Probably an easy question, but here goes:

In a concrete von Neumann algebra $M \subseteq B(H)$, every element $m \in M$ has a polar decomposition $m= p|m|$ where $p$ is a partial isometry and $|m|= \sqrt{m^*m}$. Imposing extra conditions on $p$ ensures that $p$ is unique. For example, one can ask that $\ker p = \ker m$.

Is there a way to describe the unique partial isometry $p$ without referring to the underlying Hilbert space $H$? In other words, in an abstract von Neumann algebra (= $W^*$-algebra) $M$, how would one describe the partial isometry $p$ in the decomposition $m = p|m|?$ Ideally, I hope there is some algebraic characterisation of $p$.