commutant is a weak operator closed subalgebra
Let be a Hilbert space and the algebra of bounded operators
in . Recall that the commutant of a subset is the set of all bounded operators that commute with those of , i.e.
- If , then is a subalgebra of that contains the identity operator and is closed in the weak operator topology.
: It is clear that contains the identity operator, since it commutes with all operators in and in particular with those of .
Let us now see that is a subalgebra of . Let and . We have that, for all ,
thus, , and all belong to , and therefore is a subalgebra of .
It remains to see that is weak operator closed. Suppose is a net in that converges to in the weak operator topology. Then, for all we have that . Thus, for all , we have
Hence, , so that . We conclude that is closed in the weak operator topology.