About commutative semisimple Banach algebra

Question

Is the norm on a unital semi-simple commutative Banach algebra with $\|I\|=1$, unique? ($I$ denotes the identity element)

Answer 1

Let $A := C^1([0,1])$. Then you can consider the two norms on $A$ given by $$ \lVert f \rVert_1 = \lVert f \rVert_\infty + \lVert f' \rVert _\infty $$ and $$ \lVert f \rVert_2 = \lvert f(0) \rvert + \lVert f'\rVert_\infty. $$ They both make $A$ into a Banach algebra and $C^1([0,1])$ is semisimple.