Closed maximal ideals ideals in some Fréchet algebras of holomorphic functions

Abstract

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\mathbb{D}$ such that ${\lim }_{r\to 1}(1-r{)}^{1/q}{\log }^{+}{M}_{\infty }(r,f)=0,$ 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 $\Vert f{\Vert }_{q,c}:={\sum }_{n=0}^{\infty }|a_n|\exp \left(-c{n}^{1/(q+1)}\right)<\infty$, becomes a countably normed Fréchet algebra. It is known that for every $p>1$, $F^p$ is the Fréchet 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 $\mathbb{D}$.

6 pages, no figures, no tables. arXiv admin note: substantial text overlap with arXiv:1812.11091

References in corpus (1)

• A characterization of some prime ideals in certain $F$-algebras of holomorphic functions