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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 $p$.

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I would say that the polar decomposition of $m \in M$ is the unique pair $(v,a)$ of elements in $M$ satisfying the following (algebraic) properties.

  1. $m = va$.
  2. $v$ is a partial isometry and $a$ is positive.
  3. $a^2 = m^* m$.
  4. Whenever $p \in M$ is a projection satisfying $mp = 0$, we have $vp=0$.
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