Noncommutative Geometry - Noncommutative C*-algebras, Von Neumann Algebras

Noncommutative C*-algebras, Von Neumann Algebras

(The formal duals of) non-commutative C*-algebras are often now called non-commutative spaces. This is by analogy with the Gelfand representation, which shows that commutative C*-algebras are dual to locally compact Hausdorff spaces. In general, one can associate to any C*-algebra S a topological space Ŝ; see spectrum of a C*-algebra.

For the duality between σ-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called non-commutative measure spaces.