# 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:

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

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

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

• If $\mathcal{F}$ is self-adjoint then $\mathcal{F}^{\prime}$ is self-adjoint.

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

• 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 Commutant 2013-03-22 17:21:53 2013-03-22 17:21:53 asteroid (17536) asteroid (17536) 11 asteroid (17536) Definition msc 46L10 VonNeumannAlgebra double commutant