# commutant is a weak operator closed subalgebra

Let $H$ be a Hilbert space^{} and $B(H)$ the algebra of bounded operators^{} in $H$. Recall that the commutant of a subset $\mathcal{F}\subset B(H)$ is the set of all bounded operators that commute with those of $\mathcal{F}$, i.e.

${\mathcal{F}}^{\prime}:=\{T\in B(H):TS=ST,\forall S\in \mathcal{F}\}.$ |

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

*:* It is clear that ${\mathcal{F}}^{\prime}$ contains the identity operator, since it commutes with all operators^{} in $B(H)$ and in particular with those of $\mathcal{F}$.

Let us now see that ${\mathcal{F}}^{\prime}$ is a subalgebra of $B(H)$. Let ${T}_{1},{T}_{2}\in {\mathcal{F}}^{\prime}$ and $\lambda \in \u2102$. We have that, for all $S\in \mathcal{F}$,

$S({T}_{1}+{T}_{2})=S{T}_{1}+S{T}_{2}={T}_{1}S+{T}_{2}S=({T}_{1}+{T}_{2})S$ | ||

$S(\lambda {T}_{1})=\lambda S{T}_{1}=\lambda {T}_{1}S$ | ||

$S({T}_{1}{T}_{2})={T}_{1}S{T}_{2}={T}_{1}{T}_{2}S$ |

thus, ${T}_{1}+{T}_{2}$, $\lambda {T}_{1}$ and ${T}_{1}{T}_{2}$ all belong to ${\mathcal{F}}^{\prime}$, and therefore ${\mathcal{F}}^{\prime}$ is a subalgebra of $B(H)$.

It remains to see that ${\mathcal{F}}^{\prime}$ is weak operator closed. Suppose $({T}_{i})$ is a net in ${\mathcal{F}}^{\prime}$ that converges^{} to $T$ in the weak operator topology. Then, for all $x,y\in H$ we have that $\u27e8{T}_{i}x,y\u27e9\to \u27e8Tx,y\u27e9$. Thus, for all $S\in \mathcal{F}$, we have

$\u27e8(TS-ST)x,y\u27e9$ | $=$ | $\u27e8TSx,y\u27e9-\u27e8Tx,{S}^{*}y\u27e9$ | ||

$=$ | $lim\left(\u27e8{T}_{i}Sx,y\u27e9-\u27e8{T}_{i}x,{S}^{*}y\u27e9\right)$ | |||

$=$ | $lim\u27e8({T}_{i}S-S{T}_{i})x,y\u27e9$ | |||

$=$ | $lim\u27e8({T}_{i}S-{T}_{i}S)x,y\u27e9$ | |||

$=$ | $0$ |

Hence, $TS-ST=0$, so that $T\in {\mathcal{F}}^{\prime}$. We conclude that ${\mathcal{F}}^{\prime}$ is closed in the weak operator topology. $\mathrm{\square}$

Title | commutant is a weak operator closed subalgebra |
---|---|

Canonical name | CommutantIsAWeakOperatorClosedSubalgebra |

Date of creation | 2013-03-22 18:39:32 |

Last modified on | 2013-03-22 18:39:32 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 7 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46L10 |