commutant is a weak operator closed subalgebra
: 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.
|Title||commutant is a weak operator closed subalgebra|
|Date of creation||2013-03-22 18:39:32|
|Last modified on||2013-03-22 18:39:32|
|Last modified by||asteroid (17536)|