerning the approximation property for a question on meta: Suggested tag: approximation-property. I'm pretty sure I wouldn't have found it otherwise. Oh well.) Aug 29, 2019 at 21:03
• Don't feel bad. I spent a couple of hours, with no success, trying to build a counterexample for your question. I had completely forgotten the linked question and my answer to it. Aug 29, 2019 at 22:01

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