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 $\\mathcal{M}$ . $\\square$

单词	VonNeumannAlgebrasContainTheRangeProjectionsOfItsElements
释义	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 $\\mathcal{M}$ . $\\square$
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