# Closed maximal ideals ideals in some Fr\'echet algebras of holomorphic functions

## TL;DR

This paper characterizes closed maximal ideals in certain Fréchet algebras of holomorphic functions, extending previous results by analyzing their structure via topology of uniform convergence on compact sets.

## Contribution

It extends the characterization of closed maximal ideals in the Fréchet algebras $F^p$, relating their structure to the topology of uniform convergence on compact subsets of the disk.

## Key findings

- Characterization of closed maximal ideals in $F^p$ using topology.
- Extension of previous results on algebraic structure.
- Connection between ideals and uniform convergence topology.

## Abstract

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ such that $ \lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ 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 $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty$, becomes a countably normed Fr\'{e}chet algebra. It is known that for every $p>1$, $F^p$ is the Fr\'{e}chet envelope of the Privalov space $N^p$. In this paper, we extend our study of [32] on the structure of maximal ideals in the algebras $F^p$ ($1<p<\infty$). Namely, the obtained characterization of closed maximal ideals in $F^p$ from [32] is extended here in terms of topology of uniform convergence on compact subsets of $\Bbb D$.