1. ## Semi-simple Banach Algebras

Dear Friends!
I've a little problem with this issue. How Can I show that any self-adjoint subalgebra A of n × n matrices with complex entries is semisimple. I must explain this problem step by step so any help will be highly appreciated.
Could You give me some other (maybe two simple) examples of semi-simple Banach algebras?
Best regards.

2. Originally Posted by Arczi1984
Dear Friends!
I've a little problem with this issue. How Can I show that any self-adjoint subalgebra A of n × n matrices with complex entries is semisimple. I must explain this problem step by step so any help will be highly appreciated.
Could You give me some other (maybe two simple) examples of semi-simple Banach algebras?
Best regards.
The algebra A is semisimple if its Jacobson radical J(A) is the zero ideal. The are several (equivalent) definitions of J(A), and your proof will depend on which definition you are using. I will take the definition that J(A) is the intersection of all the maximal left ideals of A.

Suppose that $0\ne a\in A$. Then $a^*a$ is a nonzero positive element of A. A nonzero positive matrix has a positive eigenvalue. If $a^*a$ has an eigenvalue $\lambda>0$ then $I-\lambda^{-1}a^*a$ is not invertible and therefore belongs to some maximal left ideal L. If also $a\in L$ then $I\in L$, contradicting the fact that L is supposed to be a proper ideal. Therefore $a\notin L$ and hence $a\notin J(A)$. Hence A is semisimple.

Each step of that argument needs to be carefully justified, and some of the steps may require a bit of thought. If you use a different definition of J(A) then the argument will need to be amended, but I think you will find that all the essential ideas are there.

Three other important examples of semisimple Banach algebras are (1) the (commutative) algebra of all bounded analytic functions on the open unit disk, with the supremum norm, (2) the algebra of all bounded operators on any Banach space, with the operator norm, (3) the algebra of all absolutely summable complex-valued functions on a group, with convolution as multiplication, and the $\ell^1$-norm.

3. Thanks for help. I've the same definition as You described. Yes, You right when I was looking for solution I found few others definition of semisimple algebras. I've one more question. I want to study this subject, that mean Banacha algebras (semisimple algebras) and maximal ideals. Could You recommend me some good books about it? Which is the best for newbies?

4. Originally Posted by Arczi1984
I want to study this subject, that mean Banach algebras (semisimple algebras) and maximal ideals. Could You recommend me some good books about it? Which is the best for newbies?
I learned the subject from Rickart's General theory of Banach algebras. That was in the 1960s, but there was a new edition of Rickart's book a few years ago, and I think it is still one of the most readable accounts of Banach algebra theory.