Maximal Ideals in Some $F$-Algebras of Holomorphic Functions

Romeo Meštrović

For $1<p<\infty$, the Privalov class $N^p$ consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ of the complex plane $\Bbb C$ such that \[ \underset{0eq req1}upıt_0^{2i}(og^+\big|f(re^{iheta}\big|)^p\frac{dheta}{2i})<+ıfty \] M. Stoll [16] showed that the space $N^p$ with the topology given by the metric $d_p$ defined as \[ d_p(f,g)=\Big(ıt_0^{2i}(og(1+|f^*(e^{iheta}-g^*(e^{iheta})|))\Big)^{1/p},\quad f,gı N^p \] becomes an $F$-algebra. Since the map $f\mapsto d_p(f,0)(f\in N^p)$ is not a norm, $N^p$ is not a Banach algebra. Here we investigate the structure of maximal ideals of the algebras $N^p(1<p<1)$. We also give a complete characterization of multiplicative linear functionals on the spaces $N^p$. As an application, we show that there exists a maximal ideal of $N^p$ which is not the kernel of a multiplicative continuous linear functional on $N^p$.