About a projection operator in cyclic von Neumann Algebra

Question

Let $M$ be a von Neumann algebra in $B(H)$ and assume $\zeta$ is a cyclic vector for $M$. Will $\mathbb{C}\zeta$, subspace generated by $\zeta$ known by $M$ i.e. will orthogonal projection of $H$ onto $\mathbb{C}\zeta$ belong to $M$?

Answer 1

There is no general answer. Take $M=B(H)$, and every projection is in $M$. Take $M$ to be any von Neumann algebra without minimal projections (for instance, a II$_1$-factor) and it cannot contain a rank-one projection.