Von Neumann Algebra - Commutative Von Neumann Algebras

Commutative Von Neumann Algebras

The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra is isomorphic to L∞(X) for some measure space (X, μ) and conversely, for every σ-finite measure space X, the * algebra 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 topology (Connes 1994).