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von Neumann algebra (Definition)

Let $ H$ be an Hilbert space and $ B(H)$ the set of bounded operators in $ H$.

A von Neumann algebra (or $ W^*$-algebra) $ \mathcal M$ is a $ C^*$-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''$, i.e. $ \mathcal M$ equals its double commutant.

The equivalence between the above conditions is given by the von Neumann double commutant theorem.

Examples:

  1. $ B(H)$ is itself a von Neumann algebra.
  2. $ L^{\infty}(\mathbb{R})$ as subalgebra of $ B(L^2(\mathbb{R}))$ is a von Neumann algebra.



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See Also: $C^*$-algebra

Other names:  $W^*$-algebra
Also defines:  double commutant theorem, von Neumann double commutant theorem, bicommutant theorem
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Cross-references: subalgebra, equivalence, double commutant, strong operator topology, weak operator topology, closed, equivalent, identity operator, contains, bounded operators, Hilbert space
There are 11 references to this entry.

This is version 9 of von Neumann algebra, born on 2007-07-04, modified 2007-12-18.
Object id is 9722, canonical name is VonNeumannAlgebra.
Accessed 1359 times total.

Classification:
AMS MSC: 46L10 (Functional analysis :: Selfadjoint operator algebras :: General theory of von Neumann algebras)
 46H35 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: Topological algebras of operators)
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