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 double commutant - Let $H$ be a Hilbert space 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. $\mathcal M = \mathcal M''$ , 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.