*(I asked this question at math stackexchange 4 months ago, but received no answers)*


Let $\{e\_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q\_k$ by 
$$
q\_k=\sum\_{n=1}^ke\_{nn}.
$$
Let $\{p\_1,p\_2,\ldots\}\subset B(H)$ be a sequence of orthogonal projections in $B(H)$ with the property that $q\_kp\_hq\_k=q\_kp\_kq\_k$ whenever $h\geq k$ (i.e. the sequence "fixes" the upper left corner as the index grows). 



> 
> **Question:** Does the sequence $\{p\_k\}$ converge strongly?
> 
> 
> 


*(my gut feeling is that it should, but after a while thinking about it I couldn't get neither a proof nor a counterexample; it is easy to show that the sequence converges weakly so it would be enough to prove that the limit is a projection, but I got nowhere through this route either)*



Strong convergence of projections in $B(H)$
$Q\_k$ is just the orthogonal projection to the linear span $L\_k$ of $e\_1,\dots,e\_k$, right? 
Let $u\_m$ be some sequence of unit vectors. Let $P\_mx=(x,u\_m)u\_m$. Now, $Q\_kP\_mQ\_k x=(Q\_kx,u\_m)Q\_ku\_m=(x,Q\_ku\_m)Q\_ku\_m$, so the condition is that $Q\_ku\_m$ stabilize for each $k$. Let's stabilize them to $\frac 12e\_1$ (just add some tails far away to get norm $1$). Take $v=e\_1$ as the test vector for strong convergence. Then $P\_m v=\frac 12 u\_m$. Now, the tails are wild and carry a lot of norm, so, alas, $u\_m$ do not converge to anything in norm... 


