Commutant - Imported Knowlpedia Content

Definition

Let $\mathcal H$ be a Hilbert space and let $S$ be a subset of the algebra $\mathcal B(\mathcal H)$ of bounded operators on $\mathcal H$ . The commutant of $S$ is

S'=\{T\in\mathcal B(\mathcal H):TA=AT\text{ for every }A\in S\}.

Thus $S'$ records all bounded symmetries of the operator family $S$ . The bicommutant is $S''=(S')'$ . This definition applies to any set of operators; $S$ need not itself be an algebra, self-adjoint, or closed in any operator topology.

Basic properties

The commutant is a unital $*$ -subalgebra of $\mathcal B(\mathcal H)$ when $S=S^*$ ; without that hypothesis it is still a unital algebra but need not be closed under adjoints. It is closed in both the weak and strong operator topologies. Inclusion reverses: $S\subseteq T$ implies $T'\subseteq S'$ , and always $S\subseteq S''$ .

Bicommutants and operator algebras

The von Neumann bicommutant theorem says that for a unital self-adjoint subalgebra $A\subseteq\mathcal B(\mathcal H)$ , the bicommutant $A''$ equals both its strong-operator and weak-operator closures Takesaki, Chapter III, §2. This makes commutants a central bridge between algebraic relations and operator-topological closure.

Examples

The commutant of all scalar multiples of the identity is $\mathcal B(\mathcal H)$ . Conversely, $\mathcal B(\mathcal H)'=\mathbb C I$ . If $P$ is an orthogonal projection, then $\{P\}'$ consists of the operators preserving both $\operatorname{ran}P$ and $\ker P$ , equivalently the block-diagonal operators for $\mathcal H=\operatorname{ran}P\oplus\ker P$ .

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III, §§1–2 on commutants and the bicommutant theorem.