example of Banach algebra which is not a -algebra for any involution
To prove the above claim we will give a proof of a more general fact about finite dimensional -algebras, which clearly shows the for a Banach algebra to be a -algebra for some involution.
Proof : Let be a finite dimensional -algebra. Let be an element of , the Jacobson radical of .
is an ideal of , so .
The Jacobson radical of a finite dimensional algebra is nilpotent, therefore there exists such that . Then, by the condition and the fact that is selfadjoint,
so and is trivial.
We now prove the above claim.
Proof of the claim: It is easy to see that is the only maximal ideal of . Therefore the Jacobson radical of is not trivial.
By the theorem we conclude that there is no involution that makes into a -algebra.
Remark - It could happen that there were no involutions in and so the above claim would be uninteresting. That’s not the case here. For example, one can see that defines an involution in (this is just the taken over the other diagonal of the matrix).
|Title||example of Banach algebra which is not a -algebra for any involution|
|Date of creation||2013-03-22 17:25:54|
|Last modified on||2013-03-22 17:25:54|
|Last modified by||asteroid (17536)|
|Defines||finite dimensional -algebras are semi-simple|