Let $M$ be a W*-algebra and $f$ be a norm one and normal functional on $M$. Polar decomposition says that, there is a unique positive linear functional, denoted by $|f|$, satisfying in: $$|f|(1)=||f||~~~, ~~\exists ||v||\leq1 ~~|f|=vf$$
Question: Let $\phi:M\to B(H)$ be a normal completely bounded linear map . What about the polar decomposition for $\phi$?
I guess, there is a $x\in M\bar{\otimes}BH$ such that $x\phi$ is a completely positive linear map and $||x\phi||_{cb}=||\phi||_{cb}$.