von Neumann double commutant theorem
The von Neumann double commutant theorem is a remarkable result in the theory of selfadjoint algebras of operators^{} on Hilbert spaces^{}, as it expresses purely topological aspects of these algebras in terms of purely algebraic properties.
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Theorem  von Neumann  Let $H$ be a Hilbert space (http://planetmath.org/HilbertSpace) and $B(H)$ its algebra of bounded operators^{}. Let $\mathcal{M}$ be a *subalgebra of $B(H)$ that contains the identity operator^{}. The following statements are equivalent^{}:

1.
$\mathcal{M}={\mathcal{M}}^{\prime \prime}$, i.e. $\mathcal{M}$ equals its double commutant.

2.
$\mathcal{M}$ is closed in the weak operator topology.

3.
$\mathcal{M}$ is closed in the strong operator topology.
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Thus, a purely topological property of a $\mathcal{M}$, 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.
Title  von Neumann double commutant theorem 
Canonical name  VonNeumannDoubleCommutantTheorem 
Date of creation  20130322 18:40:27 
Last modified on  20130322 18:40:27 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  4 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 46H35 
Classification  msc 46K05 
Classification  msc 46L10 
Synonym  double commutant theorem 
Synonym  bicommutant theorem 
Synonym  von Neumann bicommutant theorem 
Synonym  von Neumann density theorem 