Proof of the polar decomposition

Question

Suppose $M$ is a von Neumann algebra and $\phi\in M_{*}$, then there exists a unique partial isometry $v\in M$ and a positive linear functional $\psi\in M_{*}$ such that $\phi=v \psi$ and $v^*v=s(\phi)$, where $s(\phi)$ is a support projection of $\phi$.

For the proof, I met with troubles. I found a reference book, it says that "if $\phi\in M_{*}$, there exists $a\in M$ with $\|a\|\leq 1$ such that $\phi(a)=\|\phi\|$". How to prove the above statement.

I have another question about support projections of a state. If $\phi_1$ and $\phi_2$ are two states of a von Neumann algebra such that $s(\phi_1)s(\phi_2)=0$, does there exist relationships between two states

Answer 1

It's a direct consequence of the following standard result:

Result: If $X$ is normed space and $x\in X$, then $$\|x\|=\max\{|\varphi(x)|:\ \varphi\in X^*,\ \|\varphi\|=1\}.$$

Apply this to $X=M_*$ and $X^*=M$.

For your second question, I don't really know what kind of relation you expect.