Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $M\subseteq \mathcal{L}(H)$ be a von Neumann algebra, and let $x\in M$. Define
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\begin{align*}
|x|=(x^*x)^{1/2}.
\end{align*}
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Then there exists a partial isometry $v\in M$ such that
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\begin{align*}
x=v|x|.
\end{align*}
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Moreover,
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\begin{align*}
v^*v=s(|x|)
\end{align*}
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and
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\begin{align*}
vv^*=s(xx^*),
\end{align*}
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where, for a positive operator $a\in M$, $s(a)$ denotes the [orthogonal projection](/theorems/437) onto $\overline{aH}$. The partial isometry $v$ is the unique partial isometry satisfying $x=v|x|$ and $\ker v=\ker x$.