Decomposition of an operator $x\in B(H\otimes K)$ as a sum $x=\sum_{i,j} x_{ij}\otimes E_{ij}$. Let $H,K$ be Hilbert spaces. Let $\{\epsilon_i\}_{i \in I}$ be an orthonormal basis for $K$ and let $E_{ij}\epsilon_k:= \delta_{jk}\epsilon_i$ be the canonical matrix units in $B(K)$ w.r.t. the fixed orthonormal. We have canonical projection maps
$$\pi_i: H\otimes K \to H: \xi\otimes \epsilon_j \mapsto \delta_{i,j}\xi$$
and given $x\in B(H\otimes K)$, we put $x_{ij}= \pi_i x \pi_j^*\in B(H)$. I wish to prove that
$$\sum_{(i,j)\in I\times I} x_{ij}\otimes E_{ij}= x$$
where the sum converges in the strong topology.
I managed to prove the following:
If $\xi\in H$ and $k \in I$, then
$$\sum_{(i,j)\in I\times I} x_{ij}\xi\otimes E_{ij}\epsilon_k = x(\xi\otimes \epsilon_k).$$
Now, I wish to show the convergence on general elements of $H \otimes K$ (opposed to elementary tensors).
I can show this (using an $\epsilon/3$-argument) if the (finite) partial sums
$$\sum_{(i,j)\in F} x_{ij}\otimes E_{ij}$$
are uniformly bounded (in $F$). However, I'm not sure if these partial sums are bounded.
(Note that we can also notice that $x_{ij}\otimes E_{ij}= \pi_i^*\pi_i x \pi_j^*\pi_j$ and since $\sum_i \pi_i^*\pi_i = 1 = \sum_j \pi_j^*\pi_j$, the strong convergence also follows. But I'm looking for an approach that builds further on the idea that the partial sums are bounded).
Thanks in advance for your help!
 A: You have the identity$\def\e{\epsilon}$
$$\tag1
\sum_i\pi_i(\xi\otimes\e_j)\otimes \e_i=\xi\otimes\e_j.
$$
Let $\tilde y=\sum_{\ell\in G_0} \eta_\ell\otimes\e_\ell$, the sum over some finite set $G_0$. Let $G=F\cap(I\times G_0)$, and $P_G$ the projection onto $H\otimes\operatorname{span}\{\e_i:\ i\in G\}$. Then (using $(1)$ at the end)
\begin{align}
\Big(\sum_{(i,j)\in F}x_{ij}\otimes E_{ij}\Big)\,\tilde y
&=\sum_\ell\sum_{(i,j)\in F}\pi_ix\pi_{j}^*\eta_\ell\otimes\delta_{j,\ell}\,\e_i\\[0.3cm]
&=\sum_\ell\sum_{i\in G}\pi_ix\pi_{\ell}^*\eta_\ell\otimes\e_i\\[0.3cm]
&=\sum_\ell\sum_{i\in G}\pi_ix(\eta_\ell\otimes\e_\ell)\otimes\e_i\\[0.3cm]
&=\sum_{i\in G}\pi_ix\tilde y\otimes\e_i\\[0.3cm]
&=P_G\Big(\sum_{i}\pi_ix\tilde y\otimes\e_i\Big)P_G\\[0.3cm]
&=P_Gx\tilde yP_G.
\end{align}
Then
$$\tag2
\Big\|
\Big(\sum_{(i,j)\in F}x_{ij}\otimes E_{ij}\Big)\,\tilde y
\Big\|
=\|P_Gx\tilde yP_G\|
\leq\|x\tilde y\|\leq\|x\|\,\|\tilde y\|.
$$
As the elements of the form $\tilde y$ are dense in $H\otimes K$, the inequality $(2)$ holds in all of $H\otimes K$, and hence
$$
\Big\|
\sum_{(i,j)\in F}x_{ij}\otimes E_{ij}
\Big\|\leq \|x\|.
$$
