Let $A=C^1 ([0,1])$ with the norm $$ \|f\|=\|f\|_{\infty}+\|f'\|_{\infty}, \quad f\in C^1([0,1]) $$ I want to show that $A$ is semi simple.
Recall that A is semisimple if $Rad(A)=0$ where $Rad=\bigcap J$ s.t $J$ a maximal ideal.
To show that $Rad(A)=0$ is it enough to show $\sigma(f)={0}$? What is the invertible set of $A$ ?
If $t \in [0,1]$ then $\text{e}_t\colon A\to \mathbb C$ given by $\text{e}_t(f) = f(t)$ is clearly a homomorphism of algebras, so that $J_t = \text{ker}(\text{e}_t)$ is an ideal in $A$. Can you use these ideals to answer your question? (If $A$ is taken to be real-valued functions on $[0,1]$ then $e_t\colon A \to \mathbb R$, but this does not change the nature of the ideals $J_t$ significantly.)