Von Neumann algebra
A von Neumann algebra is a *-algebra of bounded Hilbert space operators which is closed in the weak operator topology, or equivalently, in the strong operator topology. Von Neumann algebras are also called a W*-algebras, the W standing for "weakly closed". Von Neumann algebras are automatically C* algebras. They are named for John von Neumann.
The von Neumann bicommutant theorem gives another description of von Neumann algebras, using algebraical rather than topological properties.
The relationship between commutative von Neumann algebras and locally compact measure spaces is analogous to that between commutative C* algebras and compact Hausdorff spaces. Every commutative von Neumann algebra is isomorphic to L∞(X) for some locally compact measure space X, and for every locally compact measure space X, conversely, L∞(X) is a von Neumann algebra.
Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of C* algebras is sometimes called noncommutative geometry.
See Quantum mechanics, Quantum field theory, Local quantum physics, C* algebra, Measure theory
Referenced By
Alain Connes | List of functional analysis topics | List of mathematical topics (V-Z)
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