Von Neumann bicommutant theorem

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In mathematics, the von Neumann bicommutant theorem in functional analysis relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory.

The formal statement of the theorem is as follows. Let <math>A</math> be an algebra of bounded operators on a Hilbert space H, containing the identity operator and closed under taking adjoints. Then the closures of <math>A</math> in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant <math>A</math> of <math>A</math>. This algebra is the von Neumann algebra generated by A.

There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If A is closed in the norm topology then it is a C^* algebra, but not necessarily a von Neumann algebra. For most other common topologies the closed *-algebras containing 1 are still von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies.