# von Neumann algebra

## Definition

Let $H$ be an Hilbert space, and let $B(H)$ be the *-algebra of bounded operators in $H$.

A (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. 1.

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

2. 2.

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

3. 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. 1.

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

2. 2.

$L^{\infty}(\mathbb{R})$ (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 2013-03-22 17:21:44 Last modified on 2013-03-22 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