A von Neumann algebra is a subspace of bounded Hilbert space operators closed under the weak/strong topology. It's also called a W* algebra. Von Neumann algebras are automatically C* algebras.
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* algebrass 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* algebrass is sometimes called noncommutative geometry.
See Quantum mechanics, Quantum field theory, Local quantum physics, C* algebra, Measure theory
