A semisimple commutative Banach algebra with a closed two-sided ideal I such that \frac{A}{I} isn’t semisimple.

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Suppose that A is the algebra C^{m}([0, 1]) of all m times continously differentiable complex-valued functions on [0, 1] with the norm ∥ f ∥= ∑_{k=0}^{m} \frac{1}{k!} sup_{x∈[0,1]} |f^{(k)}(x)|. Let I = \{f ∈ A; f(0) = f′(0) = 0\}. Then \frac{A}{I} is not semisimple, since assuming f_{∘} to be f_{∘}(x) = x, then f_{∘}^{2} ∈ I and so (f_{∘} + I)^{2} = f_{∘}^{2} + I = 0, hence r(x) = lim_{n} ∥ (f_{∘} + I)^{n} ∥^{\frac{1}{n}} = 0. Therefore f_{∘} + I ∈ Rad(\frac{A}{I}). But f_{∘} + I ≠ 0. So that \frac{A}{I} is not semisimple.