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 IF be the smallest element of F and let MF be the unique maximal ideal of C(K) containing the elements of F . Assuming the continuum hypothesis, we show that if |C(K)/IF | = 20 , and if there exists a sequence (Gn)n≥1 of subsets of F ∪ {MF} stable under unions such that F ∪{MF} = ∪n≥1Gn, 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 . AMS classification: 46H40 (primary), 46J10, 03E50 (secondary)