For a locally convex Hausdorff spaces $E$, consider the canonical map $$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$ that maps the projective tensor product to the space of weakly continuous maps on $E$ (note, that if $E$ is Banach spaces, then the latter is just the space $L(E)$ of bounded linear operators).
The question whether this map is injective is well-known to be connected to the approximation property. For example:
If $E$ is a Banach space, then $E$ has the approximation property if and only if $\overline{\psi}$ is injective.
If $E$ has is a complete locally convex space with a fundamental system of absolutely convex neighborhoods $U$ of zero such that every Banach space $E_U$ has the approximation property, then $\overline{\psi}$ is injective.
These statements can be found in Köthe's book "Topologcial Vector Spaces II".
Question: If $E$ is complete and has the approximation property, is it false/true/not known whether $\overline{\psi}$ is injective in general?
Edit: A locally convex space $X$ is said to satisfy the approximation property, if the space of finite rank operators is dense in the space $L_c(X)$, the set of continuous linear operators with the topology of uniform convergence on precompact sets. For Banach spaces, this is equivalent to saying that the finite rank operators are dense in the compact operators with respect to the operator norm.For Banach spaces, this is equivalent to saying that the finite rank operators are dense in the compact operators with respect to the operator norm.
