von Neumann algebras contain the range projections of its elements

- 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 (http://planetmath.org/PolarDecompositionInVonNeumannAlgebras) we see that $V\in\mathcal{M}$.

As $V$ is a partial isometry, $VV^{*}$ 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 $\mathcal{M}$.

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

Title von Neumann algebras contain the range projections of its elements VonNeumannAlgebrasContainTheRangeProjectionsOfItsElements 2013-03-22 17:28:57 2013-03-22 17:28:57 asteroid (17536) asteroid (17536) 5 asteroid (17536) Result msc 46L10 msc 47A05