# Polar decomposition in abstract von Neumann algebra

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 $$p$$.

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 $$vp=0$$.