Relation between support projections of two positive normal linear functionals in a von Neumann Algebra

Question

Let $M$ be a von Neumann algebra and $\phi$ be a normal linear functional on $M$. Then there is a unique positive linear functional $\psi$ on $M$ and a unique partial isometry $v$ on $M$ such that $\phi(x)=\psi(xv)$ for all $x \in M$, with $\|\phi\|=\|\psi\|$ and $S(\psi)=v^*v$, where $S(\psi)$ is the support of $\psi$. Let us now consider $\phi \ge 0,$ that is $\phi$ is positive normal linear functional on $M$. Then by aforementioned property, there is a unique positive linear functional $\psi$ on $M$ and a unique partial isometry $v$ on $M$ such that $\phi(x)=\psi(xv)$ for all $x \in M$, with $\|\phi\|=\|\psi\|$ and $S(\psi)=v^*v$, where $S(\psi)$ is the support of $\psi$. Also, since $\phi \ge 0$, the support $S(\phi)$ makes sense. I want to know the relation between $S(\phi)$ and $S(\psi)$.
Basically, $S(\phi)=S(\psi) \ne 1$. Because if $S(\phi)=1$, then $\phi$ will be faithful, but $\phi$ may not be faithful. Please help me to build a relationship between the projections $S(\phi)$ and $S(\psi)$. I think that $S(\phi)=S(\psi)$, but I am unable to prove it and also having no idea about counter-example. Thank you for your time and help.

Answer 1

If $\phi\geq0$, then $\psi=\phi$ and $v=S(\phi)$ work, so by the uniqueness they are the polar decomposition. Thus $\psi=\phi$.