Editing Von Neumann algebra (section)

==Definitions==
There are three common ways to define von Neumann algebras.

The first and most common way is to define them as [[weak operator topology|weakly closed]] [[*-algebra]]s of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by  many other [[operator topology|common topologies]] including the [[strong operator topology|strong]], [[ultrastrong topology|ultrastrong]] or [[ultraweak topology|ultraweak]] operator topologies. The *-algebras of bounded operators that are closed in the [[norm topology]] are [[C*-algebra]]s, so in particular any von Neumann algebra is a C*-algebra.

The second definition is that a von Neumann algebra is a subalgebra of the bounded operators closed under [[Semigroup with involution|involution]] (the *-operation) and equal to its double [[commutant]], or equivalently the [[commutant]] of some subalgebra closed under *. The [[von Neumann double commutant theorem]] {{harv|von Neumann|1930}} says that the first two definitions are equivalent.

The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space. {{harvtxt|Sakai|1971}} showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a [[predual]]; in other words the von Neumann algebra, considered as a [[Banach space]], is the [[dual space|dual]] of some other Banach space called the predual.  The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-algebra together with a Hilbert space and a suitable faithful unital action on the Hilbert space. The concrete and abstract definitions of a von Neumann algebra are similar to the concrete and abstract definitions of a C*-algebra, which can be defined either as norm-closed *-algebras of operators on a Hilbert space, or as [[Banach *-algebra]]s such that  <math> ||a a^*||=||a|| \ ||a^*|| </math>.