Von Neumann algebra

A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. They were believed by John von Neumann to capture abstractly the concept of an algebra of observables in quantum mechanics. Von Neumann algebras are automatically C*-algebras. The von Neumann bicommutant theorem gives another description of von Neumann algebras, using algebraic rather than topological properties.

Related Topics:
*-algebra - Bounded - Operators - Hilbert space - Weak operator topology - Strong operator topology - Pointwise convergence - John von Neumann - Observable - Quantum mechanics - C*-algebra - Von Neumann bicommutant theorem - Algebraic - Topological

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Von Neumann algebras are also called W*-algebras. The more common name was suggested by Jacques Dixmier.

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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 (highly non-commutative) von Neumann algebra. Inside this algebra we have the sub-algebra L^infty (X,mu) of bounded multiplication operators

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: psi mapsto f psi, quad psi in L^2_mu(X)

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

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