Maximal chains of closed prime ideals for discontinuous algebra norms on C(K)

Jean Esterle (1)

1 IMB - Institut de Mathématiques de Bordeaux

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)/IF | is the continuum , and if there exists a sequence (G_n ) of subsets of F U {M_F } stable under unions such that F U {MF } = UG_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.

Mots clés: continuum hypothesis; prime ideals; algebra norm; algebra of continuous functions on a Banach space

Domaines: Analyse fonctionnelle [math.FA]

Jean Esterle. Maximal chains of closed prime ideals for discontinuous algebra norms on C(K). Mathematical Proceedings of the Royal Irish Academy, 2012, 112A (2), pp.101-115. ⟨hal-00773663⟩