$\sigma$-weak closure of $B(H) \otimes B(K)$. Let $H$ and $K$ be Hilbert spaces. I'm  trying to prove that  $B(H) \otimes B(K)$ (= minimal tensor product of $C^*$-algebras) is $\sigma$-weakly dense (= weak$^*$-dense) in $B(H \otimes K)$.
My progress so far:
I managed to show:

*

*$B(H) \otimes B(K)$ is strongly dense in $B(H \otimes K)$

*$B(H) \otimes B(K)$ contains the compact operators on $H \otimes K$, hence it suffices to show that the compact operators are $\sigma$-weakly dense in $B(H \otimes K)$.

Any ideas how I can proceed?

Here is a more or less alternative argument.
By looking at what the rank-one operators look like, it is easy to see that $\mathcal K(H\otimes K)=\mathcal K(H)\otimes \mathcal K(K)$, this last product as C$^*$-algebras (no ambiguity due to nuclearity).
Then $\mathcal K(H)\otimes \mathcal K(K)$ is $\sigma$-weakly dense in $B(H\otimes K)$ because the locally compact space $\mathcal K(H\otimes K)$ is weak$^*$-dense in its double dual $B(H\otimes K)$.
