Let H be an Hilbert SpaceMathworldPlanetmath, B(H) the algebraPlanetmathPlanetmath of bounded operatorsMathworldPlanetmathPlanetmath in H and B(H).

The commutant of , usually denoted , is the subset of B(H) consisting of all elements that commute with every element of , that is


The double commutant of is just () and is usually denoted ′′.


  • If 12, then 21.

  • ′′.

  • If 𝒜 is a subalgebra of B(H), then 𝒜𝒜 is the center ( of 𝒜.

  • If is self-adjoint then is self-adjoint.

  • is always a subalgebra of B(H) that contains the identity operator and is closed in the weak operator topology.

  • If is self-adjoint then is a von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath.

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

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