von Neumann algebras contain the range projections of its elements

- Let $T$ be an operator in a von Neumann algebra $ℳ$ 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 $ℳ$.

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

By the result on the parent entry (http://planetmath.org/PolarDecompositionInVonNeumannAlgebras) we see that $V\in ℳ$.

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

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