von Neumann algebra
Definition
Let $H$ be an Hilbert space^{}, and let $B(H)$ be the *algebra of bounded operators^{} in $H$.
A von Neumann algebra^{} (or ${W}^{*}$algebra) $\mathcal{M}$ is a *subalgebra of $B(H)$ that contains the identity operator^{} and satisfies one of the following equivalent^{} conditions:

1.
$\mathcal{M}$ is closed in the weak operator topology.

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

3.
$\mathcal{M}={\mathcal{M}}^{\prime \prime}$, i.e. $\mathcal{M}$ 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.
Examples:

1.
$B(H)$ is itself a von Neumann algebra.

2.
${L}^{\mathrm{\infty}}(\mathbb{R})$ (http://planetmath.org/LinftyXDmu) as subalgebra of $B({L}^{2}(\mathbb{R}))$ is a von Neumann algebra.
Title  von Neumann algebra 
Canonical name  VonNeumannAlgebra 
Date of creation  20130322 17:21:44 
Last modified on  20130322 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 