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Von Neumann bicommutant theorem
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{{cleanup|reason=As mentioned on the talk page, the proof of item (iii) is incomplete.|date=September 2018}} In [[mathematics]], specifically [[functional analysis]], the '''von Neumann bicommutant theorem''' relates the [[closure (mathematics)|closure]] of a set of [[bounded operator]]s on a [[Hilbert space]] in certain [[operator topology|topologies]] to the [[bicommutant]] of that set. In essence, it is a connection between the [[algebra]]ic and topological sides of [[operator theory]]. The formal statement of the theorem is as follows: :'''Von Neumann bicommutant theorem.''' Let {{math|'''M'''}} be an [[Operator algebra|algebra]] consisting of bounded operators on a Hilbert space {{mvar|H}}, containing the identity operator, and closed under taking [[Hermitian adjoint|adjoint]]s. Then the [[closure (topology)|closure]]s of {{math|'''M'''}} in the [[weak operator topology]] and the [[strong operator topology]] are equal, and are in turn equal to the [[bicommutant]] {{math|'''M'''β²β²}} of {{math|'''M'''}}. This algebra is called the [[von Neumann algebra]] generated by {{math|'''M'''}}. There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If {{math|'''M'''}} is closed in the [[norm topology]] then it is a [[C*-algebra]], but not necessarily a von Neumann algebra. One such example is the C*-algebra of [[compact operator on Hilbert space|compact operator]]s (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, [[ultraweak topology|ultraweak]], [[ultrastrong topology|ultrastrong]], and *-ultrastrong topologies. It is related to the [[Jacobson density theorem]].
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