Property that follows from conditions involving slice maps on Hilbert module

Question

Let $A,B$ be $C^*$-algebras and $E$ be a right $A$-Hilbert $C^*$-module. We can form the Hilbert $A\otimes B$ (minimal tensor product) module $E \otimes B$. If $\omega \in B^*$, there is a unique bounded map $$\iota \otimes \omega: E \otimes B \to E$$ extending $\iota \odot \omega$.

Now, let $F$ be a closed submodule of $E$, and assume that $z\in E \otimes B$ satisfies $$(\iota \otimes \omega)(z)\in F$$ for all $\omega \in B^*$. Is it true that $z\in F \otimes B?$

I'm not even sure this is true if $E= A$ is a $C^*$-algebra!

Answer 1

Theorem 8 from the paper "A Pathology in the Ideal Space of $L(H)\otimes L(H)$" by Simon Wassermann states that there exists an element $x \in B(H)\otimes B(H)$ such that all slices belong to the algebra of compact operators, but $x \notin K(H) \otimes B(H)$, so in general it does not hold.

The property you are after is much more common in the setting of von Neumann algebras.