Theory and Applications of Categories, Vol. 28, No. 16, 2013, pp. 435–475.

BOUNDED ARCHIMEDEAN ℓ-ALGEBRAS AND GELFAND-NEUMARK-STONE DUALITY

GURAM BEZHANISHVILI, PATRICK J. MORANDI, BRUCE OLBERDING

Abstract. By Gelfand-Neumark duality, the category C^{∗}Alg of commutative C^{∗}- algebras is dually equivalent to the category of compact Hausdorff spaces, which by Stone duality, is also dually equivalent to the category uba ℓ of uniformly complete bounded Archimedean ℓ-algebras. Consequently, C^{∗}Alg is equivalent to uba ℓ, and this equivalence can be described through complexification.

In this article we study uba ℓ within the larger category baℓ of bounded Archimedean ℓ-algebras. We show that uba ℓ is the smallest nontrivial reflective subcategory of baℓ, and that uba ℓ consists of exactly those objects in baℓ that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for baℓ. It follows that uba ℓ is the unique nontrivial reflective epicomplete subcategory of baℓ. We also show that each nontrivial reflective subcategory of baℓ is both monoreflective and epireflective, and exhibit two other interesting reflective subcategories of baℓ involving Gelfand rings and square closed rings.

Dually, we show that Specker ℝ-algebras are precisely the co-epicomplete objects in baℓ. We prove that the category spec of Specker ℝ-algebras is a mono-coreflective subcategory of baℓ that is co-epireflective in a mono-coreflective subcategory of baℓ consisting of what we term ℓ-clean rings, a version of clean rings adapted to the ordertheoretic setting of baℓ.

We conclude the article by discussing the import of our results in the setting of complex ∗-algebras through complexification.

1. Introduction

Gelfand-Neumark duality [16] between the categories of commutative C^{∗}-algebras and compact Hausdorff spaces gives a representation of a commutative C^{∗}-algebra as the ring C(X, ℂ) of continuous complex-valued functions on a compact Hausdorff space X. As a consequence of this duality, C(X, ℂ) can be characterized by algebraic properties (involving rings with involution) along with analytic properties (involving Banach spaces). Independently, Stone [45] axiomatized the commutative rings C(X, ℝ) of continuous realvalued functions on compact Hausdorff spaces. In contemporary terminology, these rings

Received by the editors 2012-09-23 and, in revised form, 2013-06-24. Transmitted by Walter Tholen. Published on 2013-07-15. 2010 Mathematics Subject Classification: 06F25; 13J25; 54C30. Key words and phrases: Ring of continuous real-valued functions, ℓ-ring, ℓ-algebra, uniform complete-

ness, Stone-Weierstrass theorem, commutative C^{∗}-algebra, compact Hausdorff space, Gelfand-Neumark-Stone duality.

\begin{matrix}c \\ ◯ \end{matrix} Guram Bezhanishvili, Patrick J. Morandi, Bruce Olberding, 2013. Permission to copy for private use granted.

are precisely the lattice-ordered commutative ℝ-algebras that are bounded, Archimedean, and uniformly complete (all these terms are defined in Section 2). Following Banaschewski [10], we call such a ring a Stone ring. Stone rings are thus described by algebraic (ℝalgebras), order-theoretic (lattice-ordered, bounded, Archimedean), and analytic (uniformly complete) properties. Though the axiomatizations of commutative C^{∗}-algebras and Stone rings are quite different, it is not hard to see that the complexification functor provides an equivalence between the categories of Stone rings and commutative C^{∗}- algebras (see Section 7 for details). Thus, either of the dualities, that of Gelfand-Neumark or of Stone, can be deduced from the other.

Our focus in this article is on Stone’s duality rather than that of Gelfand and Neumark—that is, we focus on ℝ-algebras rather than ℂ-algebras—although in Section 7 we discuss how our results apply to the complex case. We term this duality by the collective name of Gelfand-Neumark-Stone duality, as it is often done in the literature, and note that a similar duality for the category of uniformly complete Archimedean vector lattices with strong order unit was obtained independently by Krein–Krein [31], Kakutani [28, 29], and Yosida [48].

In addition to intrinsic interest in the real case, one of our motivations is the tradition in general topology of studying properties of a topological space X via the ring C(X, ℝ), as exemplified by Gillman and Jerison in [17], where many topological properties of X are shown to correspond to algebraic and order-theoretic features of C(X, ℝ). This emphasis on algebra and order is in particular what motivates our point of view. In fact, the presence of a natural lattice order on C(X, ℝ) is one feature that distinguishes the real case from the complex case, and many of our main results rely heavily on order-theoretic properties of lattice-ordered algebras. We study a category, which we denote baℓ, that abstracts the essential algebraic and order-theoretic features of C(X, ℝ); namely, the objects in baℓ are bounded Archimedean ℓ-algebras (that is, lattice-ordered algebras over ℝ) and the morphisms in the category are ℓ-algebra homomorphisms (that is, lattice-ordered ℝ-algebra homomorphisms).

The Stone rings are the uniformly complete rings in baℓ, and a study of baℓ is useful to individuate what is special about the analytic aspect of Stone rings, i.e., uniform completeness. To this end, we study not only the algebraic and order-theoretic properties of the rings in baℓ, but also the categorical properties of baℓ. For example, in baℓ monomorphisms are simply 1-1 morphisms, while epimorphisms are not in general surjective. In fact, the Stone-Weierstrass theorem implies that a morphism α : A → B in baℓ is an epimorphism iff α(A) is uniformly dense in B. This leads to the observation that uniform completion is a reflector, and we show that the full subcategory of baℓ consisting of the Stone rings is the smallest nontrivial reflective subcategory of baℓ, and that it is the unique nontrivial epicomplete reflective subcategory of baℓ. Thus, categorical notions encode the analytic features of uniform completion. Moreover, the category baℓ has other strong features, such as the fact that every nontrivial reflective subcategory of baℓ is both monoreflective and epireflective. We exhibit two other natural reflective subcategories of baℓ related to square closure and closure with respect to bounded inversion.

BOUNDED ARCHIMEDEAN ℓ-ALGEBRAS  437

The category baℓ also has an important mono-coreflective subcategory: the category of those ℝ-algebras that are generated by their idempotents. (Unlike nontrivial reflective subcategories of baℓ, coreflective subcategories need not be mono-coreflective.) In analogy with Conrad’s usage [14] of the term Specker ℓ-group, we term these algebras Specker ℝ-algebras. We show that A is a Specker ℝ-algebra iff A is a co-epicomplete object in baℓ, which happens iff A is a von Neumann regular ring in baℓ. The algebraic features of these rings are so strong that they determine a unique order on the ring. Moreover, each ℝ-algebra homomorphism between Specker ℝ-algebras is automatically an ℓ-algebra homomorphism, and thus the category spec of Specker ℝ-algebras and ℝ-algebra homomorphisms is a subcategory of baℓ.

In unraveling the algebraic, order-theoretic, and analytic properties of Stone rings, it is of interest to determine which Stone rings in baℓ are the uniform completions of Specker ℝalgebras, for such Stone rings are structurally determined by these basic algebraic objects. These rings turn out to be precisely the clean Stone rings, and correspond dually to Stone spaces (zero-dimensional compact Hausdorff spaces). This leads us to introduce the class of ℓ-clean rings, an ℓ-ring analogue of a clean ring, and to consider the category cbaℓ of ℓ-clean rings. This is shown to be a mono-coreflective subcategory of baℓ which contains spec as a bi-coreflective subcategory. Moreover, spec forms the smallest epi-coreflective subcategory, as well as the unique epi-coreflective co-epicomplete subcategory of cbaℓ. As with reflective subcategories of baℓ, a full description of co-reflective subcategories of baℓ remains an interesting open problem. In particular, we do not know whether there is a co-reflective subcategory of baℓ that plays a role in baℓ similar to the role of spec in cbaℓ (see Question 7.7(2)).

Thus, baℓ has a rich categorical structure. Reflection serves to distinguish the “richest” objects in the category — the Stone rings, which have important algebraic, ordertheoretic, and analytic features. Coreflection, on the other hand, distinguishes within the subcategory cbaℓ the “simplest” algebraic objects in the category — the Specker ℝ-algebras. The Specker ℝ-algebras are defined only in terms of algebra, the order is implicit.

If, instead of compact Hausdorff spaces one works with the larger category of completely regular spaces, then C(X, ℝ) is no longer bounded, and the boundedness condition should be dropped from the definition of baℓ. This results in a natural generalization of baℓ, first developed by Henriksen, Isbell, and Johnson [24, 25] under the name of Φalgebras. Other natural generalizations include the categories of Archimedean vector lattices with weak order unit (see, for example, Luxemburg–Zaanen [34] and Semadini [42]) and Archimedean ℓ-groups with weak order unit, as developed by Conrad, Hager, Ball, Madden, and others (see [5, 6, 7, 8, 14, 21, 34] and the references therein). There is a large body of results for these categories characterizing epimorphisms, as well as epicomplete objects and epicompletions, and to a lesser degree, of co-epicomplete objects and co-epicompletions. In contrast, less attention has been devoted in this direction to the category of Φ-algebras, and more particularly baℓ, and the early foundational papers [24, 25] appear to provide the most direct and in-depth treatment of this category (but see also the survey article [22] and its references). Another aim of this article is to partially fill in this gap by adding to the knowledge of the category baℓ of bounded Archimedean ℓ-algebras.

Not surprisingly, the Stone-Weierstrass theorem plays a crucial role in formulating our results, as it does in Gelfand-Neumark-Stone duality. Recall that the (real version of) Stone-Weierstrass theorem asserts that if X is a compact Hausdorff space and A is an ℝsubalgebra of C(X, ℝ) that separates points of X, then A is uniformly dense in C(X, ℝ). For our purposes we require only a weak version of the Stone-Weierstrass theorem, namely, that when A is an ℓ-subalgebra of C(X, ℝ) (rather than an ℝ-subalgebra) that separates points of X, then A is uniformly dense in C(X, ℝ). The proof of this weak version requires only an elementary application of compactness. Moreover, separation of points can be reformulated to state that A is an ℓ-subalgebra of C(X, ℝ) that separates points of X iff the inclusion mapping A → C(X, ℝ) is an epimorphism. This leads to a convenient version of the Stone-Weierstrass theorem that does not require reference to the category of compact Hausdorff spaces: If A, B ∈ baℓ and α : A → B is monic and epic in baℓ, then α(A) is uniformly dense in B. We show how to deduce Gelfand-Neumark-Stone duality from this version of the Stone-Weierstrass theorem.

In the last section of the article we translate our results to the setting of complex ∗-algebras through complexification of ℝ-algebras. It is not difficult to see that the complexification and self-adjoint functors yield an equivalence between the categories of commutative ℝ-algebras and commutative complex ∗-algebras, and that under this equivalence, the image of baℓ is exactly the category of commutative complex ∗-algebras whose self-adjoint part is closed under the absolute value. Furthermore, each subcategory of baℓ is then equivalent to an appropriate subcategory of complex ∗-algebras. We describe this equivalence for several of the subcategories of baℓ we study in this article, including an equivalence between Stone rings and commutative C^{∗}-algebras, and between Specker ℝ-algebras and what we call ∗-Specker ℂ-algebras.

2. Bounded Archimedean ℓ-algebras

All rings we will consider are assumed to be commutative with 1, and all homomorphisms are assumed to be unital; that is, preserve 1. We start with the following standard definition; see, e.g., Birkhoff [13, Ch. XIII-XVII].

2.1. Definition.

1. Let A be a ring with a partial order ≤. Then A is a lattice-ordered ring, or an ℓ-ring for short, if (i) (A, ≤) is a lattice, (ii) a ≤ b implies a + c ≤ b + c for each c, and (iii) 0 ≤ a, b implies 0 ≤ ab.

2. An ℓ-ring A is Archimedean if for each a, b ∈ A, whenever na ≤ b for each n ∈ ℕ, then a ≤ 0.

3. An ℓ-ring A is bounded if for each a ∈ A there is n ∈ ℕ such that a ≤ n · 1 (that is, 1 is a strong order unit).

4. An ℓ-ring A is an f-ring if for each a, b, c ∈ A with a ∧ b = 0 and c ≥ 0, we have ac ∧ b = 0.

5. An ℓ-ring A has bounded inversion if each a ∈ A with 1 ≤ a is invertible in A.

6. An ℓ-ring A is an ℓ-algebra if it is an ℝ-algebra and for each 0 ≤ a ∈ A and 0 ≤ λ ∈ ℝ we have λa ≥ 0.

2.2. Remark. We list below several well-known facts that we will use throughout without explicit mention. They can, for example, be found in [13, Ch. XIII-XVII].

1. In each ℓ-ring A we have a + b = (a ∨ b) + (a ∧ b) and (a ∨ b) + c = (a + c) ∨ (b + c).

2. For each a ∈ A, set a^{+} = a ∨ 0 and a^{−} = (−a) ∨ 0 = −(a ∧ 0). Then a^{+}, a^{−} ≥ 0, a^{+} ∧ a^{−} = 0, and a = a^{+} − a^{−}.

3. For each a ∈ A, define the absolute value of a by |a| = a∨(−a). Then |a| = a^{+} +a^{−}.

4. Each bounded ℓ-ring A is an f-ring, so a^{2} ≥ 0 and |ab| = |a||b| for each a, b ∈ A.

5. Each bounded Archimedean ℓ-ring is commutative.

6. If A is a nonzero ℓ-algebra, then we view ℝ as an ℓ-subalgebra of A.

For ℓ-algebras A and B, a map α : A → B is an ℓ-algebra homomorphism if α is an ℝ-algebra homomorphism and a lattice homomorphism. It follows that α(|a|) = |α(a)| for each a ∈ A.

2.3. Notation. Let baℓ denote the category of bounded Archimedean ℓ-algebras and ℓ-algebra homomorphisms.

2.4. Remark. The zero ring trivially belongs to baℓ, and it is easy to see that it is the terminal object in baℓ. On the other hand, since morphisms in baℓ are unital, ℝ is the initial object in baℓ. Since most of the results presented in this article hold easily for the zero ring, in the proofs we will often skip the easy verification for the zero ring and will mostly concentrate on the nonzero objects in baℓ.

We discuss some natural examples of bounded Archimedean ℓ-algebras.

2.5. Example.

1. Probably the most natural examples are the C(X, ℝ) for X compact Hausdorff. It is well known that C(X, ℝ) is a commutative ring with 1, and if we equip C(X, ℝ) with componentwise order ≤, then C(X, ℝ) is a bounded Archimedean ℓ-algebra. Note that each C(X, ℝ) has bounded inversion. In addition, C(X, ℝ) is uniformly complete, meaning that C(X, ℝ) is complete with respect to the uniform norm on C(X, ℝ) given by

∥f∥ = sup{|f(x)| : x ∈ X}.

2. For an example of a bounded Archimedean ℓ-algebra without bounded inversion, let X = [0, 1]. We recall that f ∈ C(X, ℝ) is a piecewise polynomial function if there are closed intervals F_{1}, …, F_{n} of X and polynomials g_{1}, …, g_{n} ∈ ℝ[x] such that X = ⋃ F_{i} and f = g_{i} on F_{i}. Let P P (X, ℝ) be the set of all piecewise polynomial functions on X. Then it is not hard to verify that P P (X, ℝ) is an ℓ-subalgebra of C(X, ℝ), and so P P (X, ℝ) ∈ baℓ. Let

f(x) = \{ \begin{matrix}1 & : x ∈ [0, \frac{1}{2}] \\ x + \frac{1}{2} & : x ∈ [\frac{1}{2}, 1]. \end{matrix}

Clearly f ∈ P P (X, ℝ) and f ≥ 1. If there exists g ∈ P P (X, ℝ) such that fg = 1, then fg − 1 = 0 on [\frac{1}{2}, 1]. As g ∈ P P (X, ℝ), it is easy to see that g and hence fg is a polynomial function on an infinite closed interval [a, b] of [\frac{1}{2}, 1]. But fg − 1 can only have finitely many zeros on [a, b]. The obtained contradiction proves that P P (X, ℝ) does not have bounded inversion.

3. Let X be compact Hausdorff. We call f ∈ C(X, ℝ) piecewise constant if there exists a clopen partition {P_{1}, …, P_{n}} of X and λ_{i} ∈ ℝ such that f(x) = λ_{i} for each x ∈ P_{i}. Let P C(X, ℝ) be the set of all piecewise constant functions on X. It is straightforward to see that P C(X, ℝ) is an ℓ-subalgebra of C(X, ℝ), and so P C(X, ℝ) ∈ baℓ. In fact, P C(X, ℝ) is the ℝ-subalgebra of C(X, ℝ) generated by the idempotents of C(X, ℝ), which are the characteristic functions of clopen subsets of X. If X is not a Stone space, P C(X, ℝ) may be rather small. For example, P C([0, 1], ℝ) is isomorphic to ℝ.

Let A ∈ baℓ. We recall that an ideal I of A is an ℓ-ideal if for all a, b ∈ A, whenever |a| ≤ |b| and b ∈ I, then a ∈ I. In other words, ℓ-ideals of A are exactly the ideals of A that are convex. It follows that if x, y ∈ I, then x∨y ∈ I. Note that ℓ-ideals are the kernels of ℓ-algebra homomorphisms. Moreover, if I is an ℓ-ideal of A, then A/I is a bounded ℓ-algebra, but A/I may fail to be Archimedean. In fact, A/I is Archimedean iff I is an intersection of maximal ℓ-ideals of A. For, if I = ⋂ M_{i}, where M_{i} are maximal ℓ-ideals of A, then A/I embeds into ∏ A/M_{i}. By [25, Cor. 2.7], each A/M_{i} is isomorphic to ℝ. Therefore, ∏ A/M_{i} and hence A/I is Archimedean. Conversely, if A/I is Archimedean, then the intersection of all maximal ℓ-ideals of A/I is 0 [26, Thm. II.2.11]. But maximal ℓideals of A/I correspond to maximal ℓ-ideals of A containing I. Thus, I is the intersection of all maximal ℓ-ideals of A containing I. Consequently, if A, B ∈ baℓ and α : A → B is a morphism in baℓ, then the kernel of α is an intersection of maximal ℓ-ideals.

2.6. Example. Given a nonzero A ∈ baℓ and an ℓ-ideal I of A, this example shows that ℝ + I is also in baℓ. This will be used in Lemma 6.1 and Theorem 6.2. It is sufficient to show that ℝ + I is an ℓ-subalgebra of A. It is straightforward to see that ℝ + I is an ℝ-subalgebra of A. From the relations between ∨, ∧ and + in an ℓ-algebra, it suffices to show ℝ + I is closed under ∨. To see this, let a = λ + x and b = μ + y with λ, μ ∈ ℝ and x, y ∈ I. Without loss of generality, we may assume that λ ≤ μ. Then

a ∨ b = (λ + x) ∨ (μ + y) = μ + [(λ − μ + x) ∨ y].

Furthermore, since λ ≤ μ, we see that y ≤ (λ − μ + x) ∨ y ≤ x ∨ y. Since x, y ∈ I and I is an ℓ-ideal, x ∨ y ∈ I, and so (λ − μ + x) ∨ y ∈ I. Thus, a ∨ b ∈ ℝ + I, and hence ℝ + I ∈ baℓ.

2.7. Notation. For A ∈ baℓ, let Max(A) denote the set of maximal ideals of A, and let X_{A} denote the set of maximal ℓ-ideals of A.

Let A ∈ baℓ. Since each maximal ℓ-ideal of A has real residue field ([25, Cor. 2.7]), each maximal ℓ-ideal is a maximal ideal of A, so X_{A} ⊆ Max(A). It is well known that Max(A) can be given the topology, where the closed sets are the sets of the form Z(I) = {M ∈ Max(A) : I ⊆ M} for some ideal I of A, and that Max(A) is a subspace of Spec(A), where Spec(A) denotes the prime spectrum of A with the Zariski topology. We recall that Max(A) is a compact T_{1}-space, but it may not be Hausdorff in general. We view X_{A} as a subspace of Max(A); that is, closed sets of X_{A} are the sets

Z_{ℓ}(I) := Z(I) ∩ X_{A} = {M ∈ X_{A} : I ⊆ M},

where I is an ideal of A. As follows from [25, Thm. 2.3(i)], X_{A} is compact Hausdorff.

Since each maximal ℓ-ideal of A has real residue field, to each element a ∈ A, we may associate a real-valued function f_{a} : X_{A} → ℝ by f_{a}(M) = a + M. Moreover, since f_{a}^{−1}(λ, μ) = Z_{ℓ}((a − λ)^{+})^{c} ∩ Z_{ℓ}((μ − a)^{+})^{c} for any λ < μ in ℝ, where (−)^{c} denotes settheoretic complement, it follows that f_{a} ∈ C(X_{A}, ℝ). Therefore, if we define ϕ_{A} : A → C(X_{A}, ℝ) by ϕ_{A}(a) = f_{a}, then ϕ_{A} is an ℓ-algebra homomorphism. As ⋂ X_{A} = 0 ([26, Thm. II.2.11]), the mapping ϕ_{A} is 1-1. Collecting together these observations, we arrive at the following well-known theorem (see [25], especially p. 81, and the references therein), which is fundamental in what follows. (The last assertion of the theorem is clear since any ℓ-subalgebra of an object in baℓ is in baℓ.)

2.8. Theorem. If A ∈ baℓ, then X_{A} is a compact Hausdorff space and ϕ_{A} : A → C(X_{A}, ℝ) is a 1-1 morphism in baℓ. Conversely, if A is isomorphic to an ℓ-subalgebra of C(X, ℝ), for X compact Hausdorff, then A ∈ baℓ. Thus, an ℓ-algebra A is in baℓ iff A is isomorphic to an ℓ-subalgebra of C(X, ℝ) for some compact Hausdorff space X.

Let A, B ∈ baℓ and let α : A → B be a morphism in baℓ. Then α is monic iff α is 1-1. This follows from a more general fact concerning bounded Archimedean ℓ-groups [5, Thm. 2.2(a)], but can also be seen directly. If A is the zero ring, then it is easily checked that α is monic iff α is 1-1. So suppose that A is nonzero. Let α : A → B be monic and I = ker(α). By Example 2.6, ℝ + I is an ℓ-subalgebra of A. Let β : ℝ + I → A be the identity map and let γ : ℝ + I → A be given by γ(λ + b) = λ for λ ∈ ℝ and b ∈ I. A short calculation shows that β, γ are morphisms in baℓ, and that α ∘ β = α ∘ γ. As α is monic, we obtain β = γ. Thus, I = 0, and so α is 1-1. On the other hand, epimorphisms in baℓ may not be onto, and so there exist bimorphisms (monic and epic morphisms) in baℓ that are not isomorphisms.

Define α^{∗} : Spec(B) → Spec(A) by α^{∗}(P) = α^{−1}(P) for each P ∈ Spec(B). It is well known that α^{∗} is continuous. In fact, α^{∗} restricts to a continuous mapping from X_{B} to X_{A}. To see this, note that when B is the zero ring, the claim is clear, so suppose that B is nonzero. If N ∈ X_{B}, then N has real residue field, so B = ℝ + N, and it follows that A = ℝ + α^{−1}(N). Thus, α^{−1}(N) is a maximal ideal of A. Moreover, since α is an ℓalgebra homomorphism, α^{−1}(N) is an ℓ-ideal. This shows that α^{−1}(N) ∈ X_{A}. Therefore, the restriction of α^{∗} : Spec(B) → Spec(A) is a continuous map α^{∗} : X_{B} → X_{A}.

2.9. Lemma. Let α : A → B be a morphism in baℓ.

1. α : A → B is monic iff α^{∗} : X_{B} → X_{A} is onto.

2. α : A → B is epic iff α^{∗} : X_{B} → X_{A} is 1-1.

3. α : A → B is a bimorphism iff α^{∗} : X_{B} → X_{A} is a homeomorphism.

Proof. (1) Let α be monic and M ∈ X_{A}. Let J = {b ∈ B : |b| ≤ α(m) for some m ∈ M}. We show that J is an ℓ-ideal of B. If a, b ∈ J, then there are m, n ∈ M with |a| ≤ α(m) and |b| ≤ α(n). So |a + b| ≤ α(m + n). Thus, a + b ∈ J. Furthermore, if a ∈ J and c ∈ B, then since B is bounded, there is λ ∈ ℕ with |c| ≤ λ. Therefore, if |a| ≤ α(m) for some m ∈ M, then |ac| ≤ α(λm), and λm ∈ M. Thus, ac ∈ J, and so J is an ideal of B. The definition of J shows that it is an ℓ-ideal containing α(M), so M ⊆ α^{−1}(J). We show that J is proper. If not, then 1 ∈ J, so 1 ≤ α(m) for some m ∈ M. As α is monic, this implies 1 ≤ m. Since M is an ℓ-ideal, this forces 1 ∈ M, a contradiction to M being proper. Therefore, J is a proper ℓ-ideal of B and hence is contained in a maximal ℓ-ideal N of B. Thus, M ⊆ α^{−1}(N), so M = α^{−1}(N), and so α^{∗} is onto. Conversely, let α^{∗} be onto. If α is not monic, then there exists a ∈ A such that a ≠ 0 but α(a) = 0. Since ⋂ X_{A} = 0, there exists a maximal ℓ-ideal M of A such that a ∉ M. As α^{∗} is onto, there exists N ∈ X_{B} such that M = α^{−1}(N). But then 0 = α(a) ∉ N, a contradiction to N being an ideal. Thus, α is monic.

(2) Let α be epic and let N_{1} and N_{2} be distinct maximal ℓ-ideals of B. We let β_{1} : B → ℝ and β_{2} : B → ℝ denote the canonical morphisms in baℓ which have kernels N_{1} and N_{2}, respectively. Then β_{1} ≠ β_{2}. As α is epic, there exists a ∈ A such that β_{1}(α(a)) ≠ β_{2}(α(a)). Let M_{i} = ker(β_{i} ∘ α). Then since M_{i} ∈ X_{A}, we have A = ℝ + M_{i}, so that a = λ_{1} + m_{1} for some λ_{1} ∈ ℝ and m_{1} ∈ M_{1}. Thus, 0 = β_{1}(α(m_{1})) = β_{1}(α(a)) − λ_{1}. But if m_{1} ∈ M_{2}, then 0 = β_{2}(α(m_{2})) = β_{2}(α(a)) − λ_{1}, which forces β_{1}(α(a)) = β_{2}(α(a)), which is false. Thus, M_{1} ≠ M_{2}, so α^{∗}(N_{1}) ≠ α^{∗}(N_{2}). Therefore, m_{1} ∉ M_{2}, and hence α(m_{1}) ∈ N_{1} but α(m_{1}) ∉ N_{2}. Thus, α^{−1}(N_{1}) ≠ α^{−1}(N_{2}), and so α^{∗} is 1-1.

Conversely, let α^{∗} be 1-1. To see that α is epic, let β_{1}, β_{2} : B → C be morphisms in baℓ such that β_{1} ∘α = β_{2} ∘α. If there is b ∈ B such that β_{1}(b) ≠ β_{2}(b), then there is a maximal ℓ-ideal N of C such that β_{1}(b) − β_{2}(b) ∉ N. Let π : C → C/N be the canonical map, and define N_{i} = ker(π ∘ β_{i}). Then N_{1}, N_{2} ∈ X_{B}. If b ∈ N_{1} ∩ N_{2}, then π(β_{1}(b) − β_{2}(b)) = 0, a contradiction. Thus, N_{1} and N_{2} are distinct maximal ideals of B. As α^{∗} is 1-1, there exists a ∈ A such that α(a) ∈ N_{1} \ N_{2}. But then 0 = π(β_{1}(α(a))) = π(β_{2}(α(a))), a contradiction to α(a) ∉ N_{2}. Therefore, β_{1} = β_{2}.

(3) Apply (1) and (2).

Let A ∈ baℓ and let ϕ_{A} : A → C(X_{A}, ℝ) be the mapping defined in Theorem 2.8. It follows from [25, Cor. 2.6] that ϕ_{A}^{∗} : X_{C(X_{A},ℝ)} → X_{A} is a homeomorphism. As an immediate consequence of Lemma 2.9, we then obtain:

2.10. Proposition. ϕ_{A} : A → C(X_{A}, ℝ) is a bimorphism.

2.11. Remark. Lemma 2.9 and Proposition 2.10 can be viewed as consequences of a more general result concerning commutative bounded Archimedean ℓ-groups [5, Thm. 2.2]. This is because X_{A} coincides with the notion of the Yosida space of a commutative Archimedean ℓ-group with weak order unit. To see this, we recall (see, e.g., [13, Ch. XIII]) that an ℓ-group is a group G which is a lattice and in which every group translation is orderpreserving. An element e > 0 of G is a weak order unit if e ∧ |a| = 0 implies a = 0, and the Yosida space of an Archimedean ℓ-group G with weak order unit e is the space (with the hull-kernel topology) of ℓ-subgroups that are maximal with respect to not containing e (see, e.g., [5, Sec. 1]). If we view A ∈ baℓ as an Archimedean ℓ-group, then as A is bounded, 1 is a strong order unit of A (cf. Definition 2.1(3)), hence a weak order unit. To see then that X_{A} is the Yosida space of A, let M be an ℓ-subgroup of A maximal with respect to not containing 1, and let J = {a ∈ A : |a| ≤ m for some m ∈ M}. By an argument similar to that in the proof of Lemma 2.9(1), we see that J is an ℓ-ideal of A. If 1 ∈ J, then 1 ≤ m for some m ∈ M. Since M is convex, this yields 1 ∈ M, a contradiction. Thus, J is a proper ℓ-ideal of A, and hence is contained in a maximal ℓ-ideal N of A. Since 1 ∉ N, this forces M = N. Consequently, each ℓ-subgroup of A maximal with respect to not containing 1 is a maximal ℓ-ideal, and it follows that X_{A} is the Yosida space of A.

3. Epicompletion as a reflector in baℓ

Let A ∈ baℓ. We define the uniform norm on A by

∥a∥ = inf{λ ∈ ℝ : |a| ≤ λ}.

This is well-defined because A is bounded, and as A is Archimedean, it follows that ∥ · ∥ is a norm on A. Thus, we have the norm topology on A, and it is easy to see that A is a topological ℓ-ring with respect to this topology. Furthermore, if α : A → B is an ℓ-algebra homomorphism, then it is immediate that ∥α(a)∥ ≤ ∥a∥ for each a ∈ A. Consequently, α is continuous with respect to the norm topologies.

3.1. Definition. Let A ∈ baℓ. We call A uniformly complete if the uniform norm on A is complete. Let uba ℓ be the full subcategory of baℓ consisting uniformly complete objects in baℓ. We call objects in uba ℓ Stone rings.

3.2. Remark. Johnstone [27, p. 155] calls Stone rings C^{∗}-algebras. To avoid confusion, we follow Banaschewski [10] in naming uniformly complete objects in baℓ Stone rings.

The following proposition, which plays a crucial role throughout the remainder of the article, can be viewed as a categorical reformulation of a weak version of the Stone-Weierstrass theorem, upon which the proof depends.

3.3. Proposition. The following are equivalent for a monomorphism α : A → B in baℓ.

1. α is a bimorphism.

2. α^{∗} : X_{B} → X_{A} is a homeomorphism.

3. There is a bimorphism β : B → C(X_{A}, ℝ) such that β ∘ α = ϕ_{A}.

4. α(A) separates points of X_{B}.

5. α(A) is uniformly dense in B.

Proof. (1) ⇔ (2): This is Lemma 2.9(3).

(2) ⇒ (3): The mapping α^{∗} induces an isomorphism α̃^{∗} : C(X_{A}, ℝ) → C(X_{B}, ℝ) in baℓ, given by α̃^{∗}(f) = f ∘ α^{∗}. We thus have the commutative diagram

\xymatrix{A \ar[r]^{α} \ar[d]_{ϕ_{A}} & B \ar[d]^{ϕ_{B}} \\ C(X_{A}, ℝ) \ar[r]_{α̃^{∗}} & C(X_{B}, ℝ).}

Define β : B → C(X_{A}, ℝ) by β = α̃^{∗}^{−1} ∘ ϕ_{B}. By Proposition 2.10, ϕ_{B} is a bimorphism. Also, since α̃^{∗}^{−1} is an isomorphism, it is a bimorphism. Therefore, β, as a composition of bimorphisms, is a bimorphism. Moreover, β ∘ α = α̃^{∗}^{−1} ∘ ϕ_{B} ∘ α = α̃^{∗}^{−1} ∘ α̃^{∗} ∘ ϕ_{A} = ϕ_{A}.

(3) ⇒ (2): Since β ∘ α = ϕ_{A}, we have α^{∗} ∘ β^{∗} = ϕ_{A}^{∗}. By Proposition 2.10, ϕ_{A} is a bimorphism, and by assumption β is a bimorphism. Therefore, by Lemma 2.9, both β^{∗} and ϕ_{A}^{∗} are homeomorphisms. Thus, so is α^{∗}.

\xymatrix{A \ar[r]^{α} \ar[d]_{ϕ_{A}} & B \ar[dl]^{β} & X_{A} & X_{B} \ar[l]_{α^{∗}} \\ C(X_{A}, ℝ) &  & X_{C(X_{A},ℝ)} \ar[u]^{_{ϕ_{A}_{∗}}} \ar[ur]_{β^{∗}}}

(2) ⇒ (4): If M and N are distinct maximal ideals in X_{B}, then by (2), α^{∗}(M) = α^{−1}(M) and α^{∗}(N) = α^{−1}(N) are distinct maximal ideals in X_{A}, and hence there exists a ∈ A such that α(a) ∈ M \ N.

(4) ⇒ (5): Since A separates points of X_{A}, the Stone-Weierstrass theorem yields (5).

(5) ⇒ (1): It suffices to show that α is epic. Suppose that β_{1}, β_{2} : B → C are morphisms in baℓ with β_{1} ∘ α = β_{2} ∘ α. Then since β_{1} and β_{2} are continuous with respect to the norm topology on B and α(A) is uniformly dense in B, it follows that β_{1} = β_{2}.

Using Proposition 3.3, we characterize now Stone rings in several ways. Motivated by the terminology in [5, 6, 7, 8], we call A ∈ baℓ epicomplete if each epimorphism α : A → B in baℓ is onto.

3.4. Corollary. The following are equivalent for A ∈ baℓ.

1. A is a Stone ring.

2. Each bimorphism α : A → B in baℓ is an isomorphism.

3. A is isomorphic to C(X, ℝ) for some compact Hausdorff space X.

4. A is epicomplete in baℓ.

Proof. (1) ⇒ (2): Let α : A → B be a bimorphism. By Proposition 2.10, ϕ_{B} : B → C(X_{B}, ℝ) is a bimorphism. Therefore, ϕ_{B} ∘ α : A → C(X_{B}, ℝ) is a bimorphism. Thus, by Proposition 3.3, A is isomorphic to a uniformly dense ℓ-subalgebra of C(X_{B}, ℝ). But as A is a Stone ring, A is uniformly complete. This yields that ϕ_{B} ∘ α is an isomorphism, and hence α is an isomorphism.

(2) ⇒ (3): By Proposition 2.10, the canonical map ϕ_{A} : A → C(X_{A}, ℝ) is a bimorphism, and hence by (2), an isomorphism.

(3) ⇒ (1): Since C(X, ℝ) is uniformly complete, (1) follows.

(1) ⇒ (4): Let A be a Stone ring, let α : A → B be an epimorphism in baℓ, and let I be the kernel of α. Then I is an intersection of maximal ideals in X_{A}, and hence is a uniformly closed subset of A. As such, A/I is uniformly complete [34, Thm. 60.4]. Thus, A/I is a Stone ring, and the induced map A/I → B is a bimorphism. Since we have established already the equivalence of (1) and (2), we conclude that this mapping is an isomorphism, and hence α is onto.

\xymatrix{A \ar[rr] \ar[dr] &  & B \\  & A/I \ar@{-->}[ur]}

(4) ⇒ (2): This is clear.

As we see below, Gelfand-Neumark-Stone duality follows quickly from Corollary 3.4. To formulate the duality, we let KHaus denote the category of compact Hausdorff spaces and continuous maps. The notations for this category vary in the literature. Different authors denote it by Comp_{2}, CompHaus, and HComp. We follow Johnstone’s notation KHausSp [27], but abbreviate it to KHaus. The reader is cautioned not to confuse KHaus with the category of Hausdorff k-spaces.

We define X : baℓ → KHaus as follows. If A ∈ baℓ, then X (A) = X_{A} and if α : A → B is a morphism in baℓ, then X (α) = α^{∗}. It is elementary to see that X is a contravariant functor. On the other hand, associating C(X, ℝ) with each X ∈ KHaus, and η̃ : C(Y, ℝ) → C(X, ℝ) with each morphism η : X → Y in KHaus, where η̃(f) = f ∘ η, produces a contravariant functor C : KHaus → baℓ. Moreover, for A ∈ baℓ and X ∈ KHaus, we have hom_{baℓ}(A, C(X, ℝ)) ≃ hom_{KHaus}(X, X_{A}). Therefore, X and C define a contravariant adjunction.

3.5. Corollary. [Gelfand-Neumark-Stone duality] The functor X : baℓ → KHaus, restricted to uba ℓ, and the functor C : KHaus → baℓ yield a dual equivalence between uba ℓ and KHaus.

Proof. The contravariant adjunction X : baℓ → KHaus and C : KHaus → baℓ restricts to a contravariant adjunction between uba ℓ and KHaus. By Corollary 3.4, the unit of the contravariant adjunction is an isomorphism. On the other hand, it is easy to see that for each X ∈ KHaus, we have X is homeomorphic to X_{C(X,ℝ)}. Indeed, that x ↦ M_{x} = {f ∈ C(X, ℝ) : f(x) = 0} is a bijection can be found in [25, Cor. 2.6], and that this map is continuous is straightforward. Therefore, the counit of the contravariant adjunction is also an isomorphism. Thus, the contravariant adjunction restricts to a dual equivalence between uba ℓ and KHaus.

Let A ∈ baℓ. In analogy with [5, 6, 7, 8], we call B ∈ baℓ the epicompletion of A if B is epicomplete and there is a bimorphism α : A → B. It is well known that the uniform completion of A is the completion of the metric space A with respect to the uniform topology. From Corollary 3.4 it follows that C(X_{A}, ℝ) is isomorphic to both the epicompletion and the uniform completion of A. Thus, we have a categorical characterization of uniform completion in baℓ:

3.6. Proposition. For each A ∈ baℓ, the epicompletion of A is isomorphic to the uniform completion of A.

We turn next to a characterization of the full subcategory uba ℓ in baℓ. We recall that a subcategory R of baℓ is a reflective subcategory of baℓ if the inclusion functor R → baℓ has a left adjoint. In general, R may be neither full nor replete (closed under isomorphisms) in baℓ. We will be interested in reflective full replete subcategories of baℓ. In order to avoid adding the adjective “full replete” to our discussion, we will assume that all reflective subcategories of baℓ are full replete.

As we saw, X : baℓ → KHaus and C : KHaus → baℓ define a contravariant adjunction, with C being full and faithful. Consequently, since Stone rings are, up to isomorphism, the algebras C(X, ℝ), for X compact Hausdorff (Corollary 3.4), C ∘ X is a left adjoint to the inclusion functor uba ℓ → baℓ. Thus, uba ℓ is a reflective subcategory of baℓ. Note that the reflector is exactly the mapping ϕ_{A} : A → C(X_{A}, ℝ) for each A ∈ baℓ. Since by Proposition 2.10, ϕ_{A} is a bimorphism, uba ℓ is actually a bireflective subcategory of baℓ.

We call a subcategory of baℓ trivial if it consists of the zero ring and its isomorphic copies. If a subcategory of baℓ is not trivial, then we call it nontrivial. Clearly the trivial subcategory is the least reflective subcategory of baℓ.

3.7. Lemma. Every nontrivial reflective subcategory of baℓ is bireflective.

Proof. Let R be a nontrivial reflective subcategory of baℓ. Let r : baℓ → R be the reflector. Then, for each A ∈ baℓ, there is a morphism r_{A} : A → r(A) in baℓ such that for each B ∈ R and each morphism α : A → B in baℓ, there is a unique morphism β : r(A) → B for which β ∘ r_{A} = α.

\xymatrix{A \ar[r]^{r_{A}} \ar[dr]_{α} & r(A) \ar@{-->}[d]^{β} \\  & B}

Consider R := r(ℝ) ∈ R. Since ℝ is the initial object in baℓ and reflectors preserve colimits, R is the initial object in R. Since R is nontrivial and there is no morphism in baℓ from the zero ring to a nonzero object in baℓ, we see that R is nonzero. As hom_{R}(R, R) is a singleton and R is a full subcategory of baℓ, we also see that hom_{baℓ}(R, R) is a singleton. If M ∈ X_{R}, then we have the canonical map π_{M} : R → ℝ with kernel M. Therefore, r_{ℝ} ∘ π_{M} ∈ hom_{baℓ}(R, R), and its kernel is M. If X_{R} has two points, then we get two maps from R to R in this way, and these maps are different since their kernels are not equal. Thus, X_{R} is a singleton. Therefore, C(X_{R}, ℝ) ≅ ℝ, and since R embeds in C(X_{R}, ℝ) and both of these algebras have the same ℝ-vector space dimension, we see that R ≅ ℝ.

We now prove R is monoreflective. Let A ∈ baℓ. If A is zero, then r(A) is zero, so r_{A} is trivially monic. Suppose A is nonzero. Then for each M ∈ X_{A}, there is an onto morphism α_{M} : A → R with kernel M, and hence since R is reflective, there is a morphism β_{M} : r(A) → R with α_{M} = β_{M} ∘ r_{A}. Let N = Ker(β_{M}), and note that N ∈ X_{r(A)}. Also, β_{M}(r_{A}(M)) = α_{M}(M) = 0, so that M ⊆ r_{A}^{−1}(N). Since M is maximal, M = r_{A}^{−1}(N). It follows that r_{A}^{∗} : X_{r(A)} → X_{A} is onto. Therefore, by Lemma 2.9(1), r_{A} is monic. Consequently, R is monoreflective. Now since a monoreflective subcategory is necessarily bireflective [1, Prop. 16.3] (our convention that reflective subcategories are full is being used implicitly here), it follows that R is bireflective.

3.8. Theorem. uba ℓ is the smallest nontrivial reflective subcategory of baℓ.

Proof. As discussed above, uba ℓ is a bireflective subcategory of baℓ. Let B be a nontrivial reflective subcategory of baℓ. We claim that uba ℓ ⊆ B. Let C be a Stone ring. By Lemma 3.7, B is a bireflective subcategory of baℓ. Therefore, there exists a bimorphism C → B for some B ∈ B. By Corollary 3.4, this bimorphism must be an isomorphism. Since B is full replete, we have C ∈ B, which proves the theorem.

Theorem 3.8 distinguishes uba ℓ as the smallest nontrivial reflective subcategory of baℓ. Restricting to epicomplete subcategories (that is, subcategories whose objects are epicomplete), we obtain uniqueness:

3.9. Corollary. uba ℓ is the unique nontrivial reflective epicomplete subcategory of baℓ.

Proof. That uba ℓ is epicomplete and reflective follows from Corollary 3.4 and Theorem 3.8. If A is an epicomplete subcategory of baℓ, then by Corollary 3.4, A ⊆ uba ℓ, and if also A is nontrivial and reflective, then by Theorem 3.8, uba ℓ ⊆ A.

4. Some other reflectors in baℓ

Our purpose in this section is to exhibit two reflectors arising in a natural way in baℓ involving Gelfand rings and square closure. It remains an open problem to characterize all reflective subcategories of baℓ; see Question 7.7(1).

We recall that a commutative ring A with 1 is a Gelfand ring if for each a, b ∈ A, whenever a + b = 1, there exist r, s ∈ A such that (1 + ar)(1 + bs) = 0. It is well known (see, e.g., [37, Prop. 1.3] and the references therein) that the following conditions are equivalent:

1. A is a Gelfand ring;

2. each prime ideal of A is contained in a unique maximal ideal of A (that is, A is a pm-ring);

3. for each distinct M, N ∈ Max(A) there exist a ∉ M and b ∉ N such that ab = 0;

4. Spec(A) is a normal space.

In particular, it follows that if A is a Gelfand ring, then Max(A) is Hausdorff, but the converse is not true in general. However, if the Jacobson radical of A is 0, then the converse is also true [15, Prop. 1.2], where we recall that the Jacobson radical of a ring A is J(A) = ⋂ Max(A). Since Max(A) is always compact, it follows that if J(A) = 0, then A is a Gelfand ring iff Max(A) is compact Hausdorff. Note that if A ∈ baℓ, then J(A) = ⋂ X_{A} = 0.

4.1. Proposition. The following are equivalent for A ∈ baℓ.

1. A has bounded inversion.

2. Max(A) = X_{A}.

3. A is a Gelfand ring.

4. Max(A) is a Hausdorff space.

Proof. The equivalence of (1) and (2) can be found in [24, Lem. 1.1]. To see (2) implies (3), let Max(A) = X_{A}. Then Max(A) is Hausdorff. Moreover, since for each A ∈ baℓ we have J(A) = 0, as discussed above, A is a Gelfand ring. That (3) implies (4) also follows from this same discussion. Finally, to see (4) implies (2), let Max(A) be Hausdorff. Then, as X_{A} is a compact subset of Max(A), it is closed. Therefore, X_{A} = Z(⋂ X_{A}) = Z(0) = Max(A).

Let gbaℓ be the full subcategory of baℓ consisting of Gelfand rings in baℓ. For each A ∈ baℓ, define g(A) = A_{S}, where A_{S} is the localization of A at the multiplicatively closed subset S = {s ∈ A : 1 ≤ s}. Then, as we show in the next proposition, g defines a reflector from baℓ to gbaℓ; in particular, g(A) ∈ gbaℓ for each A ∈ baℓ.

4.2. Proposition. gbaℓ is a reflective subcategory of baℓ properly containing uba ℓ.

Proof. To see that uba ℓ is a proper subcategory of gbaℓ, note that if A ∈ uba ℓ, then by Gelfand-Neumark-Stone duality, A is isomorphic to C(X, ℝ) for some compact Hausdorff space X. Since each C(X, ℝ) has bounded inversion, we conclude that A ∈ gbaℓ, hence uba ℓ ⊆ gbaℓ. On the other hand, if X is the one-point compactification of ℕ and P C(X, ℝ) is defined as in Example 2.5(3), then it has bounded inversion, so by Proposition 4.1, it is a Gelfand ring. It is easy to see that the sequence (f_{n}) in P C(X, ℝ), where

f_{n}(m) = \{ \begin{matrix}1/(m + 1) & if & m ≤ n \\ 0 & if & m > n or m = ∞ \end{matrix}

converges in C(X, ℝ) to the function f given by

f(m) = \{ \begin{matrix}1/(m + 1) & if & m ∈ ℕ \\ 0 & if & m = ∞. \end{matrix}

Since f ∉ P C(X, ℝ), we see that (f_{n}) is Cauchy but does not have a limit in P C(X, ℝ). Therefore, P C(X, ℝ) is not uniformly complete, hence is not a Stone ring. Thus, uba ℓ is a proper subcategory of gbaℓ.

To see that gbaℓ is a reflective subcategory of baℓ, let A ∈ baℓ. First we claim that A_{S} ∈ baℓ. If A = 0, then clearly A_{S} = 0 ∈ baℓ. Thus, assume A is nonzero. Note that no element of S is a zero divisor, for if a ∈ A, s ∈ S, and as = 0, then 0 = |as| = |a|s. Since 1 ≤ s, we have 0 ≤ |a| ≤ |a|s = 0, which forces |a| = 0, so a = 0. As a consequence, if a ∈ A, s ∈ S, and as ≥ 0, then a ≥ 0; to see this, (|a| − a)s = |a|s − as = 0, so a = |a| ≥ 0. Thus, since S consists of nonzerodivisors of A, the ring A_{S} can be viewed as a subring of the complete ring of quotients of A [33, Prop. 6]. Also, since A is an f-ring that is reduced (i.e., has no nonzero nilpotent elements), the ordering on A extends to an ordering on the complete ring of quotients of A [36, Thm. 2.1]. The ring A_{S} inherits this ordering, which is given by a/s ≤ b/t if at ≤ bs, and the lattice operations are given by (a/s) ∨ (b/t) = (at ∨ bs)/st and (a/s) ∧ (b/t) = (at ∧ bs)/st. That A_{S} ∈ baℓ is then straightforward. Moreover, A_{S} has bounded inversion since if 1 ≤ a/s, then s ≤ a. This forces a ≥ 1, so a ∈ S. Therefore, a/s is a unit in A_{S}, and hence by Proposition 4.1, A_{S} is a Gelfand ring.

Next we claim that g : baℓ → gbaℓ is a reflector. Let A ∈ baℓ and let ι : A → A_{S} be the canonical mapping. Suppose that α : A → B is a morphism in baℓ with B ∈ gbaℓ. For each s ∈ S, since 1 ≤ s, we have 1 ≤ α(s). Therefore, as B has bounded inversion (Proposition 4.1), α(s) is a unit in B. Thus, we define β : A_{S} → B by β(a/s) = α(a)α(s)^{−1} for all a ∈ A and s ∈ S.

\xymatrix{A \ar@{^{(}->}[rr]^{i} \ar[dr]_{α} &  & A_{S} \ar@{-->}[dl]^{β} \\  & B}

Since it is clear that β ∘ ι = α, it follows that gbaℓ is a reflective subcategory of baℓ.

4.3. Remark. Proposition 4.2 is an analogue for baℓ of a general result due to Schwartz and Madden: The reduced partially ordered rings with bounded inversion form a monoreflective subcategory of the category of reduced partially ordered rings [41, pp. 41–42].

4.4. Remark. Proposition 4.1 and a localization argument show that when A ∈ baℓ, then X_{A_{S}} = Max(A_{S}) ≅ X_{A}. As a result, we obtain that up to isomorphism A is an ℓsubalgebra of A_{S} and A_{S} is an ℓ-subalgebra of C(X_{A}, ℝ), and that up to homeomorphism, all three ℓ-algebras A, A_{S}, and C(X_{A}, ℝ) have X_{A} as the space of maximal ℓ-ideals. More generally, if A is a reflective subcategory of baℓ, then by Lemma 3.7, A is bireflective. Thus, if r denotes the reflector, then there is a bimorphism A → r(A). By Proposition 3.3, there is a bimorphism r(A) → C(X_{A}, ℝ), and so, up to homeomorphism, A, r(A), and C(X_{A}, ℝ) all have X_{A} as the space of maximal ℓ-ideals.

As we saw, not every A ∈ baℓ is a Gelfand ring, and hence not every A ∈ baℓ is a pm-ring. We show that Proposition 4.2 implies a weaker property holds for each A ∈ baℓ:

4.5. Corollary. If A ∈ baℓ, then each prime ideal of A is contained in at most one maximal ℓ-ideal.

Proof. Let S = {a ∈ A : 1 ≤ a}. By Proposition 4.2, A_{S} ∈ gbaℓ. Let P be a prime ideal of A that is contained in M, N ∈ X_{A}. Since M and N are ℓ-ideals, M and N extend to maximal ideals MA_{S} and NA_{S} of A_{S}, both of which contain the prime ideal P A_{S}. Moreover, they are ℓ-ideals. To see that MA_{S} is an ℓ-ideal, suppose that |a/s| ≤ |b/t| ∈ MA_{S} with b/t ∈ MA_{S}. Then b ∈ M, so sb ∈ M. Furthermore, |ta| ≤ |sb|. Therefore, ta ∈ M. Thus, a/s = (ta)/(st) ∈ MA_{S}, which proves that MA_{S} is an ℓ-ideal. Since A_{S} is a Gelfand ring, hence a pm-ring, because the maximal ℓ-ideals MA_{S} and NA_{S} both contain P A_{S}, we must have MA_{S} = NA_{S}, which in turn implies that M = N.

So far we have worked with two reflective subcategories of baℓ, namely uba ℓ and gbaℓ. There are many more such subcategories; we refer to the book [41], which introduces many reflective subcategories of the larger category of partially ordered rings. Next we produce one more reflective subcategory of baℓ, which was motivated by the example of real closed rings in [41], which satisfy, among other things, that each positive element is a square.

Recall that if C ∈ uba ℓ, then for each f ∈ C with f ≥ 0, there is a unique 0 ≤ g ∈ C with f = g^{2}. Let scba ℓ (square root closed) be the full subcategory of baℓ whose objects C satisfy

{a ∈ C : a ≥ 0} = {b^{2} : b ∈ C}.  (1)

4.6. Proposition. scba ℓ is a reflective subcategory of baℓ that properly contains uba ℓ and is incomparable with gbaℓ.

Proof. For each A ∈ baℓ we set sc(A) to be the intersection of all ℓ-subalgebras of C(X_{A}, ℝ) containing ϕ_{A}(A) which satisfy Equation (1). Because each positive element of C(X_{A}, ℝ) has a unique positive square root, it follows that sc(A) ∈ scba ℓ. We first claim that sc(A) can be constructed as follows. For each ℓ-subalgebra B of C(X_{A}, ℝ), let B′ be the ℓ-subalgebra of C(X_{A}, ℝ) generated by B ∪ \{\sqrt{b} : b ∈ B, b ≥ 0\}. It is clear that B′ ∈ baℓ, that B ⊆ B′ ⊆ C(X_{A}, ℝ), and that each b ∈ B with b ≥ 0 has a positive square root in B′. We then claim sc(A) = ⋃_{n=0}^{∞} A_{n}, where A_{0} = ϕ_{A}(A) and A_{n+1} = A′_{n} for each n ≥ 0. Let C be this union. Then ϕ_{A}(A) ⊆ C, and it is clear that C ∈ baℓ. Moreover, if c ∈ C, then c ∈ A_{n} for some n. Therefore, \sqrt{c} ∈ A_{n+1} ⊆ C. Thus, C satisfies Equation (1). Consequently, sc(A) ⊆ C. However, since ϕ_{A}(A) ⊆ sc(A), it follows that A_{1} = ϕ_{A}(A)′ ⊆ sc(A). By induction, we see that C ⊆ sc(A). Thus, C = sc(A).

To see that sc : baℓ → scba ℓ is a functor, let α : A → B be a morphism in baℓ. Then there is an induced morphism α̃^{∗} : C(X_{A}, ℝ) → C(X_{B}, ℝ). For notational convenience, we will write α′ = α̃^{∗}.

\xymatrix{A \ar[r]^{α} \ar[d] & B \ar[d] \\ C(X_{A}, ℝ) \ar[r]^{α′} & C(X_{B}, ℝ)}

We claim that α′ sends sc(A) into sc(B). Let 0 ≤ x = ϕ_{A}(a) ∈ ϕ_{A}(A). Then α′(x) = ϕ_{B}(α(a)) ≥ 0 in ϕ_{B}(B). Thus, \sqrt{α′(x)} ∈ sc(B). Now, if y is the (positive) square root of x in C(X_{A}, ℝ), then y ∈ sc(A), and α′(y)^{2} = α′(x). Furthermore, as \sqrt{α′(x)} = α′(y) is the unique positive element of C(X_{B}, ℝ) whose square is α(a), we see that α′(\sqrt{x}) = \sqrt{α′(x)} ∈ sc(B). Consequently, α′ sends y = \sqrt{x} into sc(B). Since this is true for all nonnegative elements of ϕ_{A}(A), this implies α′ sends A_{1} into sc(B). An inductive argument then yields α′(sc(A)) ⊆ sc(B). We thus define sc(α) = α′|_{sc(A)}, and conclude that sc is a functor.

We now show that scba ℓ is a reflective subcategory of baℓ. We let i_{A} be the map ϕ_{A} : A → sc(A) viewed as a map into sc(A). Suppose that C ∈ scba ℓ and α : A → C is a morphism in baℓ. We need to show there is a unique morphism β : sc(A) → C in baℓ with β ∘ i_{A} = α. Now, we have sc(α) : sc(A) → sc(C) is a morphism in baℓ. Clearly, sc(C) = ϕ_{C}(C), since sc(C) is the intersection of all square closed ℓ-subalgebras of C(X_{C}, ℝ) containing ϕ_{C}(C), and C is square closed. Set β = i_{C}^{−1} ∘ sc(α). Then β : sc(A) → C is a morphism in baℓ.

\xymatrix{A \ar[r]^{i_{A}} \ar[d]_{α} & sc(A) \ar[d]^{sc(α)} \ar[dl]_{β} \\ C \ar[r]_{i_{C}} & sc(C)}

We need to show β ∘ i_{A} = α and that β is unique with respect to this property. First, by the definition of α′, we have α′ ∘ ϕ_{A} = ϕ_{C} ∘ α. Thus, β ∘ i_{A} = α, as desired. Now, suppose that γ : sc(A) → C also satisfies γ ∘ i_{A} = α. This implies β|_{A_{0}} = γ|_{A_{0}}. Let 0 ≤ a ∈ A. Then \sqrt{a} ∈ sc(A), and γ(\sqrt{a}) = β(\sqrt{a}) since both are positive elements whose squares are equal to α(a). Therefore, γ and β agree on A_{1}. An inductive argument then shows γ = β. Thus, scba ℓ is a reflective subcategory of baℓ.

We next show that gbaℓ is not a subcategory of scba ℓ, nor is scba ℓ a subcategory of gbaℓ. For the first statement, let A = P P ([0, 1], ℝ). To see that the reflection g(A) of A in gbaℓ is not in scba ℓ, consider the nonnegative function f(x) = x. If it is the square of an element of g(A), which is a localization of A, then f would be represented on an infinite closed subinterval of [0, 1] as a square of a rational function. Since this is false, g(A) is not square closed. Thus, gbaℓ is not a subcategory of scba ℓ. To see that scba ℓ is not a subcategory of gbaℓ, we need the following claim.

4.7. Claim. For every A ∈ baℓ, the reflection sc(A) is an integral extension of ϕ_{A}(A).

Proof of claim. By transitivity of integrality and the construction of sc(A), it is enough to prove, for each ℓ-subalgebra B of C(X_{A}, ℝ), that B′ is integral over B, where, as above, B′ is the ℓ-subalgebra of C(X_{A}, ℝ) generated by B and all square roots of nonnegative elements of B. Now, let D be the ℝ-subalgebra of C(X_{A}, ℝ) generated by B ∪ \{\sqrt{b} : 0 ≤ b ∈ B}. Then D is integral over B, and B′ is the sublattice of C(X_{A}, ℝ) generated by D ([23, Thm. 3.3]). Thus, all we need to show is if a, b ∈ D, then a ∨ b is integral over B. We first point out that if a is integral over B, then so is a^{2} = |a|^{2}, and so |a| is also integral over B. Next, as a ∨ b = \frac{1}{2}(a + b) + \frac{1}{2}|a + b|, if a, b are integral over B, then so is a ∨ b. This completes the proof of the claim.

Now we produce an example of C ∈ scba ℓ which does not have bounded inversion. With A = P P ([0, 1], ℝ) as above, and C = sc(A), it follows from Claim 4.7 that sc(A) is an integral extension of ϕ_{A}(A). We identify C([0, 1], ℝ) and C(X_{A}, ℝ), and thus work with the inclusions P P ([0, 1], ℝ) ⊆ C ⊆ C([0, 1], ℝ). Consider 1 + x ∈ C and suppose that 1 + x is a unit in C. Since C is integral over A, this implies that 1 + x is a unit in A, and so (1 + x)^{−1} is a piecewise polynomial function on [0, 1]. Therefore, (1 + x)^{−1} is a polynomial function on an infinite closed subinterval of [0, 1]. This is false since the ring of polynomial functions on such an interval is integrally closed in its quotient field. Thus, since 1 ≤ 1 + x, we see that sc(A) does not have bounded inversion. Consequently, scba ℓ is not a subcategory of gbaℓ. That uba ℓ is a proper subcategory of scba ℓ is now obvious; the proof of the proposition is complete.

4.8. Remark. The subcategory of baℓ of real closed rings (cf. [41, p. 135]) is not the same as scba ℓ, since, by [41, Prop. 12.4], any such algebra has bounded inversion. We chose to work with scba ℓ in this proposition since the defining condition is simpler than that for real closed rings.

4.9. Remark. The reflectors g : baℓ → gbaℓ and sc : baℓ → scba ℓ can be used to define reflective subcategories C and D of baℓ such that C ⊆ gbaℓ ∩ scba ℓ ⊆ gbaℓ ∪ scba ℓ ⊆ D. In fact, our discussion here is a special instance of a more general framework for infima and suprema of reflectors, which is due to Schwartz and Madden [41, Ch. 9]. Let C = gbaℓ ∩ scba ℓ, and let r : baℓ → gbaℓ ∩ scba ℓ be defined by r(A) = g(sc(A)) for all A ∈ baℓ. Then r is well-defined; i.e., its image consists of Gelfand, square root closed rings in baℓ. Also, since g and sc are reflectors, it follows easily that r is a reflector. Next, let D be the full subcategory of baℓ consisting of objects A ∈ baℓ such that ϕ_{A}(A) = B ∩ C, where B and C are ℓ-subalgebras of C(X_{A}, ℝ) with B ∈ gbaℓ and C ∈ scba ℓ. Define a functor t : baℓ → D by t(A) = sc(A) ∩ g(ϕ_{A}(A)) for all A ∈ baℓ. It is straightforward, using the fact that ϕ_{A}(A) is uniformly dense in C(X_{A}, ℝ), to verify that t is a reflector. Thus, we have constructed subcategories C and D such that C = gbaℓ ∩ scba ℓ ⊆ gbaℓ ∪ scba ℓ ⊆ D. Note also that uba ℓ is a proper subcategory of gbaℓ ∩ scba ℓ, since g(sc(A)), where A = P P ([0, 1], ℝ), is not a Stone ring. (For example, it does not contain the restriction of the exponential function to [0, 1].)

5. Specker ℝ-algebras, ℓ-clean rings, and Baer rings

We have established that A ∈ baℓ is a Stone ring iff A is epicomplete, that uba ℓ is the unique reflective epicomplete subcategory of baℓ, and described several other natural reflective subcategories of baℓ. Our next goal is to investigate the dual concepts of epicocomplete objects and coreflective subcategories of baℓ. For this purpose, we study Specker ℝ-algebras, ℓ-clean rings, and Baer rings in baℓ.

Specker ℓ-groups were first introduced and studied by Conrad [14]. Here we introduce an analogous concept of Specker ℝ-algebra, which is particularly suited for baℓ. A more general concept, encompassing both Specker ℓ-groups and Specker ℝ-algebras, is that of Specker R-algebra, where R is a commutative ring. It is studied in detail in [12]. Before defining Specker ℝ-algebras, we emphasize again that all rings are assumed to be commutative with 1.

5.1. Definition. Let A be an ℝ-algebra. We call A a Specker ℝ-algebra if it is generated as an ℝ-algebra by its idempotents.

5.2. Example. Let X be a Stone space. As we pointed out in Example 2.5(3), P C(X, ℝ) is an ℝ-subalgebra of C(X, ℝ) generated by the idempotents of C(X, ℝ). It follows that P C(X, ℝ) is a Specker ℝ-algebra. In Theorem 6.2 we will see that each Specker ℝ-algebra is isomorphic to P C(X, ℝ) for some Stone space X.

If e_{1}, …, e_{n} are nonzero idempotents in A such that e_{i}e_{j} = 0 when i ≠ j, then we call {e_{1}, …, e_{n}} an orthogonal set of idempotents. The next two lemmas are routine observations about decompositions in Specker algebras. For lack of a reference, we include their proofs.

5.3. Lemma. Let A be an ℝ-algebra. Suppose that {e_{1}, …, e_{k}} and {f_{1}, …, f_{n}} are orthogonal sets of idempotents in A such that λ_{1}e_{1} + ⋯ + λ_{k}e_{k} = μ_{1}f_{1} + ⋯ + μ_{n}f_{n} for nonzero distinct real numbers λ_{1}, …, λ_{k} and μ_{1}, …, μ_{n}. Then {e_{1}, …, e_{k}} = {f_{1}, …, f_{n}} and for each i, j we have λ_{i} = μ_{j} iff e_{i} = f_{j}.

Proof. Multiplying by λ_{1}^{−1}e_{1}, we have e_{1} = ∑_{i=1}^{n} λ_{1}^{−1}μ_{i}e_{1}f_{i}. Squaring both sides and using the fact that e_{1} is idempotent and {f_{1}, …, f_{n}} is an orthogonal set of idempotents yields ∑_{i=1}^{n} λ_{1}^{−1}μ_{i}e_{1}f_{i} = e_{1} = ∑_{i=1}^{n}(λ_{1}^{−1}μ_{i})^{2}e_{1}f_{i}. Using again that f_{i}’s are orthogonal, we conclude that for each i, either e_{1}f_{i} = 0 or λ_{1}^{−1}μ_{i} = (λ_{1}^{−1}μ_{i})^{2}. If e_{1}f_{i} ≠ 0, then since μ_{i} ≠ 0, it follows that μ_{i} = λ_{1}. But the μ_{i} are all distinct, so there exists exactly one i such that e_{1}f_{i} ≠ 0. After relabeling we assume e_{1}f_{1} ≠ 0, and so, from above, we get e_{1} = e_{1}f_{1} and λ_{1} = μ_{1}. A symmetrical argument shows that f_{1} = e_{i}f_{1} for some i. Thus, e_{1} = e_{1}f_{1} = e_{1}e_{i}f_{1}, so that necessarily i = 1, and hence e_{1} = e_{1}f_{1} = f_{1}. Having established that e_{1} = f_{1} and λ_{1} = μ_{1}, we have then that ∑_{i=2}^{k} λ_{i}e_{i} = ∑_{i=2}^{n} μ_{i}f_{i}, and by repeating the argument we obtain the lemma.

Motivated by the lemma, we say that an element a ∈ A has a canonical decomposition if there exist orthogonal idempotents e_{1}, …, e_{n} ∈ A and distinct nonzero real numbers λ_{1}, …, λ_{n} such that a = λ_{1}e_{1} + ⋯ + λ_{n}e_{n}. By Lemma 5.3, the λ_{i} and e_{i} are unique in such a decomposition.

5.4. Lemma. If A is a Specker ℝ-algebra, then each nonzero element of A has a canonical decomposition.

Proof. Since A is generated as an ℝ-algebra by idempotents, each element of A can be written as an ℝ-linear combination of idempotents. We prove, by induction on n, that any nonzero linear combination of n idempotents has a canonical decomposition. If n = 1, then any such linear combination has the form λe with 0 ≠ λ ∈ ℝ and e ∈ A an idempotent. The element λe clearly has a canonical decomposition. Now, suppose the result holds for n − 1, and let x = ∑_{i=1}^{n} μ_{i}f_{i} be a linear combination of n idempotents. Set y = ∑_{i=1}^{n−1} μ_{i}f_{i}. By induction, y has a canonical decomposition y = ∑_{i=1}^{m} λ_{i}e_{i}. Then x = (∑_{i=1}^{m} λ_{i}e_{i}) + μ_{n}f_{n}. Since {e_{1}, …, e_{m}} is an orthogonal set of idempotents, ∑_{i=1}^{m} e_{i} is an idempotent, and so

x = ∑_{i=1}^{m} λ_{i}e_{i}(1 − f_{n}) + ∑_{i=1}^{m} λ_{i}e_{i}f_{n} + μ_{n}f_{n} ∑_{i=1}^{m} e_{i} + μ_{n}f_{n} (1 − ∑_{i=1}^{m} e_{i})

= ∑_{i=1}^{m} λ_{i}e_{i}(1 − f_{n}) + ∑_{i=1}^{m}(λ_{i} + μ_{n})e_{i}f_{n} + μ_{n}f_{n} (1 − ∑_{i=1}^{m} e_{i})

is a linear combination of orthogonal idempotents. We then obtain a canonical decomposition of x by combining terms with the same coefficient. Thus, by induction, the result follows.

We recall that a ring A is von Neumann regular if for each x ∈ A there is y ∈ A with xyx = x. It is well known (see, e.g., [19, Thm. 1.16]) that for a commutative ring A the following conditions are equivalent:

1. A is von Neumann regular;

2. A is reduced and each prime ideal of A is maximal;

3. the localization A_{M} at M is a field for each maximal ideal M of A.

We also recall that a ring A is clean if each element of A is the sum of an idempotent and a unit. It is known (see, e.g., [37, Thm. 1.7] and the references therein) that for a commutative ring A the following conditions are equivalent:

1. A is a clean ring;

2. for each distinct M, N ∈ Max(A) there exists an idempotent e ∈ A such that e ∈ M and e ∉ N;

3. A is a Gelfand ring and Max(A) is a zero-dimensional space.

In particular, each clean ring is a Gelfand ring, and if J(A) = 0, then A is a clean ring iff Max(A) is a Stone space. Note that each von Neumann regular ring A is clean. To see this, let M, N be distinct maximal ideals of A. Then there is x ∈ M\N. Because A is von Neumann regular, there is y ∈ A with xyx = x. If e = xy, then e is an idempotent, and e ∈ M since x ∈ M. Moreover, e ∉ N since otherwise x = xe ∈ N. Thus, e ∈ M\N, and hence A is clean.

5.5. Proposition. Every Specker ℝ-algebra A is a bounded Archimedean ℓ-algebra with bounded inversion, and every ℝ-algebra homomorphism between Specker ℝ-algebras is an ℓ-algebra homomorphism. Furthermore, A is von Neumann regular, hence clean.

Proof. Let A be a Specker ℝ-algebra. As noted in [9, p. 115], it is straightforward to verify that A ∈ baℓ. Indeed, by Lemma 5.4, we may write each x ∈ A uniquely in the following canonical form x = ∑_{i} λ_{i}e_{i} with λ_{i} ∈ ℝ nonzero and {e_{1}, …, e_{n}} an orthogonal set of idempotents. We then can define ≤ on A by 0 ≤ x if each λ_{i} ≥ 0. That an ℝ-algebra homomorphism between Specker ℝ-algebras is an ℓ-algebra homomorphism follows from [12, Cor. 5.3]. To see that A is a von Neumann regular ring, let x ∈ A, and write x = ∑_{i} λ_{i}e_{i} as above. For each i set μ_{i} = λ_{i}^{−1}. Setting y = ∑_{i} μ_{i}x_{i}, we see immediately that xyx = x. Thus, A is von Neumann regular. It follows that A is a clean ring. Finally, to see that A has bounded inversion, let x ∈ A with 1 ≤ x. Then x = ∑_{i} λ_{i}e_{i} with 1 ≤ λ_{i} for each i. Thus, ∑_{i} λ_{i}^{−1}e_{i} is the multiplicative inverse of x in A.

By the proposition, every Specker ℝ-algebra is a von Neumann regular ring in baℓ. In Theorem 6.2 we show that the converse is also true, and give a number of other characterizations of Specker ℝ-algebras.

Let A ∈ baℓ be a clean ring. Then, as follows from the discussion above, A is a Gelfand ring and Max(A) is a Stone space, and Proposition 4.1 gives us that X_{A} is a Stone space. However, there exist A ∈ baℓ such that X_{A} is a Stone space, but A is not a clean ring, as we see in the next example.

5.6. Example. We modify Example 2.5(2). Let X be the Cantor space and let A be the set of all piecewise polynomial functions on [0, 1] restricted to X. Then A ∈ baℓ. Since A separates points of X, it follows that X_{A} is homeomorphic to X (see Section 3 for details). On the other hand, the same argument as in Example 2.5(2) shows that A does not have bounded inversion. Therefore, by Proposition 4.1, A is not a Gelfand ring, hence not a clean ring.

In order to characterize those A ∈ baℓ for which X_{A} is a Stone space, we will weaken the concept of a clean ring. Let A be a ring and let X be a subspace of Max(A). We say that A is X-clean if each a ∈ A can be written in the form e + v with e an idempotent and v ∉ ⋃ X. Since ⋃ Max(A) is the set of units of A, we see that A is clean iff it is Max(A)-clean.

5.7. Definition. We call A ∈ baℓ ℓ-clean if A is X_{A}-clean. Let cbaℓ be the full subcategory of baℓ consisting of ℓ-clean rings.

Note that if A ∈ baℓ is a clean ring, then A is ℓ-clean. However, Example 5.6 and Theorem 5.9 below show the converse is not true in general. In order to characterize the objects in cbaℓ, we need the following lemma whose analogue for commutative rings with trivial Jacobson radical is well-known (see, e.g., [40, Lem. 2.1]). We omit the proof because it is nearly identical to that for commutative rings. The only difference is to note that for two ℓ-ideals J and K, the ideal J + K, as a sum of two ℓ-ideals, is an ℓ-ideal, and hence, if proper, is contained in a maximal ℓ-ideal.

5.8. Lemma. Let A ∈ baℓ and let I be an ideal of A. Then Z_{ℓ}(I) is clopen in X_{A} iff I is generated by an idempotent e ∈ A. In particular, if e ∈ A is an idempotent, then Z_{ℓ}(e) is clopen in X_{A}; and if U is clopen in X_{A}, then there exists an idempotent e ∈ A such that U = Z_{ℓ}(e).

5.9. Theorem. The following conditions are equivalent for nonzero A ∈ baℓ.

1. A is ℓ-clean.

2. Each element a ∈ A can be written as a = e + v, where e is an idempotent and there exists ε ∈ ℝ such that 0 < ε < |v|.

3. If M, N ∈ X_{A} are distinct, then there is an idempotent e ∈ A with e ∈ M\N.

4. X_{A} is a Stone space.

Proof. (1) ⇒ (2): Let A be ℓ-clean and let a ∈ A. Then a = e + v for some idempotent e and v ∉ ⋃ X_{A}. It is elementary to see that the ℓ-ideal I of A generated by v is {b ∈ A : |b| ≤ n|v| for some n ∈ ℕ}. Since v is not contained in any maximal ℓ-ideal, I = A, so 1 ≤ n|v| for some natural number n. Then 1/n ≤ |v|, so (2) holds.

(2) ⇒ (3): Let M, N ∈ X_{A} be distinct. Then there is a ∈ A with a ∈ M\N. Since N is a maximal ideal, there is x ∈ A with ax − 1 ∈ N. By (2), we may write ax = e + v with e an idempotent and λ < |v| for some positive λ ∈ ℝ. Note that v ∉ M, N since, say, if v ∈ M, then as M is an ℓ-ideal, |v| ∈ M, so λ ∈ M, and hence 1 ∈ M. Because ax ∈ M and v ∉ M, we have e ∉ M. Now, ax − 1 = (e − 1) + v. Since v ∉ N but ax − 1 ∈ N, we have 1 − e ∉ N, so e ∈ N. Thus, e ∈ N\M.

(3) ⇒ (4): By Lemma 5.8, each clopen of X_{A} has the form Z_{ℓ}(e) for some idempotent e ∈ A. By (3), the family {Z_{ℓ}(e) : e an idempotent in A} separates points of X_{A}. Therefore, since X_{A} is compact, X_{A} is a Stone space (see, e.g., [27, Thm. II.4.2]).

(4) ⇒ (1): Let a ∈ A. It is clear that Z_{ℓ}(1 − a) ⊆ Z_{ℓ}(a)^{c}. Therefore, as X_{A} is a Stone space, there is a clopen set U with Z_{ℓ}(1 − a) ⊆ U ⊆ Z_{ℓ}(a)^{c}. By Lemma 5.8, there is an idempotent e ∈ A with U = Z_{ℓ}(e). Thus, Z_{ℓ}(1 − a) ⊆ Z_{ℓ}(e) ⊆ Z_{ℓ}(a)^{c}. We show that this implies a − e ∉ ⋃ X_{A}. For, suppose a − e ∈ M for some M ∈ X_{A}. If e ∈ M, then a ∈ M, which is false since Z_{ℓ}(e) ∩ Z_{ℓ}(a) = ∅. If e ∉ M, then 1 − e ∈ M. So, (a − e) − (1 − e) = a − 1 ∈ M. But, this violates Z_{ℓ}(a − 1) ⊆ Z_{ℓ}(e). Consequently, A is ℓ-clean.

5.10. Corollary. Let A ∈ baℓ. Then A is a clean ring iff A is ℓ-clean and has bounded inversion.

Proof. Since the statement is clear for the zero ring, we assume that A is nonzero. If A has bounded inversion, then any element v satisfying 0 < ε < |v| is a unit, since 1 ≤ ε^{−1}|v|, so ε^{−1}|v| is a unit. Therefore, |v| is a unit, so there is w ∈ A with |v|w = 1. Squaring gives 1 = |v|^{2}w^{2} = v^{2}w^{2}, so v is a unit. Thus, if A is ℓ-clean and has bounded inversion, then it is clean. Conversely, if A is clean, then A is ℓ-clean, and A has bounded inversion by Proposition 4.1.

From Theorem 5.9 and Corollary 5.10 we deduce Azarpanah’s characterization of clean rings of continuous real-valued functions.

5.11. Corollary. ([3, Thm. 2.5]) Let X be a compact Hausdorff space. Then C(X, ℝ) is a clean ring iff X is a Stone space.

Proof. Let A = C(X, ℝ). If X is a Stone space, then since X_{A} is homeomorphic to X, Theorem 5.9 implies that A is ℓ-clean. As A has bounded inversion, by Corollary 5.10, A is a clean ring. Conversely, if A is a clean ring, then by Corollary 5.10, A is ℓ-clean, hence by Theorem 5.9, X_{A} is a Stone space. Since X and X_{A} are homeomorphic, it follows that X is a Stone space.

Next we show that in cbaℓ bimorphisms are determined entirely by their behavior on idempotents. Let A ∈ baℓ and let Id(A) be the set of idempotents of A. It is well known (see, e.g., [27, p. 181]) that Id(A) is a Boolean algebra under the operations

e ∨ f = e + f − ef
e ∧ f = ef
¬e = 1 − e

5.12. Proposition. Let A, B ∈ baℓ and let α : A → B be a monomorphism in baℓ. Consider the two statements:

1. α is a bimorphism.

2. α induces an isomorphism of Boolean algebras Id(A) → Id(B).

Then (1) ⇒ (2). Furthermore, if B is ℓ-clean, then (2) ⇒ (1).

Proof. (1) ⇒ (2): Suppose that α : A → B is a bimorphism. It suffices to show that Id(B) = α(Id(A)). Since α is a bimorphism, by Lemma 2.9(3), α^{∗} : X_{B} → X_{A} is a homeomorphism. Take e ∈ Id(B). By Lemma 5.8, Z_{ℓ}(e) is clopen in X_{B}. Consequently, α^{∗}(Z_{ℓ}(e)) is clopen in X_{A}, so there is f ∈ Id(A) with Z_{ℓ}(f) = α^{∗}(Z_{ℓ}(e)). If e ≠ α(f), then there is M ∈ X_{B} with e − α(f) ∉ M. But, the equality Z_{ℓ}(f) = α^{∗}(Z_{ℓ}(e)) says that e ∈ M iff f ∈ α^{−1}(M), which happens iff α(f) ∈ M. This is a contradiction, so e = α(f). Therefore, Id(B) = α(Id(A)).

Next, we prove that if B is ℓ-clean, then (2) ⇒ (1). Suppose that α(Id(A)) = Id(B). Since α : A → B is monic, by Lemma 2.9(1), α^{∗} : X_{B} → X_{A} is onto. We show that α^{∗} is also 1-1. As B is an ℓ-clean ring, for distinct M, N ∈ X_{B}, there exists an idempotent in B contained in one but not in the other. This idempotent is in α(A), which yields α^{−1}(M) ≠ α^{−1}(N). Therefore, α^{∗} is 1-1, so α^{∗} is a homeomorphism, and hence by Lemma 2.9(3), α is a bimorphism.

In Example 2.5(3), for a compact Hausdorff space X, we saw that P C(X, ℝ) ∈ baℓ and that P C(X, ℝ) is the ℝ-subalgebra of C(X, ℝ) generated by its idempotents.

5.13. Corollary. Let A ∈ baℓ. Then P C(X_{A}, ℝ) is isomorphic to the ℝ-subalgebra of A generated by the idempotents of A.

Proof. The canonical map ϕ_{A} : A → C(X_{A}, ℝ) is a bimorphism by Proposition 2.10. Consequently, by Proposition 5.12, C(X_{A}, ℝ) and ϕ_{A}(A) have the same idempotents. Thus, the ℝ-subalgebra of C(X_{A}, ℝ) generated by its idempotents is the same as the ℝ-subalgebra of ϕ_{A}(A) generated by its idempotents. Since the former ℝ-subalgebra is isomorphic to P C(X_{A}, ℝ), the result follows.

We recall that an ideal I of a ring A is an annihilator ideal if there exists an ideal J of A such that I = ann(J) := {a ∈ A : aJ = 0}, and that A is a Baer ring if each annihilator ideal of A is a principal ideal generated by an idempotent. If J(A) = 0, then A is a Baer ring iff Max(A) is an extremally disconnected space [39, Thm. 2.7], where we recall that a space is extremally disconnected if the closure of each open set is open. It follows that if A is a Gelfand ring and J(A) = 0, then A is a Baer ring iff Max(A) is an extremally disconnected compact Hausdorff space. In particular, if A is Baer Gelfand with J(A) = 0, then A is clean.

In Proposition 5.14 we give one more equivalent characterization of a Baer ring A in terms of the Boolean algebra Id(A) of idempotents of A. If A is a Baer ring, then Id(A) is a complete Boolean algebra [32, p. 271]. The converse is false; for example, let A = C(X, ℝ), where X = [0, 1]. Then A has no nontrivial idempotents, so Id(A) is the two-element Boolean algebra, hence Id(A) is complete. But A is not a Baer ring. Note that A is not clean. We show that if A is a clean ring and J(A) = 0, then A is a Baer ring iff Id(A) is a complete Boolean algebra.

We recall [43] that if B is a Boolean algebra, then the space Max(B) of maximal ideals of B is a Stone space, where the closed sets are the sets of the form {M ∈ Max(B) : I ⊆ M} with I an arbitrary ideal of B.

5.14. Proposition. Let A be a clean ring and J(A) = 0. Then the following conditions are equivalent:

1. A is a Baer ring.

2. Max(A) is an extremally disconnected compact Hausdorff space.

3. Id(A) is a complete Boolean algebra.

Proof. That (1) is equivalent to (2) follows from the fact that each clean ring is Gelfand, and when J(A) = 0, then, as discussed above, A is a Gelfand ring iff Max(A) is Hausdorff, and A is a Baer ring iff Max(A) is an extremally disconnected space. To see that (2) is equivalent to (3), observe that the mapping Max(A) → Max(Id(A)) : M ↦ M ∩ Id(A) is continuous and onto [2, Sec. 2]. Moreover, since idempotents in a clean ring A separate maximal ideals in A, it follows that this mapping is a homeomorphism. It is well known that Max(Id(A)) is an extremally disconnected compact Hausdorff space iff Id(A) is a complete Boolean algebra [44, Thm. 4.7]. Thus, (2) is equivalent to (3).

As a corollary, we recover a result of Azarpanah and Karamzadeh:

5.15. Corollary. ([4, Thm. 3.5]) A compact Hausdorff space X is extremally disconnected iff C(X, ℝ) is a Baer ring.

Proof. Let A = C(X, ℝ). Then since Max(A) is homeomorphic to X, Proposition 5.14 implies that A is a Baer ring iff X is an extremally disconnected space.

Let A ∈ baℓ. We show that unlike the case when A is a clean ring, whether A is a Baer ring is determined entirely by X_{A}. For a topological space X we denote the closure and interior of S ⊆ X by \overline{S} and int(S), respectively. For lack of a reference, we include the proof of the following standard observations.

5.16. Lemma. Let A ∈ baℓ and let I be a proper ℓ-ideal of A.

1. ann(I) = ⋂ Z_{ℓ}(I)^{c}.

2. \overline{int(Z_{ℓ}(I))} = Z_{ℓ}(ann(ann(I))).

3. I is an annihilator ideal of A iff I is an intersection of maximal ℓ-ideals and Z_{ℓ}(I) is regular closed in X_{A}.

Proof. (1) Set K = ⋂ Z_{ℓ}(I)^{c} = ⋂{M ∈ X_{A} : I ⊈ M}. Since I ann(I) = 0, for each M ∈ X_{A}, either I ⊆ M or ann(I) ⊆ M. Consequently, ann(I) ⊆ K. Conversely, let M ∈ X_{A}. Then I ⊆ M or K ⊆ M. Therefore, IK ⊆ M. Thus, IK = ⋂ X_{A} = 0. This implies K ⊆ ann(I), and so ann(I) = K.

(2) Recall that if S ⊆ X_{A}, then \overline{S} = Z_{ℓ}(⋂ S). Therefore, (1) implies \overline{Z_{ℓ}(I)^{c}} = Z_{ℓ}(ann(I)), so int(Z_{ℓ}(I)) = Z_{ℓ}(ann(I))^{c}. Thus,

\overline{int(Z_{ℓ}(I))} = \overline{Z_{ℓ}(ann(I))^{c}}
= Z_{ℓ}(⋂ Z_{ℓ} (ann(I))^{c})  (by \overline{S} = Z_{ℓ}(⋂ S))
= Z_{ℓ}(ann(ann(I)))  (by (1)).

Consequently, \overline{int(Z_{ℓ}(I))} = Z_{ℓ}(ann(ann(I))).

(3) If I is an annihilator ideal, then by (1), I is an intersection of maximal ℓ-ideals. Moreover, since I = ann(ann(I)), we have \overline{int(Z_{ℓ}(I))} = Z_{ℓ}(I), and so Z_{ℓ}(I) is regular closed. Conversely, suppose that Z_{ℓ}(I) is regular closed. By (2), \overline{int(Z_{ℓ}(I))} = Z_{ℓ}(ann(ann(I))), so Z_{ℓ}(I) = Z_{ℓ}(ann(ann(I))). Thus, since I and ann(ann(I)) are intersections of maximal ℓ-ideals, we have I = ann(ann(I)).

5.17. Theorem. If A ∈ baℓ, then A is a Baer ring iff X_{A} is extremally disconnected.

Proof. Suppose that X_{A} is extremally disconnected. We identify A with its image in C = C(X_{A}, ℝ). Let I be an ideal of A. Since by Proposition 5.14, C is a Baer ring, there exists an idempotent e ∈ C such that ann_{C}(I) = eC. Therefore, ann_{A}(I) = A∩ann_{C}(I) = A ∩ eC. It follows then from Corollary 5.13 that P C(X_{A}, ℝ) ⊆ A, and in particular, A contains every idempotent of C(X_{A}, ℝ). Thus, eA ⊆ A ∩ eC. For the reverse inclusion, if a ∈ A ∩ eC, then we may write a = ec for some c ∈ C. Therefore, ea = e^{2}c = ec = a. Consequently, a ∈ eA. Thus, ann_{A}(I) = eA. This proves that A is Baer.

Conversely, let A ∈ baℓ be a Baer ring. By Lemma 5.16, a subset F of X_{A} is regular closed in X_{A} iff F = Z_{ℓ}(I) for some annihilator ideal I. Since A is a Baer ring, I = eA for some idempotent e. Therefore, F = Z_{ℓ}(e), which is clopen. Thus, X_{A} is extremally disconnected.

5.18. Corollary. If A ∈ baℓ is a Baer ring, then A is ℓ-clean.

Proof. If A ∈ baℓ is a Baer ring, then by Theorem 5.17, X_{A} is an extremally disconnected compact Hausdorff space. Therefore, X_{A} is a Stone space, and so by Theorem 5.9, A is ℓ-clean.

Let bbaℓ be the full subcategory of baℓ consisting of Baer rings. By Corollary 5.18, bbaℓ is a full subcategory of cbaℓ. To see that bbaℓ is properly contained in cbaℓ, we utilize Gleason covers of compact Hausdorff spaces. We recall that the Gleason cover of a compact Hausdorff space X is a pair (Y, π), where Y is an extremely disconnected compact Hausdorff space and π : Y → X is an irreducible map (a continuous map is irreducible if it is onto and the image of a proper closed set is not the entire space). By [18, Thms. 2.5, 3.2], the pair (Y, π) is unique up to homeomorphism and is the projective cover of X in KHaus. Gleason constructs Y as the Stone space of the (complete) Boolean algebra of regular open subsets of X.

5.19. Example. We give an example of A ∈ bbaℓ which does not have bounded inversion and is not square closed. Let X be the Cantor space and let π : Y → X be its Gleason cover. Then Y is an extremally disconnected compact Hausdorff space. Recall from Example 5.6 the ℓ-algebra P P (X, ℝ) of piecewise polynomial functions on X, which we saw does not have bounded inversion. We may view C(X, ℝ) ⊆ C(Y, ℝ) via the identification h ↦ h ∘ π. Set

A = {∑_{i=1}^{n} a_{i}e_{i} : a_{i} ∈ P P (X, ℝ), e_{i} ∈ Id(C(Y, ℝ))} .

We claim that A is an ℓ-subalgebra of C(Y, ℝ) which contains Id(C(Y, ℝ)) and does not have bounded inversion. Given these claims, since then P C(Y, ℝ) ⊆ A ⊆ C(Y, ℝ), we conclude, by the argument in the proof of Theorem 5.17, that A is Baer. Since it does not have bounded inversion, it cannot be clean.

The definition of S implies that P C(Y, ℝ) ⊆ A and P P (X, ℝ) ⊆ A. It also follows from the definition that A is closed under addition, multiplication, and scalar multiplication, since the same is true for P P (X, ℝ). Each element of A can be written in the form ∑_{i=1}^{n} a_{i}e_{i} where {e_{1}, …, e_{n}} is an orthogonal set of idempotents. To see that A is closed under ∨, let a, b ∈ A. Write a = ∑_{i} a_{i}e_{i} and b = ∑_{j} b_{j}f_{j} with a_{i}, b_{j} ∈ P P (X, ℝ) and {e_{1}, …, e_{n}} and {f_{1}, …, f_{m}} each an orthogonal set of idempotents. We may assume the sum of each set is 1 since, if not, we may adjoin 1 − ∑_{i} e_{i} or 1 − ∑_{j} f_{j} to the set. We can then write a = ∑_{i,j} a_{i}e_{i}f_{j} and b = ∑_{i,j} b_{j}e_{i}f_{j}. Therefore, a ∨ b = ∑_{i,j}(a_{i} ∨ b_{j})e_{i}f_{j} ∈ A. Thus, A is an ℓ-subalgebra of C(Y, ℝ).

Let a ∈ P P (X, ℝ) with a ≥ 1 but for which 1/a does not exist in P P (X, ℝ); such a exists since P P (X, ℝ) does not have bounded inversion. If A has bounded inversion, then 1/a ∈ A, so we may write a^{−1} = ∑_{i=1}^{n} b_{i}e_{i} with b_{i} ∈ P P (X, ℝ) and {e_{1}, …, e_{n}} an orthogonal set. Multiplying both sides by ae_{j} yields ab_{j}e_{j} = e_{j}. Consequently, ab_{j} = 1 on Z_{ℓ}(1 − e_{j}), which is a clopen set in Y. By [18], Y is the Stone space of the Boolean algebra of regular open subsets of X. Therefore, there is a regular open set U_{j} ⊆ X with Z_{ℓ}(1 − e_{j}) = π^{−1}(U_{j}). Then ab_{j} = 1 on π^{−1}(U_{j}), so ab_{j} = 1 on U_{j}. Now, since ∑_{i=1}^{n} b_{i}e_{i} is a unit, it has no zeros, so it must be the case that the π^{−1}(U_{j}) cover Y. Consequently, the U_{j} cover X. Therefore, at least one, say U_{k}, is infinite. Then ab_{k} = 1 on an infinite subset of X, which would imply that we have polynomial equation ab = 1 on an infinite subset of X. Since this cannot happen, A does not have bounded inversion.

We now show A is not square closed. Consider a ∈ P P (X, ℝ), defined by a(x) = x for all x ∈ X. It is clear that a is not a square in P P (X, ℝ). Suppose a = b^{2} for some b ∈ A. We may write b = ∑_{i=1}^{n} b_{i}e_{i} for some b_{i} ∈ P P (X, ℝ) with the e_{i} forming a set of orthogonal idempotents in C(Y, ℝ). By the argument above, we may assume that ∑_{i} e_{i} = 1. Then a = b^{2} = ∑_{i=1}^{n} b_{i}^{2}e_{i}. Let C_{i} = {y ∈ Y : e_{i}(y) = 1}. The C_{i} form a clopen partition of Y, so the closed sets π(C_{i}) cover X. Thus, there is i for which π(C_{i}) is infinite. Moreover, a and b_{i}^{2} agree on C_{i}. Consequently, a and b_{i}^{2} agree on an infinite subset of X. Since they are polynomial functions, a = b_{i}^{2} on X. This is false since a is not a square in P P (X, ℝ). This proves that A is not square closed.

Let cuba ℓ be the full subcategory of uba ℓ consisting of clean Stone rings and let buba ℓ be the full subcategory of uba ℓ consisting of Baer Stone rings. Since each A ∈ uba ℓ has bounded inversion, buba ℓ is a full subcategory of cuba ℓ. Let also Stone and ED be the full subcategories of KHaus consisting of Stone spaces and extremally disconnected spaces, respectively. Then ED is a full subcategory of Stone. We conclude this section by pointing out that, by the results of Azarpanah and Karamzadeh stated in Corollaries 5.11 and 5.15, Gelfand-Neumark-Stone duality restricts to a dual equivalence between cuba ℓ and Stone, which further restricts to a dual equivalence between buba ℓ and ED:

5.20. Proposition.

1. The functors X and C, restricted to cuba ℓ and Stone, respectively, yield a dual equivalence between cuba ℓ and Stone.

2. The functors X and C, restricted to buba ℓ and ED, respectively, yield a dual equivalence between buba ℓ and ED.

6. Epi-cocomplete objects and coreflective subcategories of baℓ

In this section we give several characterizations of Specker ℝ-algebras, including that they are epi-cocomplete objects A in baℓ, meaning that each epimorphism α : B → A in baℓ is onto. We show that the category of Specker ℝ-algebras is a mono-coreflective subcategory of baℓ, and that it is the smallest epi-coreflective subcategory of cbaℓ, which in turn is a mono-coreflective subcategory of baℓ. We start with the following auxiliary lemma.

6.1. Lemma. If A ∈ baℓ and P is a minimal prime ideal of A, then ℝ + P ∈ baℓ and the inclusion mapping ℝ + P → A is a bimorphism.

Proof. By [46, p. 196], each minimal prime ideal of A is an ℓ-ideal. Thus, by Example 2.6, ℝ + P is an ℓ-subalgebra of A. We prove that the inclusion mapping ℝ + P ⊆ A is a bimorphism. By Proposition 3.3, it suffices to show that the elements of ℝ + P separate points of X_{A}. Let N_{1}, N_{2} ∈ X_{A} be distinct. By Corollary 4.5, P cannot be contained in both N_{1} and N_{2}. We assume without loss of generality that P ⊈ N_{2}. Then P ∩ N_{1} ⊈ N_{2} since N_{1} ⊈ N_{2} and N_{2} is prime. Therefore, (P ∩ N_{1}) + N_{2} = A, so that n_{1} + n_{2} = 1 for some n_{1} ∈ P ∩ N_{1} and n_{2} ∈ N_{2}. Thus, n_{1}, n_{2} = 1 − n_{1} ∈ ℝ + P with n_{1} ∈ N_{1}\N_{2} and n_{2} ∈ N_{2}\N_{1}. This shows that ℝ + P separates points of X_{A}.

6.2. Theorem. The following are equivalent for A ∈ baℓ.

1. A is a Specker ℝ-algebra.

2. A is a von Neumann regular ring.

3. A is epi-cocomplete.

4. Each bimorphism α : B → A in baℓ is an isomorphism.

5. No proper ℓ-subalgebra of A is uniformly dense in A.

6. A is isomorphic to P C(X, ℝ) for some Stone space X.

7. A/I ∈ baℓ for each ℓ-ideal I of A.

Proof. Since the theorem is trivially true when A = 0, we assume throughout the proof that A is nonzero.

(1) ⇒ (2): Apply Proposition 5.5.

(2) ⇒ (1): Since A is von Neumann regular, it is clean, hence Gelfand, and so by Proposition 4.1, A has bounded inversion. Let B be the ℝ-subalgebra of A generated by Id(A). Then by Proposition 5.5, B ∈ baℓ. We verify that B separates points of X_{A}. Let N_{1}, N_{2} ∈ X_{A} be distinct. Since A is clean, X_{A} has a basis of clopens. Therefore, there exists a clopen set U ⊆ X_{A} such that N_{1} ∈ U and N_{2} ∉ U. By Lemma 5.8, there is an idempotent e ∈ A such that U = Z_{ℓ}(e). Thus, e ∈ N_{1}\N_{2}, which proves that B separates points of A. By Proposition 3.3, the inclusion B ⊆ A is then a bimorphism, and so the induced mapping X_{B} → X_{A} is a homeomorphism. To prove that B = A, let a ∈ A, and let I = {b ∈ B : ba ∈ B}. We show that a ∈ B by proving that I = B. If not, then there is a maximal ideal M of B containing I. By Proposition 5.5, B has bounded inversion, and by the above argument, so does A. Therefore, by Proposition 4.1, Max(B) = X_{B} and Max(A) = X_{A}. Since, as noted above, the induced mapping X_{B} → X_{A} is a homeomorphism, there exists a unique maximal ideal N of A such that M = N ∩ B. Let S = B\M. Then S is a multiplicatively closed subset of B. It follows immediately from the definition that A_{S} is von Neumann regular. Since N is the unique maximal ideal of A with N ∩B = M, we see that NA_{S} is the unique maximal ideal of A_{S}. But, since A_{S} is von Neumann regular, A_{S} = (A_{S})_{NA_{S}} is a field. Thus, NA_{S} = 0. Now, since N ∈ X_{A}, we have A = ℝ + N. Therefore, writing a = λ + n, with λ ∈ ℝ and n ∈ N, there exists b ∈ S = B\M such that bn = 0, so that ba = bλ + bn = bλ ∈ B. Therefore, b ∈ I. Thus, since b ∉ M, we see that I ⊈ M. This proves that I is not contained in any maximal ideal of B, so I = B. Therefore, a ∈ B. Since this is true for each a ∈ A, we conclude that B = A.

(1) ⇒ (3): Let A be a Specker ℝ-algebra and let α : B → A be an epimorphism in baℓ. Then \overline{α} : B/Ker(α) → A is a bimorphism, and the ℓ-algebra B/Ker(α) is in baℓ since it is isomorphic to a subalgebra of A ∈ baℓ. By Proposition 5.12, Id(A) = \overline{α}(Id(B/Ker(α))). Since A is generated over ℝ by its idempotents, this implies that α is onto.

\xymatrix{B \ar[rr]^{α} \ar[dr] &  & A \\  & B/Ker(α) \ar[ur]_{\overline{α}}}

(3) ⇒ (4): This is clear.

(4) ⇒ (2): Let P be prime ideal of A. By Lemma 6.1, the inclusion mapping ℝ+P ⊆ A is a bimorphism. Thus, by (4), ℝ + P = A, so P is a maximal ideal of A. Therefore, as A is reduced and each prime ideal of A is maximal, A is von Neumann regular.

(1) ⇒ (6): Let A be a Specker ℝ-algebra. By Proposition 5.5 and Theorem 5.9, X_{A} is a Stone space, and by Proposition 5.12, Id(ϕ_{A}(A)) = Id(C(X_{A}, ℝ)). By Corollary 5.13, P C(X_{A}, ℝ) is the ℝ-subalgebra of C(X_{A}, ℝ) generated by Id(C(X_{A}, ℝ)). Now, as each element of A is a linear combination of idempotents, the equality Id(ϕ_{A}(A)) = Id(C(X_{A}, ℝ)) shows that ϕ_{A}(A) = P C(X_{A}, ℝ). Thus, ϕ_{A} is an isomorphism between A and P C(X_{A}, ℝ).

(6) ⇒ (1): Apply Example 5.2.

(4) ⇔ (5): Apply Proposition 3.3.

(2) ⇒ (7): Let A ∈ baℓ be von Neumann regular and let I be an ℓ-ideal of A. Then A is a Specker ℝ-algebra by (2) ⇒ (1). Thus, A/I is also a Specker ℝ-algebra, because each element is an ℝ-linear combination of cosets e + I for e ∈ Id(A), and each of these elements is idempotent in A/I. By Proposition 5.5, A/I ∈ baℓ with respect to the partial order described in the proof of the proposition. We claim this is the same partial order as that inherited from A. To see this, let a ∈ A. Then a + I ≥ 0 for the inherited order if there is b ∈ A with b ≥ 0 and a + I = b + I. On the other hand, a + I ≥ 0 for the order in the proof of the proposition if a + I is a linear combination of idempotents of A/I with all positive coefficients. But, b ≥ 0 implies b = ∑_{i} λ_{i}e_{i} for some idempotents e_{i} and λ_{i} ≥ 0. Thus, the two partial orders are the same.

(7) ⇒ (2): Let P be a minimal prime ideal of A. We recall that by [46, p. 196], each minimal prime ideal of A is an ℓ-ideal, so A/P ∈ baℓ, and so P is an intersection of maximal ℓ-ideals. However, by Corollary 4.5, each prime ideal of A is contained in at most one maximal ℓ-ideal. This forces P to be maximal. Thus, as A is reduced, it is von Neumann regular.

6.3. Remark. In the literature on ℓ-groups and vector lattices, the Archimedean objects whose images in the category are Archimedean are known as hyperarchimedean or epiarchimedean. Statement (7) of the theorem shows that Specker ℝ-algebras are precisely the hyperarchimedean objects in baℓ. Conrad in [14, Cor. I to Prop. 1.2] shows that a hyperarchimedean vector lattice A with strong order unit is isomorphic to P C(X, ℝ), where X is the Yosida space of A. Statement (6) of Theorem 6.2 is the corresponding statement in the category baℓ. For a corresponding version of the equivalence of (1) and (2) for vector lattices see [35, Lem. 7] and its references.

We remark that the ℝ-subalgebra of A generated by Id(A) is the largest Specker ℝ-subalgebra of A. The next result gives alternative characterizations of this subalgebra.

6.4. Proposition. Let A ∈ baℓ be nonzero and let B be the ℝ-subalgebra of A generated by Id(A). Then B = ⋂_{P} (ℝ + P), where P ranges over the minimal prime ideals of A, and B is also the intersection of all uniformly dense ℓ-subalgebras of A.

Proof. Let B′ := ⋂_{P} (ℝ + P), where P ranges over the minimal prime ideals of A. We show that B′ is a Specker ℝ-algebra. To prove this it is enough by Theorem 6.2 to show that B′ is a von Neumann regular ring. Since B′ is reduced and each prime ideal of a commutative ring contains a minimal prime [30, p. 6, Thm. 10], this amounts to showing that each minimal prime ideal of B′ is a maximal ideal of B′. Let Q be a minimal prime ideal of B′, and let S = B′\Q. Since each ideal of A_{S} is extended from an ideal of A, there exists a minimal prime ideal P of A such that P A_{S} ≠ A_{S}. Thus, it must be that P ∩ B′ ⊆ Q, and since Q is a minimal prime ideal of B′, we have in fact that Q = P ∩ B′. Now Q ⊆ B′ ⊆ ℝ + P ⊆ A, and P is the intersection of ℝ + P with any maximal ideal of A containing P. Since each ℓ-ideal is contained in a maximal ℓ-ideal, there is a maximal ℓ-ideal M of A containing P. Then M ∩ B′ = B′ ∩ P = Q, and so Q is a maximal ideal of B′ since, by Lemma 2.9, maximal ℓ-ideals of A contract to maximal ℓ-ideals of B′. This proves that each minimal prime ideal of B′ is a maximal ideal, and hence that B′ is a von Neumann regular ring. Therefore, by Theorem 6.2, B′ is a Specker ℝ-algebra. Moreover, Id(A) ⊆ B′. For, if e is an idempotent in A, then e(e−1) = 0, so that each minimal prime ideal of A contains either e or e − 1, and hence e ∈ B′. Thus, Id(A) ⊆ B′, and since B′ is generated by its idempotents and B′ ⊆ A, it follows that B′ is the Specker ℝ-algebra generated by Id(A).

Finally, to complete the proof of the proposition, we show that B′ is equal to the intersection of all the uniformly dense ℓ-subalgebras of A. By Proposition 5.12, each uniformly dense ℓ-subalgebra of A contains Id(A), so contains B′. Thus, B′ is contained in the intersection of all uniformly dense ℓ-subalgebras of A. But, by Proposition 3.3 and Lemma 6.1, this intersection is contained in B′. Thus, B′ is the intersection of all uniformly dense ℓ-subalgebras of A.

Proposition 6.4 allows us to add two more equivalent conditions to Theorem 5.9.

6.5. Theorem. Let A ∈ baℓ be nonzero. The following conditions are equivalent to the four equivalent conditions of Theorem 5.9.

(5) P C(X_{A}, ℝ) is uniformly dense in C(X_{A}, ℝ).

(6) The intersection of any collection of uniformly dense ℓ-subalgebras of A is uniformly dense in A.

Proof. (3) ⇒ (5): From (3) it is clear that P C(X_{A}, ℝ) separates points in X_{A}. Thus, by Proposition 3.3, (5) holds.

(5) ⇒ (6): This follows immediately from Corollary 5.13 and Proposition 6.4.

(6) ⇒ (4): Let B be the Specker ℝ-subalgebra of A generated by the idempotents of A. By Proposition 6.4, B is uniformly dense in A. Thus, by Proposition 3.3, X_{B} is homeomorphic to X_{A}. However, by Theorem 6.2, X_{B} is a Stone space, so X_{A} is a Stone space.

Let spec be the category of Specker ℝ-algebras and ℝ-algebra homomorphisms. It follows from Proposition 5.5 that spec is a full subcategory of baℓ. We show that spec and cbaℓ are mono-coreflective subcategories of baℓ. For each A ∈ baℓ, let s(A) denote the largest Specker ℝ-subalgebra of A; that is, s(A) is the ℝ-subalgebra of A generated by the idempotents of A. Let s : s(A) → A be the identity map. For each A ∈ baℓ, define \overline{s}(A) to be the closure of s(A) in A with respect to the uniform topology. Alternatively, with ϕ_{s(A)} : s(A) → C(X_{s(A)}, ℝ) and s̃^{∗} : C(X_{s(A)}, ℝ) → C(X_{A}, ℝ), we have \overline{s}(A) = ϕ_{A}^{−1}(s̃^{∗}(ϕ_{s(A)}(s(A))).

\xymatrix{s(A) \ar[r]^{s} \ar[d]_{ϕ_{s(A)}} & A \ar[d]^{ϕ_{A}} \\ C(X_{s(A)}, ℝ) \ar[r]^{s̃^{∗}} & C(X_{A}, ℝ)}

6.6. Theorem. spec and cbaℓ are mono-coreflective subcategories of baℓ.

Proof. To see that spec is a mono-coreflective subcategory of baℓ, let A ∈ baℓ, B ∈ spec , and α : B → A be a morphism in baℓ. Then α(Id(B)) ⊆ Id(A) ⊆ s(A). As B is a Specker ℝ-algebra, each b ∈ B is a linear combination of idempotents in B. Therefore, α(B) ⊆ s(A), so α : B → A factors through the inclusion mapping s(A) → A, and hence spec is a mono-coreflective subcategory of baℓ.

\xymatrix{s(A) \ar[rr] \ar[dr] &  & A \\  & B \ar[ur]_{α}}

To see that cbaℓ is a mono-coreflective subcategory of baℓ, let A ∈ baℓ. We claim first that \overline{s}(A) is ℓ-clean. If A is zero, then so is \overline{s}(A); hence, we may assume that A is nonzero. Let x ∈ \overline{s}(A). Then ϕ_{A}(x) = s̃^{∗}(f) for some f ∈ C(X_{s(A)}, ℝ). Since X_{C(X_{s(A)},ℝ)} is homeomorphic to X_{s(A)}, by Corollary 5.11, C(X_{s(A)}, ℝ) is clean, so there exists an idempotent e and a unit v in C(X_{s(A)}, ℝ) such that f = e + v. By Proposition 5.12, e = ϕ_{s(A)}(e′) for some idempotent e′ ∈ s(A) ⊆ A. Therefore, s̃^{∗}(e) = ϕ_{A}(e′), which implies that s̃^{∗}(v) = ϕ_{A}(u) for some u ∈ A. Because v is a unit in C(X_{s(A)}, ℝ) and X_{s(A)} is compact, |v| is bounded away from 0. Thus, there is ε > 0 with 0 < ε ≤ |v|. Applying s̃^{∗} gives 0 < ε ≤ |u|. Since x = e′ + u with e′ ∈ A an idempotent, and u satisfies 0 < ε < |u|, Theorem 5.9 yields that \overline{s}(A) is ℓ-clean.

Now suppose that B ∈ cbaℓ and α : B → A is a morphism in baℓ. Since B is ℓclean, it follows from Theorem 5.9(2) that B′ := Im α is also ℓ-clean. Then since s(B′) is generated by idempotents in A, we have s(B′) ⊆ s(A), and hence the uniform closure of s(B′) in A is contained in \overline{s}(A). But since B′ is ℓ-clean, s(B′) is uniformly dense in B′ by Proposition 3.3 and Proposition 5.12. Consequently, α : B → A must factor through the inclusion mapping \overline{s}(A) → A, which shows that cbaℓ is a mono-coreflective subcategory of baℓ.

Neither spec nor cbaℓ, with the above coreflectors, is an epi-coreflective subcategory of baℓ. For example, if A = C([0, 1], ℝ), then s(A) = \overline{s}(A) = ℝ, and clearly the morphism ℝ → C([0, 1], ℝ) is not epic. Note also that spec is not the smallest mono-coreflective subcategory of baℓ (it properly contains the trivial subcategory consisting of copies of ℝ), nor is it the largest (it is properly contained in cbaℓ). As with reflective subcategories, we assume that coreflective subcategories are full replete.

This shows inherent nonsymmetry between uba ℓ and spec as uba ℓ is both epireflective and monoreflective in baℓ, and it is the smallest nontrivial reflective subcategory of baℓ. We show that the symmetry is restored if we restrict our attention to the subcategory cbaℓ of baℓ.

6.7. Corollary. spec is the smallest epi-coreflective subcategory of cbaℓ, and the unique epi-coreflective epi-cocomplete subcategory of cbaℓ.

Proof. We know that spec is a subcategory of cbaℓ and, for each A ∈ cbaℓ, the inclusion morphism s(A) → \overline{s}(A) = A is a bimorphism by Proposition 3.3. Thus, spec is an epicoreflective subcategory of cbaℓ. To see that it is the smallest such subcategory, suppose C is an epi-coreflective subcategory of cbaℓ, and let c : cbaℓ → C be the coreflector. Let A ∈ cbaℓ. By the statement dual to [1, Prop. 16.3], C is a bi-coreflecive subcategory of cbaℓ. Therefore, c(A) → A is a bimorphism, so by Proposition 5.12, Id(A) = Id(c(A)). Thus, s(A) ⊆ c(A). In particular, if B ∈ spec , then B = s(B) ⊆ c(B) ⊆ B. Consequently, B = c(B), so spec ⊆ C. This proves that spec is the smallest epi-coreflective subcategory of cbaℓ.

For the second statement, by Theorem 6.2, spec is an epi-cocomplete subcategory of cbaℓ. To prove uniqueness, suppose that C is an epi-coreflective epi-cocomplete subcategory of cbaℓ with coreflector c : cbaℓ → C. By the argument of the previous paragraph, s(A) ⊆ c(A), and Id(s(A)) = Id(c(A)). Since c(A) is ℓ-clean, Proposition 5.12 shows that the inclusion s(A) → c(A) is a bimorphism. Because C is epi-cocomplete, we conclude that c(A) = s(A). Consequently, C = spec .

The results we have obtained easily yield a duality between spec and Stone, and between the full subcategory bspec of spec consisting of Baer rings and ED.

6.8. Theorem.

1. spec is dually equivalent to Stone.

2. bspec is dually equivalent to ED.

Proof. (1) By Proposition 5.5, each Specker ℝ-algebra is a clean ring. Thus, by Proposition 5.20, the restriction of X to spec is a contravariant functor X : spec → Stone. It is also clear that P C : Stone → spec , associating P C(X, ℝ) with each X ∈ Stone and η̃|_{P C(Y,ℝ)} with each η ∈ hom_{Stone}(X, Y), is a contravariant functor. Thus, the theorem comes down to verifying that there are natural isomorphisms A ≅ P C(X_{A}, ℝ) and X ≅ X_{P C(X,ℝ)}. The first isomorphism follows from Corollary 5.13. For the second, observe that X_{P C(X,ℝ)} = Max(P C(X, ℝ)), which as noted in the proof of Proposition 5.14 is homeomorphic to Max(Id(P C(X, ℝ)), and this by Stone duality for Boolean algebras [43] is homeomorphic to X.

(2) First observe that A ∈ spec is a Baer ring iff Id(A) is a complete Boolean algebra. Indeed, by Proposition 5.5, A ∈ baℓ and has bounded inversion. Hence, J(A) = 0, and it is sufficient to apply Proposition 5.14. Now apply (1).

6.9. Remark. While the proof of Theorem 6.8 uses the Stone-Weierstrass theorem, the result can be proved without using Stone-Weierstrass. Indeed, in [12, Thm. 3.8], we prove that spec is equivalent to the category BA of Boolean algebras. By Stone duality for Boolean algebras, BA is dually equivalent to Stone. Combining these two results yields a proof of Theorem 6.8 that does not use Stone-Weierstrass.

By Proposition 5.20 and Theorem 6.8, spec is equivalent to cuba ℓ and bspec is equivalent to buba ℓ. An explicit construction of the functors establishing these equivalences is as follows. The functor spec → cuba ℓ (resp. bspec → buba ℓ) associates with each A ∈ spec the epicompletion (equivalently, the uniform completion) of A, and the functor cuba ℓ → spec (resp. buba ℓ → bspec ) associates with each A ∈ cuba ℓ the ℓ-subalgebra of A generated by Id(A). Thus, Theorem 6.8 can be thought of as an “economic” version of Gelfand-Neumark-Stone duality for clean and Baer cases.

We point out that cuba ℓ = cbaℓ ∩ uba ℓ, and that the missing symmetry between spec and uba ℓ is restored between spec and cuba ℓ.

We conclude this section with the following diagram which lists the categories we have studied in this article. The line segments without arrows represent category inclusions, and the arrows represent category equivalences or dual equivalences when the arrow has a d superscript. Most of the inclusions (and non-inclusions) are clear; we point out the ones which are not. Each A ∈ spec is also in scba ℓ; for, take a ∈ A positive. We may write a = ∑_{i} λ_{i}e_{i} with the λ_{i} > 0 in ℝ and the e_{i} ∈ Id(A) orthogonal. Then a = (∑_{i} \sqrt{λ_{i}}e_{i})^{2}. Thus, A is square closed. We also pointed out in Proposition 5.5 that Specker algebras have bounded inversion, hence are Gelfand. As follows from the proof of Proposition 4.2, Specker algebras are not always uniformly complete. To see that bspec is not contained in uba ℓ, observe that P C(β(ℕ), ℝ) is not uniformly complete since it is a dense proper ℓ-subalgebra of C(β(ℕ), ℝ). Since β(ℕ) ∈ ED, this shows bspec is not contained in uba ℓ. Finally, Example 5.19 shows that bbaℓ is contained in neither gbaℓ nor scba ℓ, and Proposition 4.6 shows that scba ℓ and gbaℓ are incomparable.

[figure: baℓ; scba ℓ; gbaℓ; KHaus; d; uba ℓ; cbaℓ; Stone; d; cuba ℓ; spec; bbaℓ; ED; d; buba ℓ; bspec]

7. The Complexification of baℓ

In this section we discuss briefly complexification of the category baℓ in order to indicate how our results give a context for some questions regarding complex ∗-algebras. We continue to assume all rings are commutative with 1.

7.1. There is a category equivalence between the category of ℝ-algebras with ℝ-algebra homomorphisms and the category of complex ∗-algebras with ∗-algebra homomorphisms, and hence for each subcategory of ℝ-algebras, complexification yields an equivalent subcategory of complex ∗-algebras. Recall (see, e.g., [38, Def. 9.1.1]) that a complex ∗-algebra is a ℂ-algebra B with an involution ∗ satisfying (λb)^{∗} = \overline{λ}b^{∗} for each λ ∈ ℂ and b ∈ B. If A is an ℝ-algebra, then we can form the complexification B := A ⊗_{ℝ} ℂ, which is a ℂ-algebra. Each tensor a ⊗ (λ + μi) of B has the form (λa ⊗ 1) + (μa ⊗ i). Thus, each element of B can be written in the form (a ⊗ 1) + (b ⊗ i) for some a, b ∈ A. By identifying a ∈ A with a ⊗ 1 ∈ B and 1 ⊗ i with i, we then view B = {a + bi : a, b ∈ A}. There is an involution ∗ on B given by (a + bi)^{∗} = a − bi, which then makes B into a complex ∗-algebra. If α : A → A′ is an ℝ-algebra homomorphism, then α⊗Id_{ℂ} : A⊗_{ℝ} ℂ → A′ ⊗_{ℝ} ℂ is a ∗-homomorphism (i.e., a ℂ-algebra homomorphism compatible with the involution). We thus have a functor from the category of ℝ-algebras and ℝ-algebra homomorphisms to the category of complex ∗-algebras with ∗-algebra homomorphisms.

Going backwards, if B is a complex ∗-algebra, we associate to B the ℝ-algebra of self-adjoint elements of B, and to a ∗-algebra homomorphism β : B → B′ its restriction to the self-adjoint part of B. To see that these functors yield an equivalence between the categories of ℝ-algebras and complex ∗-algebras, if A is an ℝ-algebra and B = A ⊗_{ℝ} ℂ is its complexification, then the self-adjoint subalgebra of B is {a + bi : (a + bi)^{∗} = a + bi}, which is equal to A under our identification of A as an ℝ-subalgebra of B. Conversely, if B is a complex ∗-algebra and A is its self-adjoint subalgebra, then for each x ∈ B, we have

x = (\frac{x + x^{∗}}{2} ) + (\frac{x − x^{∗}}{2i} ) i,

which represents x in the form a + bi with a, b ∈ A. Thus, B is isomorphic to the complexification of A.

7.2. The complexification of baℓ is the category of complex ∗-algebras B whose selfadjoint subalgebra is closed under absolute value. Let B be the complexification of baℓ. Thus, a complex ∗-algebra B is in B iff its self-adjoint subalgebra A is in baℓ. Any B ∈ B is then a ∗-subalgebra of C(X, ℂ) for some X ∈ KHaus. We claim that a ∗-subalgebra B of C(X, ℂ) is the complexification of some A ∈ baℓ iff B is closed under the operation x ↦ |x + x^{∗}|. If x = a + bi, then |x + x^{∗}| = |2a|, so B is closed under this operation iff its self-adjoint subalgebra A is closed under the absolute value of C(X, ℂ). Since a ∨ b = \frac{1}{2}(a + b + |b − a|), a ∧ b = \frac{1}{2}(a + b − |b − a|), and |a| = a ∨ (−a), it follows that A ∈ baℓ iff A is closed under absolute value. So, a complex ∗-algebra B is in B iff B is closed under the operation x ↦ |x + x^{∗}|, iff the self-adjoint subalgebra of B is closed under absolute value.

7.3. The complexification of uba ℓ is the category C^{∗}Alg of commutative C^{∗}-algebras. We show, by restricting the equivalence between baℓ and B, that uba ℓ is equivalent to C^{∗}Alg. First, let A ∈ uba ℓ. If we define a norm on the complexification B of A by ∥a + bi∥ = \sqrt{∥a^{2} + b^{2}∥}, then a routine argument shows that B is complete and that for all x ∈ B, ∥xx^{∗}∥ = ∥x∥^{2}. Thus, B is a commutative C^{∗}-algebra. Conversely, if B is a commutative C^{∗}-algebra, then its self-adjoint subalgebra is complete, since the involution is continuous in the norm topology, and so is in uba ℓ. This pointfree equivalence between uba ℓ and C^{∗}Alg implies, as was pointed out in the introduction, that Stone duality follows from Gelfand-Neumark duality and vice-versa.

7.4. The complexification of the category of uniformly complete Baer rings in baℓ is the category of commutative AW^{∗}-algebras. Recall (see, e.g., [11, Def. 4.1]) that a ∗- ring B is said to be a Baer ∗-ring if each annihilator in B is generated by a self-adjoint idempotent (i.e., a projection; see, e.g., [11, Def. 1.2]). An AW^{∗}-algebra is a C^{∗}-algebra which is a Baer ∗-ring ([11, Def. 4.2]). If X ∈ KHaus, then C(X, ℂ) is a ∗-Baer ring iff X ∈ ED ([11, Thm. 7.1]), and C(X, ℝ) is Baer iff X ∈ ED (Corollary 5.15). Therefore, complexification yields an equivalence between buba ℓ and the category of commutative AW^{∗}-algebras.

7.5. The complexification of spec is the category of complex ∗-algebras generated as ℂalgebras by projections. If A ∈ spec , then each element of A is an ℝ-linear combination of idempotents. Let B be the complexification of A. Then idempotents of A are projections in B. As each element of B has the form a + bi with a, b ∈ A, we see that each element is a ℂ-linear combination of projections. Conversely, suppose that B is a ∗-algebra in which each element is a linear combination of projections. Let x ∈ B with x^{∗} = x. Since there is a unique representation x = ∑_{i} λ_{i}e_{i}, where the e_{i} form a set of pairwise disjoint projections and the λ_{i} are distinct, x = x^{∗} = ∑_{i} \overline{λ_{i}}e_{i}. Because this representation is unique, we see that \overline{λ_{i}} = λ_{i} for each i, so each λ_{i} ∈ ℝ. Therefore, the self-adjoint subalgebra A of B is generated over ℝ by idempotents, so A ∈ spec . Thus, the complexification of spec is the category of ∗-Specker ℂ-algebras; that is, the complex ∗-algebras generated as ℂ-algebras by projections.

7.6. Define A ∈ baℓ to be Pythagorean if for each a, b ∈ A, there is c ∈ A with a^{2} + b^{2} = c^{2}. The complexification of the subcategory pbaℓ of baℓ consisting of Pythagorean objects in baℓ is the subcategory of B consisting of those objects in B that are closed under the absolute value. Recall from 7.2 that, since B is the complexification of baℓ, each B ∈ B is a subalgebra of C(X, ℂ) for some X ∈ KHaus. We consider those B which are closed under the absolute value of C(X, ℂ). If h ∈ C(X, ℂ) and we write h = f + gi with f, g ∈ C(X, ℝ), then |h| = \sqrt{f^{2} + g^{2}}. Therefore, if A is the self-adjoint subalgebra of B and B is closed under the absolute value, then \sqrt{f^{2} + g^{2}} ∈ A for each f, g ∈ A. It follows that the complexification of pbaℓ is the subcategory of B consisting of those objects in B that are closed under the absolute value. It is easy to see that scba ℓ is a subcategory of pbaℓ. Furthermore, a small modification of Proposition 4.6 shows that pbaℓ is a reflective subcategory of baℓ.

We conclude by mentioning several interesting open problems.

7.7. Question.

1. Describe all reflective subcategories of baℓ. By Theorem 3.8, uba ℓ is the smallest nontrivial reflective subcategory of baℓ. Does there exist a largest proper such subcategory? In [20], Hager describes all the monoreflective subcategories of W that are closed under homomorphic images, where W is the category of Archimedean ℓ-groups with weak order unit. Hager parameterizes the set (a fortiori it is a set) of all such subcategories by the subsets of C(ℝ^{ω}, ℝ) [20, pp. 166-169]. Can Hager’s method be adapted to parameterize all the reflective subcategories of baℓ? (Recall that by Lemma 3.7, nontrivial reflective subcategories of baℓ are monoreflective.)

2. Is scba ℓ a proper subcategory of pbaℓ? We conjecture that the Pythagorean reflection of P P ([0, 1], ℝ) is not square closed, which would give an affirmative answer to the question.

3. What is the complexification of gbaℓ? In particular, is there a suitable notion of ∗-Gelfand algebra such that the complexification of gbaℓ is the category consisting of those subalgebras of commutative C^{∗}-algebras that are ∗-Gelfand?

4. We recall that a ∗-algebra B is ∗-clean if each element of B is the sum of a unit and a projection (see, e.g., [47, Def. 1]). It is not hard to show that A ∈ baℓ is clean iff the complexification of A is ∗-clean. What is the complex analogue of A ∈ baℓ being ℓ-clean?

5. Describe all coreflective subcategories of baℓ. In particular, does there exist a proper bi-coreflective subcategory of baℓ? This question is motivated by the goal of finding a canonical choice of uniformly dense ℓ-subalgebra B of each A ∈ baℓ. Corollary 6.7 shows that in cbaℓ, Specker ℝ-algebras play this role. On the other hand, the piecewise polynomials in C([0, 1]) serve as a canonical choice of uniformly dense ℓ-subalgebra for a specific ring not in cbaℓ. Both of these examples are “free” constructions, and it would be interesting to determine whether a similar free construction can be given for any ring in baℓ. Determining whether there exists a proper bi-coreflector for baℓ would be a step in this direction.

6. In Theorem 6.6, it is shown that cbaℓ is a mono-coreflective subcategory of baℓ that contains spec . Is cbaℓ the largest proper mono-coreflective subcategory of baℓ? If so, then it follows that Question (5) has a negative answer. For if as in (5), there exists a proper bi-coreflective subcategory C of baℓ, and cbaℓ is the largest proper mono-coreflective subcategory of baℓ, then every object of C is an ℓ-clean ring, and the bi-coreflectivity of C forces then every A ∈ baℓ to be ℓ-clean, a contradiction.

7. By Lemma 3.7, each nontrivial reflective subcategory of baℓ is bireflective. On the other hand, not every nontrivial coreflective subcategory of baℓ is bi-coreflective. Is each nontrivial coreflective subcategory of cbaℓ bi-coreflective?

Acknowledgments

We are very grateful to the referee who has provided us with many helpful comments, which have improved considerably both the content and exposition of the article. In particular, Section 7 is based on the suggestions of the referee. We are also thankful to Tony Hager and Warren McGovern for suggestions and pointers to the recent literature on vector lattices and ℓ-groups.

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Department of Mathematical Sciences, New Mexico State University, Las Cruces NM 88003-8001

Email: gbezhani@nmsu.edu, pmorandi@nmsu.edu, olberdin@nmsu.edu

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ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/16/28-16.{dvi,ps,pdf}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:

gavin seal@fastmail.fm

Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Ieke Moerdijk, Radboud University Nijmegen: i.moerdijk@math.ru.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca