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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.

Remarks

The von Neumann algebras and/or (more generally) C*-algebras are, for example, employed to define locally compact quantum groups by equipping such algebras with a co-associative multiplication and also with associated, both left- and right- Haar measures, defined by two semi-finite normal weights [1].

Bibliography

1
Leonid Vainerman. 2003. Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians., Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh & Co: Berlin.



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See Also: $C^*$-algebra, algebra classification, C*-algebras and quantum compact groupoids, weak Hopf C*-algebra, locally compact quantum group, groupoid C*-dynamical system, Hilbert space, classification of Hilbert spaces, proof of classification of separable Hilbert spaces, quantum groups and von Neumann algebras, H $*$ -algebra

Other names:  $W^*$-algebra
Also defines:  double commutant theorem, von Neumann double commutant theorem, bicommutant theorem
Keywords:  operator algebras, C*-algebras, groupoid C*-dynamical systems, Hilbert spaces
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Cross-references: weights, normal, Haar measures, multiplication, algebras, C*-algebras, subalgebra, equivalence, double commutant, strong operator topology, weak operator topology, closed, equivalent, identity operator, contains, bounded operators, Hilbert space
There are 15 references to this entry.

This is version 18 of von Neumann algebra, born on 2007-07-04, modified 2008-09-08.
Object id is 9722, canonical name is VonNeumannAlgebra.
Accessed 2029 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)
 22D25 (Topological groups, Lie groups :: Locally compact groups and their algebras :: $C^*$-algebras and $W$*-algebras in relation to group representations)
 46C15 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Characterizations of Hilbert spaces)
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