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 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 are the zero subspace and H itself. Then the closures of in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant of .