Let $M$ be a cyclic von Neumann algebra in $B(H)$ with cyclic vector $\zeta$, that is $H=\overline{M\zeta}$. We may assume the norm of $\zeta$ is $1$. Let $\eta$ be an arbitrary norm one vector in $H$ and consider the the normal functional $\phi_{\zeta,\eta}$ on $M$ given by $\phi(x)=\langle x\zeta,\eta\rangle$. Clearly $\phi$ is non-zero on $M$, because $\zeta$ is cyclic.
My questions are concerned normal extensions of $\phi$ on $B(H)$, denoted by ext($\Phi$) ! Certainty ext($\Phi$) is non-empty, since $\Phi(y)=\langle y\zeta,\eta\rangle$ clearly extends $\phi$ on $B(H)$ (as a normal functional).
1- Is ext($\Phi$) singleton?
2- One may easily check that $|\Phi|$, the absolute value of $\Phi$, is the normal positive linear functional on $B(H)$ given by $y\to \langle y\eta,\eta\rangle$. Therefore the support of $\Phi$ is the rank one projection $\eta\otimes\eta$. True or false: the support of $|\phi|$ is the projection $e_{\eta}\in M$ where $e_{\eta}$ projects on $\overline{M'\eta}$?