A characterization of maximal ideals in the Fréchet algebras of holomorphic functions $F^p$ $(1<p<\infty$

Abstract

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\mathbb{D}$ for which ${\lim }_{r\to 1}(1-r{)}^{1/q}{\log }^{+}{M}_{\infty }(r,f)=0,$ where $M_{\infty }(r,f)={\max }_{|z|\le r}|f(z)|$ with $0<r<1$. Stoll [5, Theorem 3.2] proved that the space $F^p$ with the topology given by the family of seminorms ${\left\{\Vert \cdot {\Vert }_{q,c}\right\}}_{c>0}$ defined for $f\in F^q$ as $\Vert f{\Vert }_{q,c}:={\sum }_{n=0}^{\infty }|a_n|\exp \left(-c{n}^{1/(q+1)}\right)<\infty ,$ is a countably normed Fréchet algebra. Notice that for each $p>1$, $F^p$ is the Fréchet envelope of the Privalov space $N^p$. In this paper we study the structure of maximal ideals in the algebras $F^p$ ($1<p<\infty$). In particular, we give a complete characterization of closed maximal ideals in $F^p$. Moreover, we characterize multiplicative linear functionals on $F^p$.

6 pages, no figures, no tables