Maximal ideals in some F-algebras of holomorphic functions

Abstract

For 1 < p < ?, the Privalov class Np consists of all holomorphic functions f on the open unit disk D of the complex plane C such that sup 0?r<1?2?0 (log+ |f(rei?)j|p d?/2? < + ? M. Stoll [16] showed that the space Np with the topology given by the metric dp defined as dp(f,g) = (?2?0 (log(1 + |f*(ei?) - g*(ei?)|))p d?/2?)1/p, f,g ? Np; becomes an F-algebra. Since the map f ? dp(f,0) (f ? Np) is not a norm, Np is not a Banach algebra. Here we investigate the structure of maximal ideals of the algebras Np (1 < p < ?). We also give a complete characterization of multiplicative linear functionals on the spaces Np. As an application, we show that there exists a maximal ideal of Np which is not the kernel of a multiplicative continuous linear functional on Np.