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There are two basic examples of von Neumann algebras to keep in mind. Firstly, if ''X'' is a space with a \sigma -finite measure \mu and L^2(X,\mu) is the Hilbert space of complex-valued square-integrable functions on ''X'', then the space B(L^2(X,\mu)) of bounded linear operators on this space is a von Neumann algebra. Inside this algebra we have the sub-algebra L^\infty (X,\mu) of bounded multiplication operators

: \psi \mapsto f \psi, \quad \psi \in L^2_\mu(X)

which in fact is the most general example of a commutative von Neumann algebra as is stated below.


DEFINITIONS


There are three common ways to define von Neumann algebras.

  • algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by almost any other common topology other than the norm topology,

  • algebras of bounded operators that are closed in the norm topology are C--- algebras, so in particular any von Neumann algebra is a

  • algebra.)


  • and equal to its double commutator, or equivalently the commutator of some subset closed under ---. The

    • Von Neumann Bicommutant Theorem says that the first two definitions are equivalent.


  • algebras that have a Predual ; in other words the von Neumann algebra, considered as a Banach space, is the dual of some other Banach space called the predual. The predual of a von Neumann algebra is unique up to isomorphism.