3
$\begingroup$

Let $M$ be a cyclic von Neumann algebra in $B(H)$. Does there exist any vector $\zeta$ in $H_1$ with $H_1=\overline{M_1\zeta}$? where $H_1$ and $M_1$ are the closed unit ball $H$ and $M$ respectively.

$\endgroup$
0
$\begingroup$

Take $H=\mathbb C^2$ and $M$ diagonal matrices. If $\zeta=(a,b)$ is a cyclic vector with $\|\zeta\|\leq 1$, then $(1,0)$ is not in $\overline{M_1\zeta}=M_1\zeta=\{(x,y)\in\mathbb C^2:|x|\leq|a|,|y|\leq |b|\}$ because $|a|<1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.