Characterizations of Ideals in Intermediate 𝐶-Rings 𝐴(𝑋) via the 𝐴-Compactifications of 𝑋

Abstract

Let 𝑋 be a completely regular topological space. An intermediate ring is a ring 𝐴(𝑋) of continuous functions satisfying 𝐶^∗(𝑋)⊆𝐴(𝑋)⊆𝐶(𝑋). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences 𝒵_𝐴 and ℨ_𝐴 are defined between ideals in 𝐴(𝑋) and 𝑧-filters on 𝑋, and it is shown that these extend the well-known correspondences studied separately for 𝐶^∗(𝑋) and 𝐶(𝑋), respectively, to any intermediate ring. Moreover, the inverse map 𝒵^←_𝐴 sets up a one-one correspondence between the maximal ideals of 𝐴(𝑋) and the 𝑧-ultrafilters on 𝑋. In this paper, we define a function 𝔎_𝐴 that, in the case that 𝐴(𝑋) is a 𝐶-ring, describes ℨ_𝐴 in terms of extensions of functions to realcompactifications of 𝑋. For such rings, we show that ℨ^←_𝐴 maps 𝑧-filters to ideals. We also give a characterization of the maximal ideals in 𝐴(𝑋) that generalize the Gelfand-Kolmogorov theorem from 𝐶(𝑋) to 𝐴(𝑋).

1. Introduction

Let 𝑋 be a completely regular space and 𝐴(𝑋) an intermediate ring of continuous real-valued functions; that is, 𝐶^∗(𝑋)⊆𝐴(𝑋)⊆𝐶(𝑋). It is well known that there is a natural correspondence 𝐙 between ideals of 𝐶(𝑋) and 𝑧-filters on X as described in [1, pages 26-27]. Such a correspondence 𝐄 also exists for 𝐶^∗(𝑋) [1, Problem 2L]. In [2], a correspondence 𝒵_𝐴 between the ideals of any 𝐴(𝑋) and the 𝑧-filters on 𝑋 was introduced, and its properties were further investigated in [3–5]. In [6], another correspondence ℨ_𝐴 between ideals of any 𝐴(𝑋) and 𝑧-filters on 𝑋 is introduced. It is shown in [6] that the correspondences 𝒵_𝐴 and ℨ_𝐴 extend the correspondences 𝐄 and 𝐙 from 𝐶^∗(𝑋) and 𝐶(𝑋), respectively, to all intermediate rings, and an explicit formula is stated that relates the two correspondences. In this paper, we give a characterization (Definition 3 and Theorem 6) of the correspondence ℨ_𝐴 for intermediate 𝐶-rings 𝐴(𝑋) in terms of the 𝐴-compactifications of 𝑋 introduced in [7]. In this setting, we show (Theorem 14) that the inverse map ℨ^←_𝐴 of the set map ℨ_𝐴 maps ideals in 𝐴(𝑋) to 𝑧-filters on 𝑋. We also give a characterization of the maximal ideals in 𝐴(𝑋). This characterization generalizes from 𝐶(𝑋) to 𝐴(𝑋) the Gelfand-Kolmogorov theorem (Theorem 8). We follow the notation in [1, 6].

2. Preliminaries

For convenience we state some of the definitions and results needed in this paper.

Following the notation in [1], we set 𝐙(𝑋)[𝐶(𝑋=𝐙𝑓(𝑋=𝐙)]∣𝑓∈𝐶) (1) to be the collection of the zero sets 𝐙(𝑓)=𝑓⁻¹({0}) of all functions 𝑓∈𝐶(𝑋). In this paper, we generally work with functions 𝑓 on a fixed set 𝑋, as well as some extensions 𝑔 of 𝑓 to larger domains. As expected, 𝐙(𝑔) then denotes the zero set of 𝑔 on the larger domain.

A 𝑧-filter (𝑧-ultrafilter, resp.) on 𝑋 is the intersection of 𝐙(𝑋) with a filter (ultrafilter, resp.) on 𝑋. The kernel 𝑘 of a set 𝒮 of 𝑧-ultrafilters is defined by 𝑘(𝒮)={𝒰∣𝒰∈𝒮}. (2)

One can verify that the kernel of a set of 𝑧-ultrafilters is a 𝑧-filter. The hull ℎ of a 𝑧-filter ℱ is defined by ℎ(ℱ)={𝒰∣𝒰isa𝑧-ultrafilteron𝑋,𝒰⊇ℱ}. (3) Given a set 𝐸⊆𝑋, let ⟨𝐸⟩ denote the set of all zero sets 𝐹⊆𝑋, such that 𝐸⊆𝐹.

Given any two functions 𝑓 and 𝑔 on a set 𝑋 and 𝐻⊆𝑋, we write 𝑓≡𝑔 on 𝐻 if 𝑓(𝑥)=𝑔(𝑥) for all 𝑥∈𝐻. For each intermediate ring 𝐴(𝑋) of continuous functions, each noninvertible 𝑓∈𝐴(𝑋), and each 𝐻⊆𝑋, we say that 𝑓 is 𝐻-regular in 𝐴(𝑋) if there exists 𝑔∈𝐴(𝑋) such that 𝑓𝑔≡1 on 𝐻. We just say 𝑓 is 𝐻-regular if 𝐴(𝑋) is understood by context. For any set 𝐸⊆𝑋, let 𝐸^𝑐 be the complement of 𝐸 in 𝑋.

For each 𝑓∈𝐴(𝑋), set 𝒵_𝐴𝑓=𝐸∈𝐙(𝑋)∣𝑓is𝐸^𝑐-regular,ℨ_𝐴𝑓=𝐸∈𝐙(𝑋)∣𝑓is𝐻-regularforeachzero-set𝐻⊆𝐸^𝑐. (4) For an ideal 𝐼⊂𝐴(𝑋), we set 𝒵_𝐴[𝐼]=⋃{𝒵_𝐴(𝑓)∣𝑓∈𝐼} and ℨ_𝐴[𝐼]=⋃{ℨ_𝐴(𝑓)∣𝑓∈𝐼}. Several properties of 𝒵_𝐴 and ℨ_𝐴 are proved in [3, 5, 6]. In particular, we have the following lemma, which we state here for convenience.

Lemma 1. Let 𝐴(𝑋) be an intermediate ring. Then the following hold.(a)For any noninvertible 𝑓∈𝐴(𝑋), both 𝒵_𝐴(𝑓) and ℨ_𝐴(𝑓) are 𝑧-filters on 𝑋. If 𝑓 is invertible, then 𝒵_𝐴(𝑓)=ℨ_𝐴(𝑓)=𝐙[𝑋], the set of all zero sets in 𝑋. (b)For an ideal 𝐼⊂𝐴(𝑋), both 𝒵_𝐴[𝐼] and ℨ_𝐴[𝐼] are 𝑧-filters on 𝑋. (c)For any 𝑓∈𝐴(𝑋), we have ℨ_𝐴(𝑓)=𝑘ℎ𝒵_𝐴(𝑓). (d)If ℱ is a 𝑧-filter on 𝑋, then 𝒵_𝐴(𝑓)⊆ℱ if and only if lim_ℱ𝑓ℎ=0 for all ℎ∈𝐴(𝑋).

Item (a) is from [2, 6], Item (b) is from [3, 6], Item (c) is from [6], and Item (d) is from [2].

The Stone-Čech compactification of 𝑋, denoted 𝛽𝑋, is any topological space homeomorphic to the space of 𝑧-ultrafilters on 𝑋 topologized with the hull-kernel closure operator as follows: the closure of any set 𝑈 of 𝑧-ultrafilters is ℎ𝑘(𝑈). Throughout this paper, we will in particular take 𝛽𝑋 to consist of a superset of 𝑋, whose points, which we denote 𝑝,𝑞,…, can be viewed as indices of 𝑧-ultrafilters on 𝑋. Let 𝒰 be the map which associates every 𝑝∈𝛽𝑋 with a 𝑧-ultrafilter 𝒰_𝑝: 𝑝⟼^𝒰𝒰_𝑝, (5) such that for each 𝑝∈𝑋, 𝒰_𝑝=⟨{𝑝}⟩ is the fixed 𝑧-ultrafilter containing 𝑝, and for each 𝑝∈𝛽𝑋⧵𝑋,𝒰_𝑝 is a unique free 𝑧-ultrafilter, such that 𝒰 is a one-to-one correspondence between 𝛽𝑋 and the set of 𝑧-ultrafilters on 𝑋. The topology on 𝛽𝑋 is defined in such a way that the map 𝒰 is a homeomorphism. Making use of the fact that the zero sets form a base for the collection of closed sets [1, page 38], one can check that 𝒰 maps 𝑋 homeomorphically onto the subspace of fixed 𝑧-ultrafilters, and hence 𝑋 is a subspace of 𝛽𝑋.

A 𝑧-filter ℱ on 𝑋 is called 𝐴-stable if for every 𝑓∈𝐴(𝑋) there exists a set in ℱ on which 𝑓 is bounded (see [7, 8]). Following [7], for each 𝐴(𝑋) we define the 𝐴-compactification 𝜐_𝐴𝑋 of 𝑋 as the subspace of 𝛽𝑋 where 𝜐_𝐴𝑋=𝑝∈𝛽𝑋∣𝒰_𝑝is𝐴-stable. (6) From [5, Theorem 4.6], it holds that 𝜐_𝐴𝑋 is a realcompactification of 𝑋. Note that if 𝐴(𝑋)=𝐶^∗(𝑋), then 𝜐_𝐴𝑋=𝛽𝑋. If 𝐴(𝑋)=𝐶(𝑋), then 𝜐_𝐴𝑋=𝜐𝑋 is the Hewitt realcompactification.

To refine our understanding of the topology on 𝜐_𝐴𝑋, we define the 𝐴-stable hull ℎ^𝐴 of a 𝑧-filter ℱ by ℎ^𝐴(ℱ)={𝒰∣𝒰isan𝐴-stable𝑧-ultrafilteron𝑋,𝒰⊇ℱ}. (7) It is immediate from the definition of the subspace topology that 𝜐_𝐴𝑋 is homeomorphic (via 𝒰∣_𝜐𝐴𝑋) to the hull-kernel topology restricted to 𝐴-stable 𝑧-ultrafilters, that is, the topology with the following closure operator: the closure of any set 𝑈 is ℎ^𝐴𝑘(𝑈). It follows that 𝑝∈cl_𝜐𝐴𝑋𝐸iff𝐸∈𝒰_𝑝. (8)

From [7], we have that the space 𝜐_𝐴𝑋 consists of the points of 𝛽𝑋 to which every function 𝑓∈𝐴(𝑋) can be continuously extended. We denote the extension of 𝑓 to 𝜐_𝐴𝑋 by 𝑓^𝜐𝐴. From [7, Theorem 9], we have the value of 𝑓^𝜐𝐴 at a point 𝑝∈𝜐_𝐴𝑋 is given by 𝑓^𝜐𝐴𝑝=lim_𝒰𝑝𝑓. (9)

In [7], a ring 𝐴(𝑋) of continuous functions is called a 𝐶-ring if there is a completely regular space 𝑌 such that 𝐴(𝑋) is isomorphic to 𝐶(𝑌). Clearly 𝐶(𝑋) and 𝐶^∗(𝑋) are 𝐶-rings (with 𝐶^∗(𝑋) isomorphic to 𝐶(𝛽𝑋)). We use the following result from [7, Theorem 7].

Lemma 2. Let 𝐴(𝑋) be an intermediate ring. Then the following hold. (a)𝐴(𝑋) is a 𝐶-ring if and only if 𝐴(𝑋) is isomorphic to 𝐶(𝜐_𝐴𝑋).(b)If 𝐴(𝑋) is a 𝐶-ring, then 𝑓∈𝐴(𝑋) is invertible in 𝐴(𝑋) if and only if 𝐙(𝑓^𝜐𝐴)=∅.

In addition, it is shown in [5, Theorem 4.7] that there is a bijective correspondence between the realcompactifications of 𝑋 and the 𝐶-rings on 𝑋.

3. Characterizations Using Realcompactifications

In this section, we utilize the realcompactifications of 𝑋 to provide a new description of the function ℨ_𝐴 and of maximal ideals of 𝐴(𝑋), when 𝐴(𝑋) is an intermediate 𝐶-ring. The new description of the maximal ideals of 𝐴(𝑋) generalizes the Gelfand-Kolmogorov theorem [1, page 102] for 𝐶(𝑋) to all intermediate 𝐶-rings.

3.1. A New Description of ℨ_𝐴 for 𝐶-Rings

While the definition of ℨ_𝐴 is essentially algebraic (using the property of local invertibility), we now define a function 𝔎_𝐴 that provides a “topological description" (using realcompactifications of 𝑋) of a mapping from ideals of an intermediate 𝐶-ring 𝐴(𝑋) to 𝑧-filters on 𝑋, and we will show (Theorem 6) that 𝔎_𝐴 coincides with ℨ_𝐴 when 𝐴(𝑋) is an intermediate 𝐶-ring.

Definition 3. Let 𝐴(𝑋) be an intermediate ring and 𝑓∈𝐴(𝑋). We set 𝔎_𝐴𝑓=𝐸∈𝐙(𝑋)∣𝐙𝑓^𝜐𝐴⊆cl_𝜐𝐴𝑋𝐸. (10) For an ideal 𝐼⊂𝐴(𝑋), we set 𝔎_𝐴[𝐼]=⋃{𝔎_𝐴(𝑓)∣𝑓∈𝐼}.

That 𝔎_𝐴 is indeed a mapping from ideals of an intermediate 𝐶-ring 𝐴(𝑋) to 𝑧-filters on 𝑋 will follow from our main result of the section that establishes that 𝔎_𝐴=ℨ_𝐴 when 𝐴(𝑋) is an intermediate 𝐶-ring. We also show that 𝔎_𝐴 does not necessarily map ideals in 𝐴(𝑋) to 𝑧-filters on 𝑋 when 𝐴(𝑋) is not an intermediate 𝐶-ring (Example 7).

First, to illustrate some connections that motivated the development of 𝔎_𝐴, let us now observe a similarity between 𝔎_𝐴 and a simple function 𝐙 that we define in terms of 𝐙. We define 𝐙_𝑋 on 𝐶(𝑋) by 𝐙_𝑋∶𝑓⟼𝐙𝑓=𝐸∈𝐙(𝑋)∣𝐙𝑓⊆𝐸. (11) We drop the subscript 𝑋 when it is understood by context. It is easy to see that given an ideal 𝐼,𝐙[𝐼]=𝐙[𝐼].

We now turn our attention to 𝐶^∗(𝑋) and we note that 𝐶^∗(𝑋) is an intermediate 𝐶-ring, as 𝐶^∗(𝑋) is isomorphic to 𝐶(𝛽𝑋). In light of this, we have for 𝑓∈𝐶(𝑋) that 𝐙_𝛽𝑋∶𝑓^𝛽⟼𝐙𝑓^𝛽=𝐸_𝛽∈𝐙𝛽𝑋∣𝐙𝑓^𝛽⊆𝐸_𝛽. (12) Ultimately, we would like to find a map defined on 𝐶^∗(𝑋) itself rather than 𝐶(𝛽𝑋) and that maps to 𝑧-filters on 𝑋 rather than 𝛽𝑋 (it is possible that a 𝑧-filter on 𝛽𝑋 may contain a set that does not meet 𝑋; for example, 𝛽ℕ⧵ℕ is the zero set of the function 𝑓^𝛽ℕ, where 𝑓(𝑛)=1/𝑛). Observe that 𝐸_𝛽∈𝐙𝛽𝑋∣𝐙𝑓^𝛽⊆𝐸_𝛽=𝐙𝑓^𝛽=cl_𝛽𝑋𝐸∣𝐸∈𝐙(𝑋),𝐙𝑓^𝛽⊆cl_𝛽𝑋𝐸, (13) where the first equality is immediate from the fact that 𝐙(𝑓^𝛽)∈𝐙(𝛽𝑋), and the second equality holds because the sets of the form cl_𝛽𝑋𝐸 are a base for the closed sets in 𝛽𝑋 (see [1, page 94]). This motivates the following definition that relates 𝐙(𝑓^𝛽) to 𝑋: 𝔎_𝐶^∗∶𝑓⟼𝐸∈𝐙(𝑋)∣𝐙𝑓^𝛽⊆cl_𝛽𝑋𝐸. (14) We then generalize 𝔎_𝐶^∗ to all intermediate rings 𝐴(𝑋) to arrive at Definition 3.

In order to prove our main theorem that 𝔎_𝐴=ℨ_𝐴 whenever 𝐴 is an intermediate 𝐶-ring, we need some lemmas. We first show that the zero set of 𝑓^𝜐𝐴, viewed as a set of 𝑧-ultrafilters, is the 𝐴-stable hull of 𝒵_𝐴(𝑓).

Lemma 4. If 𝐴(𝑋) is an intermediate ring of continuous functions, 𝐙𝑓^𝜐𝐴≡ℎ^𝐴𝒵_𝐴𝑓, (15) where the symbol ≡ indicates that one set is the homeomorphic image of the other. In particular, if 𝑓∈𝐴(𝑋), then 𝑝∈𝐙(𝑓^𝜐𝐴) if and only if 𝒰_𝑝∈ℎ^𝐴(𝒵_𝐴(𝑓)).

Proof. We observe that the following are equivalent:(i)𝑝∈𝐙(𝑓^𝜐𝐴), (ii)𝒰_𝑝 is 𝐴-stable and lim_𝒰𝑝𝑓=0, (iii)𝒰_𝑝 is 𝐴-stable and lim_𝒰𝑝𝑓ℎ=0 for all ℎ∈𝐴(𝑋), (iv)𝒰_𝑝 is 𝐴-stable and 𝒰_𝑝⊇𝒵_𝐴(𝑓), that is, 𝒰_𝑝∈ℎ^𝐴(𝒵_𝐴(𝑓)). The equivalence (i)⇔(ii) follows from (6) and (9). The equivalence (ii)⇔(iii) follows from the fact that 𝒰_𝑝 is 𝐴-stable, and hence ℎ is bounded on some set in 𝒰_𝑝. The equivalence (iii)⇔(iv) follows from Lemma 1(d).

We use the notation 𝑘ℎℱ to denote the kernel of the hull of the 𝑧-filter ℱ, that is, the intersection of the set of 𝑧-ultrafilters containing ℱ.

Lemma 5. Let 𝐴(𝑋) be an intermediate 𝐶-ring and 𝑓∈𝐴(𝑋). Then 𝑘ℎ𝒵_𝐴𝑓=𝑘ℎ^𝐴𝒵_𝐴𝑓. (16)

Proof. Without loss of generality, we may assume that 𝑓≥0 (because 𝒵_𝐴(𝑓)=𝒵_𝐴(𝑓²)). Of course 𝑘ℎ𝒵_𝐴(𝑓)⊆𝑘ℎ^𝐴𝒵_𝐴(𝑓). For the other containment, suppose 𝐸∈𝑘ℎ^𝐴𝒵_𝐴(𝑓), so 𝐸 belongs to every 𝐴-stable 𝑧-ultrafilter containing 𝒵_𝐴(𝑓). We first note that by Lemma 4, 𝒰_𝑝∈ℎ^𝐴𝒵_𝐴(𝑓) if and only if 𝑝∈𝐙(𝑓^𝜐𝐴). In other words, for any 𝐸∈𝐙[𝑋], we have 𝐸∈𝑘ℎ^𝐴𝒵_𝐴(𝑓) if and only if 𝐸∈𝒰_𝑝 for every 𝑝 in 𝐙(𝑓^𝜐𝐴) if and only if cl_𝜐𝐴𝑋𝐸⊇𝐙(𝑓^𝜐𝐴). Now suppose 𝒰_𝑞 is any 𝑧-ultrafilter (𝐴-stable or not) containing 𝒵_𝐴(𝑓). We show that 𝐸 belongs to 𝒰_𝑞. If not, then there exists 𝐹∈𝒰_𝑞 such that 𝐸∩𝐹=∅. (17) Since 𝐸 and 𝐹 are disjoint zero sets in 𝑋, they are completely separated (see [1, page 17]). So there is a continuous function ℎ that takes the value 1 on 𝐸 and 0 on 𝐹 and 0≤ℎ≤1, so ℎ∈𝐴(𝑋). Consider the function 𝑘=𝑓+ℎ∈𝐴(𝑋). Note that 𝑘(𝑥)=𝑓(𝑥) for 𝑥∈𝐹. Also, since 𝑘≥1 on 𝐸 and since cl_𝜐𝐴𝑋𝐸⊇𝐙(𝑓^𝜐𝐴), it follows that 𝐙(𝑘^𝜐𝐴)=∅. By Lemma 2(b), 𝑘 is invertible, and as 𝑘≡𝑓 on 𝐹, it follows that 𝑘⁻¹𝑓≡1 on 𝐹. Since 𝐹∈𝒰_𝑞, it also follows that lim_𝒰𝑞𝑘⁻¹𝑓≠0, but this contradicts the fact that 𝒰_𝑞⊇𝒵_𝐴(𝑓) because of Lemma 1(d). This contradiction stems from the assumption that 𝐸 does not belong to 𝒰_𝑞. So 𝐸∈𝒰_𝑞 for every 𝒰_𝑞⊇𝒵_𝐴(𝑓), that is, 𝐸∈𝑘ℎ𝒵_𝐴(𝑓).

Recall that for any intermediate ring 𝐴(𝑋), Lemma 1(c) gives the relationship ℨ_𝐴(𝑓)=𝑘ℎ𝒵_𝐴(𝑓). For intermediate 𝐶-rings, we can characterize ℨ_𝐴 topologically as 𝔎_𝐴.

Theorem 6. Let 𝐴(𝑋) be an intermediate 𝐶-ring and 𝑓∈𝐴(𝑋). Then ℨ_𝐴(𝑓)=𝔎_𝐴(𝑓).

Proof. Suppose 𝐸∈ℨ_𝐴(𝑓). By Lemma 1(c), 𝐸∈𝑘ℎ𝒵_𝐴(𝑓). We show that in this case 𝑝∈𝐙(𝑓^𝜐𝐴) implies 𝑝∈cl_𝜐𝐴𝑋𝐸. Now if 𝑝∈𝐙(𝑓^𝜐𝐴), then by Lemma 4, 𝒰_𝑝∈ℎ^𝐴(𝒵_𝐴(𝑓)). Since 𝐸∈𝑘ℎ𝒵_𝐴(𝑓), it follows that 𝐸∈𝒰_𝑝, and hence by (8), we have that 𝑝∈𝑐𝑙_𝜐𝐴𝑋𝐸.
For the other containment, suppose 𝐸∈𝔎_𝐴(𝑓), that is, 𝐙(𝑓^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝐸. We show that 𝐸 belongs to every 𝐴-stable 𝑧-ultrafilter containing 𝒵_𝐴(𝑓). First, let 𝒰_𝑝 be an 𝐴-stable 𝑧-ultrafilter containing 𝒵_𝐴(𝑓). Then by Lemma 4, 𝑝∈𝐙(𝑓^𝜐𝐴), so by hypothesis, 𝑝∈cl_𝜐𝐴𝑋𝐸, and hence 𝐸∈𝒰_𝑝. Thus 𝐸 belongs to every 𝐴-stable 𝑧-ultrafilter containing 𝒵_𝐴(𝑓); in other words, 𝐸∈𝑘ℎ^𝐴𝒵_𝐴(𝑓). Since 𝐴(𝑋) is an intermediate 𝐶-ring, we can apply Lemma 5 so that 𝐸∈𝑘ℎ𝒵_𝐴(𝑓), and hence by Lemma 1(c), 𝐸∈ℨ_𝐴(𝑓).

The theorem does not hold if the assumption that 𝐴(𝑋) being an intermediate 𝐶-ring is removed. In fact, 𝔎_𝐴 need not map ideals in 𝐴(𝑋) to 𝑧-filters on 𝑋 when 𝐴(𝑋) is not a 𝐶-ring, as the following example shows.

Example 7. Let 𝑋=[1,∞). Let 𝐴(𝑋) be the smallest ring of continuous functions containing both 𝐶^∗(𝑋) and 𝑓(𝑥)=𝑥. Then each function ℎ∈𝐴(𝑋) has the form ℎ(𝑥)=∑^𝑛_𝑘=0𝑓_𝑘(𝑥)𝑥^𝑘 for 𝑓_𝑘∈𝐶^∗(𝑋). Note that any function ℎ∈𝐴(𝑋) is therefore bounded by some function of the form 𝑀𝑥^𝑛 for 𝑀∈ℝ and 𝑛∈ℕ, where 𝑀 is 𝑛+1 times a common bound for all of 𝑓_𝑘. We now observe that 𝑒^𝑥∉𝐴(𝑋), since 𝑒^𝑥 cannot be bounded by any function of the form 𝑀𝑥^𝑛 for 𝑀∈ℝ and 𝑛∈ℕ. Let 𝑔=𝑒^−𝑥. Then 𝑔 is in 𝐴(𝑋) but is not invertible in 𝐴(𝑋). Thus 𝐴(𝑋) and 𝐶(𝑋) are not isomorphic.
Since every set of a free 𝑧-ultrafilter on 𝑋=[1,∞) must be unbounded, the identity function 𝑓 is not bounded on any such set either. Hence any free 𝑧-ultrafilter on 𝑋 is not 𝐴-stable, and 𝜐_𝐴(𝑋)=𝑋. Then by Lemma 2(a), 𝐴(𝑋) is not a 𝐶-ring, since 𝐶(𝜐_𝐴(𝑋))=𝐶(𝑋) and 𝐶(𝑋)≠𝐴(𝑋).
Finally, since 𝑔^𝜐𝐴=𝑔 and 𝐙(𝑔)=∅, the set 𝔎_𝐴(𝑔)={𝐸∈𝐙(𝑋)∣𝐙(𝑔^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝐸} consists of all zero sets of 𝑋 (hence is not a 𝑧-filter), while ℨ_𝐴(𝑔) is the 𝑧-filter consisting of all zero sets in 𝑋 whose complement in 𝑋 has an upper bound. Thus 𝔎_𝐴(𝑔)≠ℨ_𝐴(𝑔), which is in contrast to the conclusion of Theorem 6.

We leave open the question as to precisely what rings 𝐴(𝑋) are such that 𝔎_𝐴=ℨ_𝐴. We also leave open the question as to whether there exists a ring 𝐴(𝑋) such that 𝔎_𝐴 does map ideals in 𝐴(𝑋) to 𝑧-filters on 𝑋, but 𝔎_𝐴≠ℨ_𝐴.

3.2. Characterizing Maximal Ideals in 𝐶-Rings

The following characterization for maximal ideals in 𝐴(𝑋) is proved in [6]. Every maximal ideal in 𝐴(𝑋) is of the form 𝑀^𝑝_𝐴=𝑓∈𝐴(𝑋)∣ℨ_𝐴𝑓⊆𝒰_𝑝, (18) for 𝑝∈𝛽𝑋. By Lemma 1(c), ℨ_𝐴(𝑓)⊆𝒰_𝑝 whenever 𝒵_𝐴(𝑓)⊆𝒰_𝑝, and hence by (9) and Lemma 1(d), we have 𝑀^𝑝_𝐶∗=𝑓∈𝐶^∗(𝑋)∣𝑓^𝛽𝑝=0. (19)

Since ℨ_𝐶(𝑓)=⟨𝐙(𝑓)⟩, we have by (8) that 𝑀^𝑝_𝐶=𝑓∈𝐶(𝑋)∣𝐙𝑓⊆𝒰_𝑝=𝑓∈𝐶(𝑋)∣𝑝∈cl_𝛽𝑋𝐙𝑓. (20)

The characterizations in (19) and (20) that we obtained from (18) agree with those given in [1, pages 101-102]; the latter characterization of 𝑀^𝑝_𝐶 is called the Gelfand-Kolmogorov theorem [1, page 102]. The following theorem provides a characterization of 𝑀^𝑝_𝐴 that we see extends both (20) and (19) to all intermediate 𝐶-rings 𝐴(𝑋).

Theorem 8. Let 𝐴(𝑋) be an intermediate 𝐶-ring. Then each maximal ideal in 𝐴(𝑋) is of the form 𝑀^𝑝_𝐴=𝑓∈𝐴(𝑋)∶𝑝∈cl_𝛽𝑋𝐙𝑓^𝜐𝐴, (21) where 𝑝∈𝛽𝑋.

Proof. 𝑓∈𝑀^𝑝_𝐴 if and only if 𝒵_𝐴(𝑓)⊆𝒰_𝑝 if and only if 𝒰_𝑝∈ℎ(𝒵_𝐴(𝑓)). By identifying a set in 𝛽𝑋 with its image under 𝒰, it follows from the definition of closure in 𝛽𝑋, Lemmas 4 and 5, that cl_𝛽𝑋𝐙𝑓^𝜐𝐴=ℎ𝑘𝐙𝑓^𝜐𝐴=ℎ𝑘ℎ^𝐴𝒵_𝐴𝑓=ℎ𝑘ℎ𝒵_𝐴𝑓=ℎ𝒵_𝐴𝑓. (22) The result now follows from the fact that ℎ(𝒵_𝐴(𝑓))=cl_𝛽𝑋𝐙(𝑓^𝜐𝐴).

We now verify that this theorem generalizes the Gelfand-Kolmogorov Theorem [1, page 102] to intermediate 𝐶-rings. If 𝐴(𝑋)=𝐶(𝑋), then 𝜐_𝐴𝑋=𝜐𝑋, the Hewitt realcompactification of 𝑋. Now using the fact that cl_𝜐𝑋𝐙(𝑓)=𝐙(𝑓^𝜐) [1, page 118] and Theorem 8, we have 𝑀^𝑝_𝐶=𝑓∈𝐶(𝑋)∣𝑝∈cl_𝛽𝑋𝐙𝑓^𝜐=𝑓∈𝐶(𝑋)∣𝑝∈cl_𝛽𝑋𝑐𝑙_𝜐𝑋𝐙𝑓=𝑓∈𝐶(𝑋)∣𝑝∈cl_𝛽𝑋𝐙𝑓. (23)

For the case where 𝐴(𝑋)=𝐶^∗(𝑋), we have 𝑀^𝑝_𝐶∗=𝑓∈𝐶^∗(𝑋)∣𝑝∈cl_𝛽𝑋𝐙𝑓^𝛽=𝑓∈𝐶^∗(𝑋)∣𝑓^𝛽𝑝=0. (24) So Theorem 8 simultaneously generalizes the results of (20) for 𝐶(𝑋) and (19) for 𝐶^∗(𝑋) to all intermediate 𝐶-rings.

4. The Map ℨ^←_𝐴 and Ideals in 𝐴(𝑋)

Recall that ℨ_𝐴 maps ideals to 𝑧-filters (Lemma 1(a)). Here we show that for an intermediate 𝐶-ring 𝐴(𝑋), the inverse map ℨ^←_𝐴, defined by ℨ^←_𝐴[ℱ]=𝑓∈𝐴(𝑋)∶ℨ_𝐴𝑓⊆ℱ (25) maps 𝑧-filters on 𝑋 to ideals in 𝐴(𝑋) (Theorem 14). The corresponding result for the maps 𝐙 (for 𝐶(𝑋)) and 𝒵_𝐴 (for any intermediate ring 𝐴(𝑋)) is proved, respectively, in [1] and [3]. Our proof for ℨ_𝐴 makes use of Theorem 6.

We need some lemmas concerning meets and joins on the lattice of 𝑧-filters. Recall that ℱ∨𝒢 is the smallest 𝑧-filter containing both the 𝑧-filters ℱ and 𝒢. Similarly, ℱ∧𝒢 is the largest 𝑧-filter contained in both the 𝑧-filters ℱ and 𝒢.

The following lemma is from [6].

Lemma 9. Let 𝐴(𝑋) be an intermediate ring and 𝑓,𝑔∈𝐴(𝑋).(a)𝒵_𝐴(𝑓𝑔)=𝒵_𝐴(𝑓)∧𝒵_𝐴(𝑔). (b)𝒵_𝐴(𝑓+𝑔)⊆𝒵_𝐴(𝑓)∨𝒵_𝐴(𝑔). (c)If 𝑓,𝑔≥0, then 𝒵_𝐴(𝑓+𝑔)=𝒵_𝐴(𝑓)∨𝒵_𝐴(𝑔).

The following is a special case of [6, Lemma 4.2(a)].

Lemma 10. For all 𝑓,𝑔∈𝐴(𝑋), 𝑘ℎ(𝒵_𝐴(𝑓)∧𝒵_𝐴(𝑔))=𝑘ℎ(𝒵_𝐴(𝑓))∧𝑘ℎ(𝒵_𝐴(𝑔)).

To obtain the analog of Lemma 10 for joins, we need the following lemma.

Lemma 11. If 𝐸 is a zero set in 𝑋 and if 𝐙(𝑓^𝜐𝐴)∩𝐙(𝑔^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝐸 then there exist zero sets 𝑍₁ and 𝑍₂ in 𝑋 such that 𝐙(𝑓^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝑍₁, 𝐙(𝑔^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝑍₂, and 𝑍₁∩𝑍₂=𝐸.

Proof. Let 𝐻=cl_𝜐𝐴𝑋𝐸 and let 𝑌=𝜐_𝐴𝑋⧵𝐻. Now, the sets 𝐙(𝑓^𝜐𝐴)∩𝑌 and 𝐙(𝑔^𝜐𝐴)∩𝑌 are disjoint zero sets in 𝑌, so are contained in disjoint zero set neighborhoods 𝑊₁ and 𝑊₂ in 𝑌. Moreover, since 𝑊₁ is a neighborhood of 𝐙(𝑓^𝜐𝐴)∩𝑌, it follows that 𝐙𝑓^𝜐𝐴∩𝑌⊆cl_𝜐𝐴𝑋𝑊₁=cl_𝜐𝐴𝑋𝑊₁∩𝑋. (26) Similarly 𝐙(𝑔^𝜐𝐴)∩𝑌⊆cl_𝜐𝐴𝑋(𝑊₂∩𝑋). Let 𝑍₁=𝑊₁∩𝑋∪𝐸,𝑍₂=𝑊₂∩𝑋∪𝐸. (27) Since 𝑊₁∩𝑋 is a zero set in 𝑋⧵𝐸, it follows that 𝑍₁ is a zero set in 𝑋 by the fact that the cozero set of a cozero set is a cozero set [9, Proposition 1.1]. Similarly, 𝑍₂ is a zero set in 𝑋. Also, 𝐙(𝑓^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝑍₁, because 𝐙𝑓^𝜐𝐴=𝐙𝑓^𝜐𝐴∩𝑌∪𝐙𝑓^𝜐𝐴⊆∩𝐻cl_𝜐𝐴𝑋𝑊₁∩𝑋∪cl_𝜐𝐴𝑋𝐸=cl_𝜐𝐴𝑋𝑊₁=∩𝑋∪𝐸cl_𝜐𝐴𝑋𝑍₁. (28) Similarly, 𝐙(𝑔^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝑍₂. Finally, 𝑍₁∩𝑍₂=𝐸 because 𝑊₁∩𝑋 and 𝑊₂∩𝑋 are disjoint.

The next lemma shows that the kernel-hull operation distributes over the join operation on the lattice of 𝑧-filters.

Theorem 12. If 𝐴(𝑋) is an intermediate 𝐶-ring, then 𝑘ℎ𝒵_𝐴𝑓∨𝒵_𝐴𝑔=𝑘ℎ𝒵_𝐴𝑓∨𝑘ℎ𝒵_𝐴𝑔. (29)

Proof. The containment 𝑘ℎ(𝒵_𝐴(𝑓)∨𝒵_𝐴(𝑔))⊇𝑘ℎ(𝒵_𝐴(𝑓))∨𝑘ℎ(𝒵_𝐴(𝑔)) is a special case of [6, Lemma 4.2(b)].
We show that the other containment is equivalent to Lemma 11 and hence must hold. First, we show the equivalence of the premises by showing that the following are equivalent. Recall that 𝒵_𝐴(𝑓)=𝒵_𝐴(𝑓²), and hence we can assume without loss of generality that 𝑓,𝑔≥0.(i)𝐸∈𝑘ℎ(𝒵_𝐴(𝑓)∨𝒵_𝐴(𝑔)). (ii)𝐸∈𝑘ℎ(𝒵_𝐴(𝑓+𝑔)). (iii)𝐙((𝑓+𝑔)^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝐸. (iv)𝐙(𝑓^𝜐𝐴)∩𝐙(𝑔^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝐸.
The equivalence (i)⇔(ii) follows from Lemma 9(c). The equivalence (ii)⇔(iii) follows from Theorem 6. The equivalence (iii)⇔(iv) follows from the assumption that 𝑓,𝑔≥0. This establishes the equivalence between the premises of left-to-right containment of this lemma and Lemma 11.
For the equivalence of the conclusions, note that 𝐸∈𝑘ℎ(𝒵_𝐴(𝑓))∨𝑘ℎ(𝒵_𝐴(𝑔)) if and only if 𝐸 is an intersection of a set in 𝑘ℎ(𝒵_𝐴(𝑓)) with a set in 𝑘ℎ(𝒵_𝐴(𝑔)). In other words, there must exist 𝑍₁∈𝑘ℎ(𝒵_𝐴(𝑓)) and 𝑍₂∈𝑘ℎ(𝒵_𝐴(𝑔)) such that 𝑍₁∩𝑍₂=𝐸. By Lemmas 4 and 5, this is equivalent to the statement that there exist zero sets 𝑍₁, 𝑍₂ such that 𝐙(𝑓^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝑍₁, 𝐙(𝑔^𝜐𝐴)⊆cl_𝜐𝐴𝑋𝑍₂, and 𝑍₁∩𝑍₂=𝐸, which is the conclusion of Lemma 11.

Corollary 13. Let 𝐴(𝑋) be an intermediate ring of continuous functions and 𝑓,𝑔∈𝐴(𝑋). Then(a)ℨ_𝐴(𝑓𝑔)=ℨ_𝐴(𝑓)∧ℨ_𝐴(𝑔), (b)ℨ_𝐴(𝑓+𝑔)⊆ℨ_𝐴(𝑓)∨ℨ_𝐴(𝑔).

Proof. Item (a) immediately follows from Lemmas 10 and 9(a).
Item (b) immediately follows from Theorem 12 and Lemma 9(b).

We are now ready to prove the main result of this section.

Theorem 14. Let 𝐴(𝑋) be an intermediate 𝐶-ring. If ℱ is a 𝑧-filter on 𝑋, then ℨ^←_𝐴[ℱ] is an ideal in 𝐴(𝑋).

Proof. Let ℱ be a 𝑧-filter and 𝐼=ℨ^←_𝐴[ℱ]. If 𝑓∈𝐼 and 𝑔∈𝐴(𝑋), then ℨ_𝐴(𝑓)⊆ℱ. By Corollary 13(a), ℨ_𝐴(𝑓𝑔)=ℨ_𝐴(𝑓)∧ℨ_𝐴(𝑔)⊆ℱ, so 𝑓𝑔∈𝐼. If 𝑓,𝑔∈𝐼, then both ℨ_𝐴(𝑓)⊆ℱ and ℨ_𝐴(𝑔)⊆ℱ. By Corollary 13(b), ℨ_𝐴(𝑓+𝑔)⊆ℨ_𝐴(𝑓)∨ℨ_𝐴(𝑔)⊆ℱ, so 𝑓+𝑔∈𝐼. Thus 𝐼 is an ideal.

Acknowledgment

Joshua Sack was partially supported by The Netherlands Organisation for Scientific Research VIDI Project 639.072.904.