Exercise 6, § 5

Let B be the algebra of complex continuously differentiable functions on $[0,1]$, endowed with the norm $\|f\|=$ sup$|f|+$ sup$|f'|$. Let A be the closed subalgebra of B consisting of the functions $f\in B$ such that $f(\frac{1}{2})= 0$. Let I be the closed ideal of A consisting of the $f\in A$ such that $f'(\frac{1}{2})= 0$.

a) The space $\mathsf{X}(A)$ is identified with $[0,1]-\{\frac{1}{2}\}$, and A is regular.

b) The ideal I is a maximal ideal of A which is not regular.