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In mathematics, specifically closure of a set of bounded operators on a Hilbert space in certain operator topologyn Neumann algebra]] generated by . There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If 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 operators (on an infinite dimensional Hilbert space). 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. It is related to the Jacobson density theorem. == Proof == Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital subalgebra of . (this means that contains the adjoints of its members, and the identity operator on ) The theorem is equivalent to the combination of the following three statements: :(i) :(ii) :(iii) where the and subscripts stand for closures in the weak and strong operator topologies, respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Von Neumann bicommutant theorem」の詳細全文を読む スポンサード リンク
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