A non-maximal primary ideal in a unital commutative Banach algebra A.

******************************

Suppose that A is the algebra C^{m}([0, 1]) of the complex valued m times continously differentiable functions on [0, 1] with the norm ∥ f ∥= ∑_{k=0}^{m} \frac{1}{k!} sup_{x∈[0,1]} |f^{(k)}(x)|. Let x_{0} ∈ [0, 1] and I = {f ∈ A; f(x_{0}) = f′(x_{0}) = 0}. Then I is a closed twosided ideal contained in only one maximal ideal; i.e. {f ∈ A : f(x_{0}) = 0}. Note that the maximal ideals of A are of the form I_{x} = {f ∈ A; f(x) = 0}, x ∈ [0, 1]. A conclusion is that C^{m}([0, 1]) is not spectral synthesis, i.e. it has a closed two-sided ideal which is not the intersection of maximal ideals containing this ideal.

Comment. The disk algebra contains a nonmaximal prime ideal, namely {0}.