Von Neumann bicommutant theorem

In mathematics, specifically functional analysis, 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.

If M is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra.

The theorem is equivalent to the combination of the following three statements: where the W and S subscripts stand for closures in the weak and strong operator topologies, respectively.

For any x and y in H, the map T → is continuous in the weak operator topology, by its definition.