Polar decompostion for the operator algebras

I find that most of books discussing the polar decompostion at the W*-algebras, but not C*-algebras. I guess the rough reason is that the element of W*algebras has the well supported set, but I want to know more details.

1) What is the basic difference between the C*-algebras and W*-algebras such that the W*-algebras has the well supported set.

2) If we insist to do the polar decompostion for a noninvertible element in the C*-algebras, what will happen?

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For example, consider the identity map $f : [-1,1]\to [-1,1]$ as a member of the C*-algebra $C[-1,1]$. It's easy to see that if $f = |f| v$ is the polar decomposition of $f$ (i.e. $v$ is a partial isometry), then $v$ must take the values $-1$ on $[-1,0)$ and $+1$ on $(0,1]$. You can find such a $v$ in $L^\infty[-1,1]$ but not $C[-1,1]$.