Commutant

Summary

Let $S$ be a set of operators on a Hilbert space. Its commutant $S'$ is the set of all bounded operators that commute with every element of $S$. It is the operator version of the centralizer in an algebra. The commutant of a self-adjoint set is always a von Neumann algebra. By the bicommutant theorem, the von Neumann algebras are the self-adjoint unital algebras equal to their double commutant.