closed ideals in -algebras are self-adjoint
Theorem - Every closed (http://planetmath.org/ClosedSet) two-sided ideal (http://planetmath.org/IdealOfAnAlgebra) of a -algebra (http://planetmath.org/CAlgebra) is self-adjoint (http://planetmath.org/InvolutaryRing), i.e.
if then .
Proof : Let .
Since is closed and the involution mapping is continuous, it follows that is also closed.
We claim that is also a of . To see this let , and . Then
is a -subalgebra of (it is a norm-closed, involution-closed, subalgebra of ).
Let be an approximate identity for with the above :
each is positive (hence self-adjoint) and
We now prove is self-adjoint:
Let . We have that
Taking limits in both we obtain
since and is an approximate identity for .
As we see that .
We conclude from the limit above that is in the closure of . Therefore .
Hence, is self-adjoint.
|Title||closed ideals in -algebras are self-adjoint|
|Date of creation||2013-03-22 17:30:42|
|Last modified on||2013-03-22 17:30:42|
|Last modified by||asteroid (17536)|