Let $x\_{n}$ be a sequence of operators in vN algebra $M$, $\Omega$ is a cyclic vector for $M$, if $x\_{n}\Omega$ converges in $\mathcal{H}$, can we say there exist a subsequence $\{y\_{n}\}$ of $\{x\_{n}\}$ are uniformly bounded in operator norm?



On boundedness of sequence of operators in vN algebra
An easy counterexample:


$M=M\_{2}(\mathbb C)$,  

$H=\mathbb C^2$,  

$\Omega=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$,  

$x\_n=\left(\begin{smallmatrix}0 & 0\\0 & n\end{smallmatrix}\right)$.


