Von Neumann bicommutant theorem
|
|
The von Neumann bicommutant theorem 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 a C* algebra of bounded operators on a Hilbert space H, such that the only closed subspaces of H left invariant by every operator in <math>A<math> are the zero subspace and H itself. 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.