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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}'$, is the subset of $ B(H)$ consisting of all elements that commute with every element of $ \mathcal{F}$, that is

$ \mathcal{F}'=\{T \in B(H):\; T\cdot S=S\cdot T \;\;\; \forall S \in \mathcal{F}\}$

The double commutant of $ \mathcal{F}$ is just $ (\mathcal{F}')'$ and is usually denoted $ \mathcal{F}''$.

Properties:

  1. If $ \mathcal{F}$ is self-adjoint then $ \mathcal{F}'$ is self-adjoint.
  2. $ \mathcal{F}'$ is always a subalgebra of $ B(H)$ that contains the identity operator and is closed in the weak operator topology.
  3. If $ \mathcal{F}$ is self-adjoint then $ \mathcal{F}'$ is a von Neumann algebra.

Remark: The commutant is a particular case of the more general definition of centralizer.



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Also defines:  double commutant
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Cross-references: centralizer, von Neumann algebra, weak operator topology, closed, identity operator, contains, subalgebra, self-adjoint, properties, subset, bounded operators, algebra, Hilbert space
There are 5 references to this entry.

This is version 3 of commutant, born on 2007-07-04, modified 2007-07-05.
Object id is 9725, canonical name is Commutant.
Accessed 591 times total.

Classification:
AMS MSC: 16-00 (Associative rings and algebras :: General reference works )
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