Von Neumann Algebras

Commutant and Bicommutant

Suppose is some Hilbert space. is a set of bounded operators on . The commutant of , denoted , is the set . The bicommutant of is the commutant of the commutant, denoted .

Proposition

If , are subsets of , then implies .

Proof For any , also commutes with all operators in .

Von Neumann Algebra