von Neumann algebra


Let H be an Hilbert spaceMathworldPlanetmath, and let B(H) be the *-algebra of bounded operatorsMathworldPlanetmathPlanetmath in H.

A von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath (or W*-algebra) is a *-subalgebra of B(H) that contains the identity operatorMathworldPlanetmath and satisfies one of the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath conditions:

  1. 1.

    is closed in the weak operator topology.

  2. 2.

    is closed in the strong operator topology.

  3. 3.

    =′′, i.e. equals its double commutant.

The equivalence between the above conditions is given by the von Neumann double commutant theorem.

Since the weak and strong operator topology are weaker than the norm topology, it follows that every von Neumann algebra is a norm closed *-subalgebra of B(H). Thus, von Neumann algebras are a particular class of C*-algebras (http://planetmath.org/CAlgebra) and the results and tools from the C* theory are also applicable in the setting of von Neumann algebras. Nevertheless, the philosophy behind von Neumann algebras is quite different from that of C*-algebras and the tools and techniques for each theory turn out to be different as well.


  1. 1.

    B(H) is itself a von Neumann algebra.

  2. 2.

    L() (http://planetmath.org/LinftyXDmu) as subalgebra of B(L2()) is a von Neumann algebra.

Title von Neumann algebra
Canonical name VonNeumannAlgebra
Date of creation 2013-03-22 17:21:44
Last modified on 2013-03-22 17:21:44
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 29
Author asteroid (17536)
Entry type Definition
Classification msc 46C15
Classification msc 46H35
Classification msc 46L10
Synonym W*-algebra
Related topic CAlgebra
Related topic TopologicalAlgebra
Related topic Commutant
Related topic GroupoidCDynamicalSystem
Related topic Algebras2
Related topic CAlgebra3
Related topic WeakHopfCAlgebra2
Related topic HAlgebra
Related topic LocallyCompactQuantumGroup
Related topic QuantumGroupsAndVonNeumannAlgebras