Are all Semi-simple commutative Banach algebras finite-dimensional?

Question

I have the following two theorems from functional analysis:

1. All norms on a finite-dimensional vector space are equivalent.
2. All norms on a semi-simple commutative Banach algebra are equivalent.

I was then given the following question: does this mean that all semi-simple commutative Banach algebras are finite-dimensional?

My instinct tells me NO, but I need to find a counterexample.

Can anybody please give me an example of a semi-simple commutative Banach algebra which is not finite-dimensional?

Answer 1

Try $C(K)$, the continuous functions on an infinite compact Hausdorff space.