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 self-adjoint then $\mathcal{F}^{\prime}$ is self-adjoint.
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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 self-adjoint 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	2013-03-22 17:21:53
Last modified on	2013-03-22 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