commutant
Definition
Let $H$ be an Hilbert Space^{}, $B(H)$ the algebra^{} of bounded operators^{} in $H$ and $\mathcal{F}\subset B(H)$.
The commutant of $\mathcal{F}$, usually denoted ${\mathcal{F}}^{\prime}$, is the subset of $B(H)$ consisting of all elements that commute with every element of $\mathcal{F}$, that is
${\mathcal{F}}^{\prime}=\{T\in B(H):TS=ST,\forall S\in \mathcal{F}\}$
The double commutant of $\mathcal{F}$ is just ${({\mathcal{F}}^{\prime})}^{\prime}$ and is usually denoted ${\mathcal{F}}^{\prime \prime}$.
Properties:

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If ${\mathcal{F}}_{1}\subseteq {\mathcal{F}}_{2}$, then ${\mathcal{F}}_{2}^{\prime}\subseteq {\mathcal{F}}_{1}^{\prime}$.

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$\mathcal{F}\subseteq {\mathcal{F}}^{\prime \prime}$.

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If $\mathcal{A}$ is a subalgebra of $B(H)$, then $\mathcal{A}\cap {\mathcal{A}}^{\prime}$ is the center (http://planetmath.org/CenterOfARing) of $\mathcal{A}$.

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If $\mathcal{F}$ is selfadjoint then ${\mathcal{F}}^{\prime}$ is selfadjoint.

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${\mathcal{F}}^{\prime}$ is always a subalgebra of $B(H)$ that contains the identity operator and is closed in the weak operator topology.

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If $\mathcal{F}$ is selfadjoint then ${\mathcal{F}}^{\prime}$ is a von Neumann algebra^{}.
Remark: The commutant is a particular case of the more general definition of centralizer^{}.
Title  commutant 

Canonical name  Commutant 
Date of creation  20130322 17:21:53 
Last modified on  20130322 17:21:53 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  11 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46L10 
Related topic  VonNeumannAlgebra 
Defines  double commutant 