# 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, $V{V}^{*}$ 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 $\mathcal{M}$.

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

Title | von Neumann algebras contain the range projections of its elements |
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Canonical name | VonNeumannAlgebrasContainTheRangeProjectionsOfItsElements |

Date of creation | 2013-03-22 17:28:57 |

Last modified on | 2013-03-22 17:28:57 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Result |

Classification | msc 46L10 |

Classification | msc 47A05 |