Maximal chains of Closed Prime Ideals for Discontinuous Algebra Norms on C(K)

Let K be an infinite compact space, let C(K) be the algebra of continuous complex-valued functions of K, let F be a well-ordered chain of nonmaximal prime ideals of C(K), let I F be the smallest element of F and let M F be the unique maximal ideal of C(K) containing the elements of F .Assuming the continuum hypothesis, we show that if |C(K)/I F | = 2 ℵ0 , and if there exists a sequence (G n ) n≥1 of subsets of F ∪ {M F } stable under unions such that F ∪ {M F } = ∪ n≥1 G n , then there exists a discontinuous algebra norm p on C(K) such that the set of all nonmaximal prime ideals of C(K) which are closed with respect to p equals F .