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!
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 $G_0\subset I$. 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\in G_0}\sum_{(i,j)\in F}\pi_ix\pi_{j}^*\eta_\ell\otimes\delta_{j,\ell}\,\e_i\\[0.3cm] &=\sum_{\ell\in G_0}\sum_{(i,\ell)\in G}\pi_ix\pi_{\ell}^*\eta_\ell\otimes\e_i\\[0.3cm] &=\sum_{\ell\in G_0}\sum_{(i,\ell)\in G}\pi_ix(\eta_\ell\otimes\e_\ell)\otimes\e_i\\[0.3cm] &=\sum_{(i,j)\in G}\pi_ix\tilde y\otimes\e_i\\[0.3cm] &=P_G\Big(\sum_{i}\pi_ix\tilde y\otimes\e_i\Big)\\[0.3cm] &=P_Gx\tilde y. \end{align} Then $$\tag2 \Big\| \Big(\sum_{(i,j)\in F}x_{ij}\otimes E_{ij}\Big)\,\tilde y \Big\| =\|P_Gx\tilde y\| \leq\|x\tilde y\|\leq\|x\|\,\|\tilde y\|. $$ 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 $$ \Big\| \sum_{(i,j)\in F}x_{ij}\otimes E_{ij} \Big\|\leq \|x\|. $$