# von Neumann double commutant theorem

The von Neumann double commutant theorem is a remarkable result in the theory of self-adjoint 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. 1.

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

2. 2.

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

3. 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 2013-03-22 18:40:27 Last modified on 2013-03-22 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