Maximal ideals in some F-algebras of holomorphic functions

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∫2π0 (log+ |f(reiθ)j|p dθ/2π < + ∞ M. Stoll [16] showed that the space Np with the topology given by the metric dp defined as dp(f,g) = (∫2π0 (log(1 + |f*(eiθ) - g*(eiθ)|))p dθ/2π)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.