Cyclic von Neumann algebras

Question

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.

Answer 1

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 $|a|<1$.