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单词 VonNeumannDoubleCommutantTheorem
释义

von Neumann double commutant theorem


The von Neumann double commutant theorem is a remarkable result in the theory of self-adjoint algebras of operatorsMathworldPlanetmath on Hilbert spacesMathworldPlanetmath, as it expresses purely topological aspects of these algebras in terms of purely algebraic properties.

Theorem - von Neumann - Let H be a Hilbert space (http://planetmath.org/HilbertSpace) and B(H) its algebra of bounded operatorsMathworldPlanetmathPlanetmath. Let be a *-subalgebra of B(H) that contains the identity operatorMathworldPlanetmath. The following statements are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

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

  2. 2.

    is closed in the weak operator topology.

  3. 3.

    is closed in the strong operator topology.

Thus, a purely topological property of a , as being closed for some operator topology, is equivalent to a purely algebraic property, such as being equal to its double commutant.

This result is also known as the bicommutant theorem or the von Neumann density theorem.

Titlevon Neumann double commutant theorem
Canonical nameVonNeumannDoubleCommutantTheorem
Date of creation2013-03-22 18:40:27
Last modified on2013-03-22 18:40:27
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id4
Authorasteroid (17536)
Entry typeTheorem
Classificationmsc 46H35
Classificationmsc 46K05
Classificationmsc 46L10
Synonymdouble commutant theorem
Synonymbicommutant theorem
Synonymvon Neumann bicommutant theorem
Synonymvon Neumann density theorem
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