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?