vector states from elementary tensors $h\otimes k\in H\otimes K$ separate all of $\mathbb{B}(H\otimes K)$

Question

In the book "C* algebras and finite dimensional approximations",there is a statement in the proof of lemma4.1.8,$A\otimes B\subset\mathbb{B}(H\otimes K)$ and vector states from elementary tensors $h\otimes k\in H\otimes K$ separate all of $\mathbb{B}(H\otimes K)$,Who can give me a little hints on how to prove the statement,thanks!

Answer 1

All you need to show is that if $T\in B(H\otimes K)$ satisfies $T(h\otimes k)=0$ for all $h\in H$, $k\in K$, then $T=0$.

This is straighforward: given $\xi\in H\otimes K$, we may write $$ \xi=\sum_{k,j} \alpha_{kj}\,e_k\otimes f_j $$ for orthonormal bases $\{e_k\}$ of $H$ and $\{f_j\}$ of $K$. Then, since $T$ is bounded, $$ T\xi=\sum_{k,j} \alpha_{kj}\,T(e_k\otimes f_j)=0. $$ So $T=0$.