Proposition - Let $T$ be an operator in a von Neumann algebra $\mathcal{M}$ acting on an Hilbert space $H$ . Then the orthogonal projection onto the range of $T$ and the orthogonal projection onto the kernel of $T$ both belong to $\mathcal{M}$ .

Proof : Let $T=VR$ be the polar decomposition of $T$ with $KerV=KerR$ .

By the result on the parent entry we see that $V \in \mathcal{M}$ .

As $V$ is a partial isometry, $VV^*$ is the (orthogonal) projection onto the range of $T$ , and $I-V^*V$ is the (orthogonal) projection onto the kernel of $T$ , where $I$ is the identity operator in $\mathcal{M}$ .

Therefore the (orthogonal) projections onto the range and kernel of $T$ both belong to $\mathcal{M}$ . $\square$