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.

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 $\Bbb D$.

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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 $\mathbb{D}$.

†† Mathematics Subject Classification (2010). 30H05, 46J15, 46J20.

1. Introduction, Preliminaries and Results

Let $\mathbb{D}$ denote the open unit disk in the complex plane and let $\mathbb{T}$ denote the boundary of $\mathbb{D}$. Let $L^{q}(\mathbb{T})$ $(0<q\leq\infty)$ be the familiar Lebesgue space on the unit circle $\mathbb{T}$.

The Privalov class $N^{p}$ $(1<p<\infty)$ is defined as the set of all holomorphic functions $f$ $f$ on $\mathbb{D}$ such that

[TABLE]

holds, where $\log^{+}|a|=\max\{\log|a|,0\}$. These classes were firstly considered by Privalov in [1, pages 93–10], where $N^{p}$ is denoted as $A_{q}$.

Recall the condition (1) with $p=1$ defines the Nevanlinna class $N$ of holomorphic functions in $\mathbb{D}$. The Smirnov class $N^{+}$ is the set of all functions $f$ holomorphic on $\mathbb{D}$ such that

[TABLE]

where $f^{*}$ is the boundary function of $f$ on $\mathbb{T}$, i.e.,

[TABLE]

is the radial limit of a function $f$ which exists for almost every $e^{i\theta}\in\mathbb{T}$. We denote by $H^{q}$ $(0<q\leq\infty)$ the classical Hardy space on $\mathbb{D}$.

The following inclusion relations hold true (see [2, 3, 4]):

[TABLE]

where the all containment relations are proper.

The study of the spaces $N^{p}$ $(1<p<\infty)$ was continued in 1977 by M. Stoll [5] (with the notation $(\log^{+}H)^{\alpha}$ in [5]). Further, the topological and functional properties of these spaces have been extensively studied by several authors (see [2], [6], [7], [8] and [9]–[23]).

M. Stoll [5, Theorem 4.2] showed that for every $p>1$ the space $N^{p}$ (with the noatation $(\log^{+}H)^{\alpha}$ in [5]) equipped with the topology given by the metric $d_{p}$ defined by

[TABLE]

becomes an $F$-algebra. This means that $N^{p}$ is an $F$-space (a complete metrizable topological vector space with the invariant metric) in which multiplication is continuous.

Observe that the function $d_{1}=d$ defined on the Smirnov class $N^{+}$ by (5) with $p=1$ induces the metric topology on $N^{+}$. N. Yanagihara [24] proved that under this topology, $N^{+}$ is an $F$-space.

In connection with the spaces $N^{p}$ $(1<p<\infty)$, Stoll [5] (see also [6] and [18, Section 3]) also studied the spaces $F^{q}$ $(0<q<\infty)$ (with the notation $F_{1/q}$ in [5]), consisting of those functions $f$ holomorphic on $\mathbb{D}$ such that

[TABLE]

where

[TABLE]

In this paper, we will need some Stoll’s results concerning the spaces $F^{q}$ only with $1<q<\infty$. Accordingly, in the sequel, we will assume that $q=p>1$ be a fixed real number.

Theorem 1 (see [5, Theorem 2.2]). Suppose that $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ is a holomorphic function on $\mathbb{D}$. Then the following statements are equivalent:

• (a)

$f\in F^{p}$;

• (b)

there exists a sequence $\{c_{n}\}_{n}$ of positive real numbers with $c_{n}\to 0$ such that

[TABLE]

• (c)

for any $c>0$,

[TABLE]

Remark. Note that in view of the equivalence (a)$\Leftrightarrow$(c) of Theorem 1, by (9) it is well defined the family of seminorms $\left\{\|\cdot\|_{p,c}\right\}_{c>0}$ on $F^{p}$.

Recall that a locally convex $F$-space is called a Fréchet space, and a Fréchet algebra is a Fréchet space that is an algebra in which multiplication is continuous. Stoll [5] also proved the following result.

Theorem 2 (see [5, Theorem 3.2]). The space $F^{q}$ $(0<q<\infty)$ equipped with the topology given by the family of seminorms $\left\{\|\cdot\|_{q,c}\right\}_{c>0}$ defined for $f\in F^{q}$ as

[TABLE]

is a countably normed Fréchet algebra.

Moreover, Stoll [5] defined the family of seminorms $\{|\|\cdot\|_{p,c}|\}_{c>0}$ on $F^{p}$ given as

[TABLE]

where

[TABLE]

For our purposes, we will also need the following result.

Theorem 3 (see [5, Proposition 3.1]). For each $c>0$, there is a constant $A$ depending only on $p$ and $c$, such that

[TABLE]

with $c_{1}=c^{p/(p+1)}$ and $c_{2}=(c/12)^{p/(p+1)}$.

Consequently, $\left\{\|\cdot\|_{p,c}\right\}_{c>0}$ and $\left\{|\|\cdot\||_{p,c}\right\}_{c>0}$ are equivalent families of seminorms.

It is known that the Privalov space $N^{p}$ $(1<p<\infty)$ is not locally convex (see [6, Theorem 4.2] and [14, Corollary]), and thus, $N^{p}$ is properly contained in $F^{p}$. Furthermore, $N^{p}$ is not locally bounded space (see [19, Theorem 1.1]). Moreover, Stoll proved ([5, Theorem 4.3]) that for every $p>1$, $N^{p}$ is a dense subspace of $F^{p}$ and the topology on $F^{p}$ equiped by the family of seminorms defined by (10) is weaker than the topology on $N^{p}$ induced by the metric $d_{p}$ defined by (5). Recall that Eoff proved [6, Theorem 4.2, the case $p>1$] that the space $F^{p}$ is the Fréchet envelope of $N^{p}$. For more information on the notion of Fréchet envelope, see [25, Theorem 1], [22, Section 1] and [26, Corollary 22.3, p. 210].

Remark. For $p=1$, the space $F_{1}$ has been denoted by $F^{+}$ and has been studied by N. Yanagihara in [27, 24]. It was proved in [27, 24] that $F^{+}$ is actually the containing Fréchet space for $N^{+}$, i.e., $N^{+}$ with the initial topology embeds densely into $F^{+}$, under the natural inclusion, and $F^{+}$ and the Smirnov class $N^{+}$ have the same topological duals.

Note that the space $F^{p}$ topologised by the family of seminorms $\left\{\|\cdot\|_{p,c}\right\}_{c>0}$ given by (10) is metrizable by the metric $\lambda_{p}$ defined as $\lambda_{p}(f,g)=\sum_{n=1}^{\infty}2^{-n}\frac{\|f-g\|_{p,1/n^{p/(p+1)}}}{1+\|f-g\|_{p,1/n^{p/(p+1)}}}$ with $f,g\in F^{p}.$

Since Privalov space $N^{p}$ and its Fréchet envelope $F^{p}$ $(1<p<\infty)$ are algebras, they can be also considered as rings with respect to the usual ring’s operations addition and multiplication. Note that these two operations are continuous on the spaces $N^{p}$ and $F^{p}$ in view of the facts that the spaces $N^{p}$ and $F^{p}$ are $F$-algebras.

Motivated by numerous results concerning the ideal structure of some spaces of holomorphic functions given in [28] [2], [12] and [29]-[36], related investigations on the spaces $N^{p}$ $(1<p<\infty)$ and their Fréchet envelopes were given in [2], [9], [12], [37], [18], [23] and [38]. Notice that a survey of these results was given in [39]. The $N^{p}$-analogue of the famous Beurling’s theorem for the Hardy spaces $H^{q}$ $(0<q<\infty)$ [29] was formulated and proved in [37]. Moreover, it was showed in [9, Theorem B]) that $N^{p}$ $(1<p<\infty)$ is a ring of Nevanlinna–Smirnov type in the sense of Mortini [36]. The structure of closed weakly dense ideals in $N^{p}$ was described in [18]. The ideal structure of $N^{p}$ and the multiplicative linear functionals on $N^{p}$ were studied in [2] and [23, Theorem ]. These results are similar to those obtained by Roberts and Stoll [30] for the Smirnov class $N^{+}$.

Motivated by results of Roberts and Stoll given in [31, Section 2] concerning a characterization of multiplicative linear functionals on $F^{+}$ and closed maximal ideals in $F^{+}$, in [32] the author of this paper proved the analogous results for the spaces $F^{p}$ $(1<p<\infty)$. These results are given by Proposition 5, Proposition 6, Theorem 7 and Theorem 8 in [32]. Namely, if $\lambda\in\mathbb{D}$ and $\gamma_{\lambda}$ is a functional on $F^{p}$ defined as

[TABLE]

for every $f\in F^{p}$, then by [32, Proposition 5], $\gamma_{\lambda}$ is a continuous multiplicative linear functional on $F^{p}$. Furthermore, if for any fixed $\lambda\in\mathbb{D}$ we define a set ${\mathcal{M}}_{\lambda}$ as

[TABLE]

then by [32, Proposition 6], ${\mathcal{M}}_{\lambda}$ is a closed maximal ideal in $F^{p}$. Moreover, if $\gamma$ is a nontrivial multiplicative linear functional on $F^{p}$, it is showed in [32, Theorem 7] that there exists $\lambda\in\mathbb{D}$ such that

[TABLE]

for every $f\in F^{p}$, and in addition, $\gamma$ is a continuous map. Finally, it is proved in [32, Theorem 8] that every closed maximal ideal in $F^{p}$ is of the form ${\mathcal{M}}_{\lambda}$ for some $\lambda\in\mathbb{D}$.

Motivated by a result of Igusa [40], here we extend Theorem 8 in [32] by the following result.

Theorem 4. Let $p>1$ and let ${\mathcal{M}}$ be a maximal ideal in $F^{p}$. Then the following statements about ${\mathcal{M}}$ are equivalent$:$

• (i)

The set ${\mathcal{M}}$ is closed with respect to the topology of uniform convergence on compact subsets of $\mathbb{D}$;**

• (ii)

$F^{p}/{\mathcal{M}}\cong{\bf C}$;**

• (iii)

There exists $\lambda\in\mathbb{D}$ such that ${\mathcal{M}}={\mathcal{M}}_{\lambda}$;**

• (iv)

${\mathcal{M}}$* is a closed ideal in $F^{p}$ with respect to the topology induced on $F^{p}$ by the family of seminorms $\{\|\cdot\|_{p,c}\}_{c>0}$ defined by $(9)$ and*

• (v)

${\mathcal{M}}$* is a closed ideal in $F^{p}$ with respect to the topology induced on $F^{p}$ by the family of seminorms $\{|\|\cdot\||_{p,c}\}_{c>0}$ defined by $(11)$.*

2. Proof of Theorem 4

In order to obtain a characterization of a maximal ideal space of the algebra $F^{p}$ with respect to the topology of uniform convergence on compact subsets of $\mathbb{D}$, we will need a result of Yanagihara [41, Lemma 10] concerning the topological algebras described below.

Let $A$ be a topological algebra over the field $\mathbb{C}$ with identity 1, locally convex and commutative. The topology of $A$ is defined by a countable family of seminorms $\{\|\cdot\|_{\alpha}\}_{\alpha\in S}$ ($S=\mathbb{N}$ or $S$ is a finite subset of the set of positive integers $\mathbb{N}$) for which $\|1\|_{\alpha}=1$ and for $a,b\in A$ there holds

[TABLE]

For an $\alpha\in S$ let $E_{\alpha}=\{a\in A:\,\|a\|_{\alpha}=0\}$. $E_{\alpha}$ is obviously an ideal of the algebra $A$. For any $a\in A$ we define the coset $\tilde{a}$ as $\tilde{a}=a+E_{\alpha}\in A/{E_{\alpha}}$. The quotient space $A/{E_{\alpha}}$ is a normed space with the assciated norm $\|\tilde{a}\|_{\alpha}=\|a\|_{\alpha}$, $\alpha\in S$. By (17) we have

[TABLE]

The completion of the space $A/{E_{\alpha}}$ with respect to the norm $\|\cdot\|_{\alpha}$ is denoted by $A_{\alpha}^{*}$. Then we have the following result.

Lemma 5. ([41, Lemma 10]). Let $A$ be a topological algebra described above. Then for any $f\in A$ there exists a complex number ${\lambda}_{f}$ depending on $f$ such that ${\lambda}_{f}-f=({\lambda}_{f}1-f)$ is not invertible element of the algebra $A$.

Proof of Theorem 4. (i)$\Rightarrow$(ii). Obviously, there holds $F^{p}/{{\mathcal{M}}}\supseteq\mathbb{C}$. For $f\in F^{p}$ denote

[TABLE]

Now we define the family of seminorms $\{\|\cdot\|_{r}\}$ $(0\leq r<1)$ in $F^{p}/{{\mathcal{M}}}$ as follows:

[TABLE]

Then obviously, we have

[TABLE]

By Lemma 5, to each $[f]\in F^{p}/{{\mathcal{M}}}$ corresponds a number $\lambda\in\mathbb{C}$ such that $\lambda-[f]$ is not invertible element of $F^{p}/{{\mathcal{M}}}$. However, by the maximality of the ideal ${\mathcal{M}}$, we conclude that $F^{p}/{{\mathcal{M}}}$ is a field, and thus, $\lambda-f$ must belong to ${\mathcal{M}}$, i.e., $\lambda\in[f]$. Hence, we obtain (cf. [32, Proof of Theorem 8] with application of Arens’s result [42])

[TABLE]

(ii)$\Rightarrow$(iii). Let $\lambda$ be in the coset $[z]\in F^{p}/{{\mathcal{M}}}$. Then $z-\lambda\in{\mathcal{M}}$, and hence, $\lambda\in\mathbb{D}$.

For each $f\in F^{p}$ we have

[TABLE]

It is easy to see that $A(z)\in F^{p}$, and thus, $f(z)-f(\lambda)\in{\mathcal{M}}$. Therefore, if $f(z)\in{\mathcal{M}}$, then $f(\lambda)\in{\mathcal{M}}$, whence it follows that $f(\lambda)=0$. This shows that ${\mathcal{M}}={\mathcal{M}}_{\lambda}$.

(iii)$\Rightarrow$(i). This implication is evident from the theorem of Hurwitz.

(iii)$\Rightarrow$(iv). This implication immediately follows from [32, Proposition 6].

(iv)$\Rightarrow$(iii). This implication immediately follows from [32, Theorem 8].

(iv)$\Leftrightarrow$(v). This equivalence immediately follows from Theorem 3. This completes the proof of Theorem 4.