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 . Then is a nonzero positive element of A. A nonzero positive matrix has a positive eigenvalue. If has an eigenvalue then is not invertible and therefore belongs to some maximal left ideal L. If also then , contradicting the fact that L is supposed to be a proper ideal. Therefore and hence . 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 -norm.