proof of quotients in -algebras
Proof: We have that is self-adjoint (http://planetmath.org/InvolutaryRing), since it is a closed ideal of a -algebra (http://planetmath.org/CAlgebra) (see this entry (http://planetmath.org/ClosedIdealsInCAlgebrasAreSelfAdjoint)). Hence, the involution in induces a well-defined involution in by .
Recall that, since is closed, the quotient norm is indeed a norm in that makes a Banach algebra (see this entry (http://planetmath.org/QuotientsOfBanachAlgebras)). Thus we only have to prove the to prove that is a -algebra.
Recall that -algebras have approximate identities (http://planetmath.org/CAlgebrasHaveApproximateIdentities). Notice that itself is a -algebra and pick an approximate identity in such that
each is positive.
Let denote the quotient norm in . We claim that for every :
We will prove the above equality as a lemma at the end of the entry. Assuming this result, it follows that for every
Since each is positive and we know that its spectrum lies on the interval . Hence is also positive and its spectrum also lies on the interval . Thus, . Therefore:
Since is a Banach algebra, we also have and so
which proves that is a -algebra.
We now prove equality (1) as a lemma.
Lemma - Suppose is a -algebra with identity element . Let be a closed ideal and be an approximate identity in such that each is positive and . Then
for every in .
Proof: Since for every it follows that
Therefore, taking the infimum over all we obtain:
Also, since ,
and this proves the lemma.
|Title||proof of quotients in -algebras|
|Date of creation||2013-03-22 17:41:56|
|Last modified on||2013-03-22 17:41:56|
|Last modified by||asteroid (17536)|