DEDEKIND COMPLETIONS of BOUNDED ARCHIMEDEAN C

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Journal of Algebra and Its Applications Vol. 12, No. 1 (2013) 1250139 (16 pages) c World Scientiﬁc Publishing Company DOI: 10.1142/S0219498812501393

DEDEKIND COMPLETIONS OF BOUNDED ARCHIMEDEAN -ALGEBRAS

GURAM BEZHANISHVILI∗, PATRICK J. MORANDI† and BRUCE OLBERDING‡ Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003-8001, USA ∗[email protected] †[email protected] ‡[email protected]

Received 19 July 2011 Accepted 28 February 2012 Published 10 December 2012

Communicated by D. Mundici

All algebras considered in this paper are commutative with 1. Let ba be the category of bounded Archimedean -algebras. We investigate Dedekind completions and Dedekind complete algebras in ba . We give several characterizations for A ∈ ba to be Dedekind complete. Also, given A, B ∈ ba , we give several characterizations for B to be the Dedekind completion of A. We prove that unlike general Gelfand-Neumark-Stone duality, the duality for Dedekind complete algebras does not require any form of the Stone– Weierstrass Theorem. We show that taking the Dedekind completion is not functorial, but that it is functorial if we restrict our attention to those A ∈ ba that are Baer rings. As a consequence of our results, we give a new characterization of when A ∈ ba is a C∗-algebra. We also show that A is a C∗-algebra if and only if A is the inverse limit of an inverse family of clean C∗-algebras. We conclude the paper by discussing how to derive Gleason’s theorem about projective compact Hausdorﬀ spaces and projective covers of compact Hausdorﬀ spaces from our results.

Keywords: -ring; -algebra; Baer ring; uniform completion; Dedekind completion; essential closure; injective hull; C∗-algebra; compact Hausdorﬀ space; Gelfand-Neumark- Stone duality; extremally disconnected space; projective cover.

Mathematics Subject Classiﬁcation 2000: 06F25, 13J25, 54C30, 54G05

1. Introduction In the theory of completions of -groups and f-rings, uniform completions and Dedekind completions are the most studied. Let ba be the category of (commutative and unital) bounded Archimedean -algebras. It is known (see, e.g., [13, The- orem 11.5]) that uniformly complete objects in ba are precisely the commutative

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real C∗-algebras with identity involution. Therefore, following [17, Ch. IV.4], we call uniformly complete objects in ba C∗-algebras. By [4, Theorem 3.4], A ∈ ba is uniformly complete if and only if A is epicomplete. Also, for A, B ∈ ba ,itis known that the following conditions are equivalent: (i) B is the uniform completion of A; (ii) B is uniformly complete and A is isomorphic to a uniformly dense -subalgebra of B; (iii) B is isomorphic to C(XA), where XA is the (compact and Hausdorff) space of maximal -ideals of A and C(XA)isthe-algebra of all continuous real-valued functions on XA. In this paper we obtain similar results for Dedekind complete algebras and Dedekind completions in ba .ForA ∈ ba , we prove that the following conditions are equivalent: (i) A is Dedekind complete; (ii) A is a Baer C∗-algebra; (iii) A is an injective object in ba ;(iv)A is essentially closed. Also, for A, B ∈ ba ,weprove that the following conditions are equivalent: (i) B is isomorphic to the Dedekind completion of A; (ii) B is the injective hull of A in ba ; (iii) B is the essential closure of A;(iv)ThespaceXB of maximal -ideals of B is the Gleason cover of the space XA of maximal -ideals of A and B isomorphic to C(XB ). Some of these results already follow from the work of Conrad [5] and Banaschewski and Hager [2], however our approach is different. We also show that unlike general Gelfand-Neumark-Stone duality, the duality for Dedekind complete algebras does not require any form of the Stone–Weierstrass Theorem; that taking the Dedekind completion is not functorial in general; but that it is functorial if we restrict our attention to those A ∈ ba that are Baer rings. As a consequence of our results, we prove that A ∈ ba is a C∗-algebra if and only if every bimorphism from A into an -subalgebra of the Dedekind completion D(A)ofA is onto, which provides an improvement of [4, Theorem 3.4]. We also show that A is a C∗-algebra if and only if A is the inverse limit of an inverse family of clean C∗-algebras, each of which is isomorphic to an -subalgebra of D(A). We conclude the paper by discussing how to derive Gleason’s theorem about projective compact Hausdorff spaces and projective covers of compact Hausdorff spaces from our results.

2. Preliminaries All algebras considered in this paper are assumed to be commutative with 1. Let A be a ring. We denote by Max(A) the set of maximal ideals of A, and recall that the closed sets of the Zariski topology on Max(A)arethesetsZ(I):={M ∈ Max(A):I ⊆ M},whereI is an ideal of A.ItiswellknownthatMax(A)isa T compact 1-space, but that it is not Hausdorﬀ in general. The Jacobson radical of A is J(A)= Max(A), and an ideal I of A is an annihilator ideal if there exists an ideal J of A such that I =Ann(J):={a ∈ A : aJ =0}. We recall that A 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; that A is a clean ring if each element of A is the sum of an idempotent and a unit; and that A is a Baer ring

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if each annihilator ideal of A is a principal ideal generated by an idempotent. As follows from [4,Secs.4and5],ifJ(A)=0,thenA is a Gelfand ring if and only if Max(A) is a compact Hausdorff space; A is a clean ring if and only if Max(A)isa zero-dimensional compact Hausdorff space (a Stone space); and A is a Baer Gelfand ring if and only if Max(A) is an extremally disconnected compact Hausdorff space. An -ring is a ring A with a partial order ≤ such that (A, ≤) is a lattice, a ≤ b and c ≤ d imply a + c ≤ b + d,anda ≤ b and 0 ≤ c imply ac ≤ bc.LetA be an -ring. Then: (1) A is bounded if for each a ∈ A there is n ∈ N such that a ≤ n · 1. (2) A is Archimedean if a, b ∈ A and na ≤ b for each n ∈ N imply a ≤ 0. (3) A is an f-ring if a ∧ b =0andc ≥ 0implyac ∧ b =0. (4) A has bounded inversion if each a ∈ A with 1 ≤ a is invertible in A. For an -ring A and a ∈ A,leta+ = a ∨ 0, a− = −a ∨ 0and|a| = a ∨ (−a). Then a+,a− ≥ 0, a+ ∧ a− =0,a = a+ − a− and |a| = a+ + a−.Moreover,ifA is an f-ring, then a2 ≥ 0and|ab| = |a||b|. Since each bounded -ring is an f-ring, these also hold for bounded -rings. We call an -ring A an -algebra if A is an R-vector lattice; that is, A is an R-vector space and for each 0 ≤ a ∈ A and 0 ≤ λ ∈ R we have λa ≥ 0. For -algebras A and B,amapα : A → B is an -algebra homomorphism if α is a (unital) R-algebra homomorphism and a lattice homomorphism. We denote by ba the category of bounded Archimedean -algebras and -algebra homomorphisms. We note that a morphism in ba is monic if and only if it is one-to-one, but not every epimorphism in ba is onto (see, e.g. [4, Sec. 3]). Let A ∈ ba .AnidealI of A is an -ideal if for all a, b ∈ A, whenever |a|≤|b| and b ∈ I,thena ∈ I.LetXA denote the set of all maximal -ideals of A.Then XA ⊆ Max(A), and we view XA as a subspace of Max(A), so closed sets of XA Z I Z I ∩ X {M ∈ X I ⊆ M} I A are the sets ( ):= ( ) A = A : ,where is an ideal of . By [14, Theorem 2.3(i)], the space XA is compact and Hausdorff. As XA =0, which follows from the proof of [14, Theorem 2.3], we have that J(A)=0foreach A ∈ ba . Let A ∈ ba . 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.WecallA uniformly complete if the uniform norm on A is complete. Since each maximal -ideal of A has residue field isomorphic to R and the intersection of all maximal -ideals is 0, to each element a ∈ A, we may associate a real-valued function fa : XA → R,givenbyfa(M)=λa,whereλa is the unique real number such that a − λa ∈ M. This produces the following important representation theorem. If A ∈ ba , then the map φA : A → C(XA), given by φ(a)=fa, is a monomorphism in ba .Conversely,ifA is isomorphic to an -subalgebra of

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C(X), for X compact Hausdorff, then A ∈ ba .Thus,A ∈ ba if and only if A is isomorphic to an -subalgebra of C(X), where X is compact Hausdorff. This representation is functorial. If α : A → B is a morphism in ba ,then ∗ ∗ −1 α : XB → XA,givenbyα (M)=α (M), is well-defined and is continuous. Therefore, associating with each A ∈ ba the compact Hausdorff space XA,and ∗ with each morphism α : A → B the continuous map α : XB → XA, defines a contravariant functor X from ba to the category KHaus of compact Hausdorff spaces and continuous maps. Moreover, associating C(X)witheachX ∈ KHaus, and ϕ : C(Y ) → C(X) with each continuous map ϕ : X → Y ,whereϕ(f)=f ◦ ϕ, produces a contravariant functor C : KHaus → ba .Furthermore,X and C define a contravariant adjunction [4, Sec. 3]. We recall (see, e.g. [17, Chap. IV.4] or [4, Definition 3.1]) that a C∗-algebra is a bounded Archimedean -algebra that is uniformly complete. Let C∗Alg denote the category of C∗-algebras and -algebra homomorphisms. Then C∗Alg is a full subcategory of ba .ForA ∈ ba ,wehavethatφA(A) separates points of XA. Thus, by the Stone–Weierstrass Theorem, φA(A) is uniformly dense in C(XA). In fact, as was shown in [4, Sec. 3], for this a weak version of the Stone–Weierstrass ∗ Theorem is sufficient. If A is a C -algebra, then φA(A) is also closed in C(XA), hence φA : A → C(XA) is an isomorphism, and we arrive at the celebrated Gelfand- Neumark-Stone duality:

Gelfand-Neumark-Stone Duality Theorem: C∗Alg is dually equivalent to KHaus. The dual equivalence is established by the functors C : KHaus → C∗Alg and the restriction of X : ba → KHaus to C∗Alg.

In fact, the composition C◦X: ba → C∗Alg is a reflector, and so C∗Alg is a reflective subcategory of ba .In[4, Theorem 3.10] we showed that C∗Alg is the smallest reflective subcategory of ba . (Note that we assume that reflective subcategories are full and closed under isomorphisms.) We recall that A ∈ ba is epicomplete if each epimorphism α : A → B in ba is onto. As follows from [4, Theorem 3.4], A ∈ ba is a C∗-algebra if and only if A is epicomplete, which happens if and only if each bimorphism (monic and epic morphism) α : A → B in ba is an isomorphism. It follows [4, Corollary 3.11] that C∗Alg is the unique reflective epicomplete subcategory of ba . Gelfand-Neumark-Stone duality restricts nicely to the case of clean and Baer C∗-algebras. We call a C∗-algebra A a clean C∗-algebra if A is a clean ring, and we call A a Baer C∗-algebra if A is a Baer ring. Let CC∗Alg denote the category of clean C∗-algebras and -algebra homomorphisms, and let BC∗Alg denote the category of Baer C∗-algebras and -algebra homomorphisms. Let also Stone denote the category of Stone spaces and continuous maps, and ED denote the category of extremally disconnected compact Hausdorff spaces and continuous maps. Then Gelfand-Neumark-Stone duality for the clean case establishes that CC∗Alg is dually equivalent to Stone, and for the Baer case it establishes that BC∗Alg is

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dually equivalent to ED. Both dualities are obtained by restricting the functors X and C [4, Theorem 5.26].

3. Dedekind Completions and Dedekind Algebras We recall that a poset P is Dedekind complete if each subset of P which has an upper bound (respectively, lower bound) has a least upper bound (respectively, greatest lower bound). It is well known that each poset P can be embedded into its Dedekind completion D(P ), which is a Dedekind complete lattice, the embedding preserves all existing meets and joins in P , and is both join-dense and meet-dense. Nakano [20, Sec. 31] showed that if A is an Archimedean R-vector lattice, then the Dedekind completion of A is a unique up to isomorphism Dedekind complete Archimedean R-vector lattice D(A)inwhichA is both join-dense and meet-dense. Furthermore, if A is a bounded Archimedean -algebra satisfying the following condition: If bi =0 and a ≥ 0, then abi =0, (∗) i i then D(A) is a Dedekind complete bounded Archimedean -algebra. Johnson [16] veriﬁed that Condition (∗) holds for every Archimedean f-ring, so each A ∈ ba satisﬁes Condition (∗). Thus, we arrive at the following theorem. Theorem 3.1 (Nakano [20, Sec. 31] and Johnson [16, p. 493]). If A ∈ ba , then D(A) ∈ ba . Moreover, there is an embedding δA : A → D(A) in ba such that δA preserves all existing meets and joins in A, and δA(A) is meet-dense in D(A). Furthermore,D(A) is unique in the sense that if β : A → B is a monomorphism in ba , where B is Dedekind complete and β(A) is meet-dense in B, then there is an -algebra isomorphism γ : D(A) → B such that γ ◦ δA = β.

Remark 3.1. The correspondence a →−a gives that A is meet-dense in B if and only if A is join-dense in B. Therefore, in Theorem 3.1 we only need that A is meet-dense (equivalently, join-dense) in D(A)becauseitimpliesthatA is also join-dense (equivalently, meet-dense) in D(A). In the category of modules over a ring (see, e.g. [18, Sec. 1.3]), essential extensions are fundamental for studying injective hulls. In analogy with corresponding deﬁnitions for various subcategories of -groups (see, e.g. [2, 5, 6]), we deﬁne a monomorphism α : A → B in ba to be essential if for each nonzero -ideal I of B, the ideal α−1(I)ofA is nonzero.

Remark 3.2. An alternative, more categorical definition would be to say that a monic morphism α is essential if for each morphism β, whenever β ◦α is monic, then β is monic. While it may appear that this is a weaker definition since not all -ideals are kernels of morphisms in ba ,Conrad[5, Theorem 3.7] proves that these two definitions are equivalent for Archimedean -groups. We will use this equivalence in the proof of Theorem 3.3 below.

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For X, Y ∈ KHaus, we recall that an onto continuous map ϕ : Y → X is irreducible if ϕ(F ) is a proper subset of X foreachproperclosedsubsetF of Y .

Theorem 3.2. Let α : A → B be a monomorphism in ba . Then the following statements are equivalent. (1) α is essential. (2) α(A) is meet-dense in B. (3) There exists a monomorphism β : B → D(A) in ba such that β ◦ α is the canonical mapping δA : A → D(A). ∗ (4) The induced mapping α : XB → XA is irreducible.

Proof. Without loss of generality, we assume throughout the proof that A is an -subalgebra of B and α is the inclusion mapping. (1) ⇒ (2): Let b ∈ B. We claim that b is the meet of S := {a ∈ A : b ≤ a}. Clearly b is a lower bound of S.Letc be another lower bound of S, and suppose that b

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∗ (1) ⇒ (4): Suppose that F is a closed subset of XB such that α (F )=XA.Let I F be the intersection of all the maximal -ideals in . Then, as an intersection of ∗ −1 -ideals, I is also an -ideal. Since α (F )=XA, it follows that α (I)= XA =0. Thus, by (1), I =0,sothatsinceF = Z(I), it must be that F = XB,proving that α∗ is irreducible. (4) ⇒ (1): Suppose α∗ is irreducible and I is a nonzero -ideal of B.Then ∗ ∗ ∗ −1 Z(I) = XB.Asα is irreducible, α (Z(I)) = XA.Butα (Z(I)) = Z(α (I)). −1 −1 Therefore, Z(α (I)) = XA, yielding that α (I) is nonzero.

Remark 3.3. The equivalence of (1) and (2) can also be derived from [5,Proposi- tion 3.1], but we included our proof because it is more direct. The equivalence of (2), (3) and (4) was proved in [11, Theorem 1] for C∗Alg using Gelfand-Neumark-Stone duality and the existence of projective covers of compact Hausdorﬀ spaces. How- ever, this duality weakens to an adjunction when extended to ba (see [4, Sec. 3]), and hence is too coarse for our setting. Instead, an algebraic approach is needed for our version of the theorem.

Corollary 3.1. Every bimorphism in ba is essential.

Proof. Let α : A → B be a bimorphism in ba .By[4, Lemma 3.3], since α is a ∗ bimorphism, the induced mapping α : XB → XA is a homeomorphism, and hence is irreducible. Thus, by Theorem 3.2, α is essential.

Deﬁnition 3.1. A Dedekind algebra is a Dedekind complete bounded Archimedean -algebra. Let DA be the full subcategory of ba consisting of all Dedekind algebras.

Remark 3.4. The requirement that a Dedekind algebra be Archimedean is in fact superﬂuous since every Dedekind complete -algebra is necessarily Archimedean. More generally, as remarked in [7, p. 182], every Dedekind complete -group is Archimedean.

Remark 3.5. It is well known and easy to see that if A is a Dedekind algebra, then ∗ A is uniformly complete, hence a C -algebra. Indeed, if {an} is a Cauchy sequence A {a } in , then it is easy to see that n is bounded from the Cauchy condition. Set sm = n≥m an and tm = n≥m an. Obviously sm and tm exist because A is a {s } {t } Dedekind algebra. Moreover, n is an increasing sequence, n is a decreasing sequence, and sn ≤ an ≤ tn for all n ∈ N.Sets = n sn and t = n tn.Since{an} is Cauchy, it is an elementary analysis argument that s = t = limn→∞ an.Thus, {an} converges.

Let A, B ∈ ba . Following [5], we say that B is an essential extension of A if there is an essential monomorphism α : A → B in ba ,thatB is essentially closed

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if it admits no proper essential extension in ba ,andthatB is the essential closure of A if B is an essential extension of A and B is essentially closed. In [10], Gleason proved that the projective objects in KHaus are precisely the extremally disconnected compact Hausdorff spaces, and that for each compact Hausdorff space X there exists an extremally disconnected compact Hausdorff space Y andanirreduciblemapϕ : Y → X such that for each extremally disconnected compact Hausdorff space Z and irreducible map η : Z → X, there is a homeomorphism σ : Y → Z such that η ◦ σ = ϕ. Thus, every compact Hausdorff space has a projective cover, also known as the Gleason cover. In particular, if X ∈ KHaus, Y ∈ ED,andϕ : Y → X is an irreducible map, then (up to homeomorphism) Y is the Gleason cover of X. Gleason covers have played an important role since their introduction in 1958 (see, e.g. [17, 21]).

Theorem 3.3. Let A ∈ ba . Then the following conditions are equivalent: (1) A is a Dedekind algebra. (2) A is a Baer C∗-algebra. (3) A is an injective object in ba . (4) A is essentially closed.

Proof. (1) ⇒ (2): Since A is a Dedekind algebra, A is a C∗-algebra, and hence the canonical mapping φA : A → C(XA) is an isomorphism. Therefore, as A is Dedekind complete, by the Stone–Nakano Theorem [22, 19] (see also [23], [8, Theorem 5.1], [9, p. 52], [22, p. 662], or [21, p. 662]), XA is an extremally disconnected space. Thus, by [4, Theorem 5.15], A is a Baer ring. (2) ⇒ (3): Let A be a Baer C∗-algebra. Suppose that α : B → C and β : B → A are morphisms in ba with α monic. Since C∗Alg is a reﬂective subcategory of ba (see, e.g. [4, Sec. 3]), there is a morphism γ : C(XB ) → A in ba such ∗ that γ ◦ φB = β.WealsohaveφC ◦ α = α ◦ φB . Consider the continuous maps γ∗ X → X ∼ X α∗ X → X α∗ α A : A C(XB ) = B and : C B,where is onto as is monic. As is Baer, by [4, Theorem 5.15], XA is extremally disconnected. By Gleason’s theorem, XA is projective in KHaus. Therefore, there exists a continuous map ϕ : XA → XC ∗ ∗ such that α ◦ ϕ = γ .AsA is isomorphic to C(XA), by Gelfand-Neumark-Stone ∗ duality, there is a morphism : C(XC ) → A in ba with ◦ α = γ. This implies that ◦ φC : C → A is a morphism in ba such that ( ◦ φC ) ◦ α = β.Consequently, A is injective in ba . α / B FF C FF FFφB FF φC F# αf∗ β C X /C X ( B ) j ( C ) xx jjj γ xx jjjj xx jjj xx jjj uj{xjjj A

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(3) ⇒ (4): Let A be injective in ba and let α : A → B be an essential morphism in ba .SinceA is injective, A is a retract of B, so there is an onto morphism β : B → A in ba with β ◦ α =idA.Asα is essential, by Remark 3.2, β is monic. Therefore, β is an isomorphism, which implies α is an isomorphism. Thus, A is essentially closed. (4) ⇒ (1): By Theorem 3.2, the canonical map δA : A → D(A) is essential. Thus, ∼ as A is essentially closed, A = D(A), and so A is a Dedekind algebra.

Remark 3.6. Theorem 3.3 implies that the category BC∗Alg of Baer C∗-algebras is isomorphic to DA. Therefore, by [4, Theorem 5.26.2], DA is dually equivalent to ED. We show that unlike Gelfand-Neumark-Stone duality for C∗-algebras, the duality between DA and ED does not require the use of analysis in the form of the Stone–Weierstrass Theorem (SW). (For a discussion of the relationship between versions of the Stone–Weierstrass Theorem and Gelfand-Neumark-Stone duality, see [4, Sec. 3].) The crucial step in the proof of the statement is showing that for a Dedekind algebra A, the canonical mapping φA : A → C(XA) is an isomorphism. A weak version of (SW) states that if A ∈ ba and φA : A → C(XA) is the canonical ∗ mapping, then φA(A) is uniformly dense in C(XA). Thus, when A is a C -algebra, φA must be an isomorphism. For a Dedekind algebra A,weobservethatφA is an isomorphism without the use of even this weak version of (SW).ForletA ∈ DA and φA : A → C(XA) be the canonical bimorphism. By Corollary 3.1, φA is essential. Therefore, by Theorem 3.2, there is a monomorphism β : C(XA) → D(A)inba such that β ◦ φA = δA.ButasA is a Dedekind algebra, A is Dedekind complete, hence δA is an isomorphism. Thus, φA is an isomorphism.

Let A, S ∈ ba and let α : A → S be a monomorphism in ba . Following the terminology in the category of modules over a ring (see, e.g. [18, Sec. 1.3]), we say that S is the injective hull of A if S is an injective object in ba and α is essential.

Corollary 3.2. For A, B ∈ ba , the following conditions are equivalent.

(1) B is isomorphic to the Dedekind completion of A. (2) B is the injective hull of A in ba . (3) B is the essential closure of A. (4) B is isomorphic to C(XB ) and XB is the Gleason cover of XA.

∼ Proof. (1) ⇒ (2): Suppose that B = D(A). Then B is Dedekind complete. There- fore, by Theorem 3.3, B is injective. Also, by Theorem 3.2,themapδA : A → D(A) is essential, and so B is the injective hull of A. (2) ⇒ (3): Since B is injective, it is essentially closed by Theorem 3.3, and since there is an essential monomorphism A → B, it follows that B is the essential closure of A.

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(3) ⇒ (4): By assumption there is an essential monomorphism α : A → B,and ∗ hence by Theorem 3.2, α : XB → XA is an irreducible map. By Theorem 3.3, B ∗ ∼ is a Baer C -algebra, and hence B = C(XB), with XB extremally disconnected, where the last assertion follows from the fact that C ∈ ba is a Baer ring if and only if XC is extremally disconnected [4, Theorem 5.15]. Thus, XB is the Gleason cover of XA. (4) ⇒ (1): Since XB is the Gleason cover of XA,byTheorem3.2, α(A) is meet- ∗ dense in B.AsB is isomorphic to C(XB), we have that B is a Baer C -algebra. Therefore, by Theorem 3.3, B is Dedekind complete. Thus, by Theorem 3.1, B is isomorphic to the Dedekind completion of A.

Remark 3.7. Parts of Theorem 3.3 and Corollary 3.2 can be derived from Conrad [5, Remark, p. 159] and Banaschewski and Hager [2, p. 122], but our approach is diﬀerent. Gleason used projective covers and Gelfand-Neumark-Stone duality to prove that injective hulls exist in C∗Alg (see [10, Theorem 5.1]). With this insight, several results about injective hulls in C∗Alg were obtained by Gonshor [11, 12]. Corollary 3.2 is stronger in that it shows that injective hulls exist already in ba .

Remark 3.8. The argument in the proof of Corollary 3.2 can be extended to show that whenever α : A → B is a monomorphism in ba ,thenXB is the Gleason cover of XA if and only if B is a Baer ring and α is essential.

One of the key themes of this paper is associating to each A ∈ ba the Dedekind algebra D(A). In [4], we associated other “nice” objects to A (e.g. C∗-algebras, Gelfand rings, -clean rings and Specker algebras), but where the present construc- tion diﬀers is that unlike the aforementioned examples, the assignment A → D(A) is not functorial. To see this, we need the following proposition, which is of indepen- dent interest. Let bba be the full subcategory of ba consisting of those objects in ba that are Baer rings.

Proposition 3.1. Let A ∈ ba . Then the following conditions are equivalent:

(1) δA : A → D(A) is an epimorphism. (2) δA : A → D(A) is a bimorphism. (3) A ∈ bba.

Proof. Since δA is always monic, it is clear that (1) is equivalent to (2). ∗ (2) ⇒ (3): Let δA : A → D(A) be a bimorphism. By [4, Lemma 3.3], δA : XD(A) → XA is a homeomorphism. This forces XA to be extremally disconnected. Therefore, by [4, Theorem 5.15], A ∈ bba. (3) ⇒ (2): Let A ∈ bba.By[4, Theorem 5.15], XA ∈ ED.ByTheorem3.2, ∗ ∗ δA : XD(A) → XA is irreducible. This forces δA tobeahomeomorphism,soby [4, Lemma 3.3], δA is a bimorphism.

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We are ready to show that under some natural conditions D : ba → DA is not a functor, but that restricting D to bba becomes a functor. Assume that D : ba → DA is a functor satisfying the following two natural conditions: (i) δB ◦ α = D(α) ◦ δA for each morphism α : A → B in ba and (ii) D(δA):D(A) → D(D(A)) is an epimorphism. Then by [1, Lemma 3.3], δA : A → D(A)isalsoan epimorphism. Therefore, by Proposition 3.1, A ∈ bba. The obtained contradiction proves that D : ba → DA is not a functor. On the other hand, for A ∈ bba, we have D(A) is isomorphic to C(A). To see this, let A ∈ bba.By[4,Theo- ∗ rem 5.15], XA is an extremally disconnected space. Therefore, C(XA)isaBaerC - algebra, hence by Theorem 3.3, C(XA) is a Dedekind algebra. As φA : A → C(XA) φ∗ X → X is monic and A : C(XA) A is a homeomorphism, Theorem 3.2 yields that φA(A) is meet-dense in C(XA). Thus, by Theorem 3.1, C(XA) is isomorphic to D(A). Consequently, the assignment bba → DA : A → D(A) can be viewed as an alternative expression of the functor A → C(XA). In [4, Theorem 3.10 and Corol- lary 3.11], it is shown that C∗Alg is the smallest reflective subcategory of ba ,as well as the unique epicomplete reflective subcategory of ba (we use the conven- tion that reflective subcategories are full and closed under isomorphisms). It follows that DA is the smallest reflective subcategory of bba and the unique epicomplete reflective subcategory of bba. The subcategory DA is not the only proper reflective subcategory of bba: The full subcategory of Gelfand rings in bba is also a proper reflective subcategory of bba; this follows from [4, Proposition 4.2 and Example 5.17].

4. A Characterization of C∗-Algebras In [4, Theorem 3.4] it is shown that A ∈ ba is a C∗-algebra if and only if each bimorphism α : A → B in ba is onto. This characterization requires quantiﬁcation over all bimorphisms α : A → B in ba . As an application of the results obtained in the previous section, we show that this quantiﬁcation can be restricted to the bimorphisms of the form α : A → B,whereB is an -subalgebra of D(A).

Theorem 4.1. Let A ∈ ba .ThenA is a C∗-algebra if and only if every bimorphism α : A → B, where B is an -subalgebra of D(A), is onto.

Proof. Suppose ﬁrst that A is a C∗-algebra. By [4, Theorem 3.4], each bimorphism α : A → B in ba is onto. In particular, each bimorphism α : A → B,whereB is an - subalgebra of D(A), is onto. Conversely, it suﬃces to show that every bimorphism A → B is onto. Let α : A → B be a bimorphism in ba . By Corollary 3.1, α is essential, so that by Theorem 3.2, there exists a monomorphism β : B → D(A)in ba such that β ◦ α = δA. We can then view δA as a morphism A → β(B), and since it is a composition of bimorphisms, it is a bimorphism. Because β(B)isan -subalgebra of D(A), we obtain that δA(A)=β(B). As β is monic, we conclude that α is onto.

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Next we show that each C∗-algebra A is the inverse limit of an inverse family of clean C∗-algebras, each of which is isomorphic to an -subalgebra of D(A). As a result, we obtain that A ∈ ba is a C∗-algebra if and only if A is an inverse limit of clean C∗-algebras; that is, algebras of the form C(X)withX a Stone space. We ﬁrst show that inverse limits exist in ba , and that inverse limits in ba of C∗ C∗ {A i ∈ I} ba -algebras are -algebras. Let i : be a family from .Withpointwise operations, it is clear that i∈I Ai is an Archimedean -algebra. However, the product may not be bounded. If we set A = (ai) ∈ Ai :thereisn ∈ N with |ai|≤n for all i ,

then A ∈ ba .

Proposition 4.1. (1) Inverse limits exist in ba . (2) Inverse limits in ba of C∗-algebras are C∗-algebras.

Proof. (1) Let {Ai,αij } be an inverse family in ba ,andlet

B = (ai) ∈ Ai : αij (aj)=ai for j ≥ i . i We let A be the bounded -subalgebra of B;thatis,

A = {a ∈ B : |a|≤n for some n ∈ N}.

Then A ∈ ba and it is easy to check that A is the inverse limit of {Ai,αij }. Thus, inverse limits exist in ba . (2) Since C∗Alg is a reﬂective subcategory of ba [4, Sec. 3], this follows from [15, Theorem 36.16].

Lemma 4.1. Let X be a Stone space and let F be a ﬁnite subset of X.IfZ is the quotient space of X obtained by identifying the points in F, then Z is a Stone space.

Proof. Let X be a Stone space, F be a finite subset of X,andZ be the quotient space of X obtained by identifying the points in F .Let∼F be the equivalence relation on X whose only nontrivial equivalence class is F .Ifx, y ∈ X and x ∼F y, then as X is a Stone space and F is finite, it is easy to find a clopen set U of X

such that U separates x and y and either F ⊆ U or F ∩ U = ∅.AsZ = X/∼F ,it follows from [3, Lemma 10] that Z is a Stone space.

Theorem 4.2. Each compact Hausdorﬀ space X is the direct limit of a direct family of Stone spaces, each of which is a quotient space of the Gleason cover of X.

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Proof. Let X be a compact Hausdorff space. We let Y be the Gleason cover of X and π : Y → X be the corresponding irreducible map. For each x ∈ X, the fiber of −1 x in Y is π (x). Let F be the set of finite subsets of Y of the form Fx1 ∪···∪Fxn , −1 where Fxi ⊆ π (xi). Clearly (F, ⊇) is a directed set. For F, G ∈Fwith G ⊇ F , the canonical map πFG : YF → YG is obviously continuous where YH = Y/∼H for each H ∈F,andforH ⊇ G ⊇ F we have πGH ◦ πFG = πFH.Thus,{YF ,πFG} ∼ Y is a directed family of Stone spaces. Let be the equivalence relation on that identifies all the fibers. Then X is homeomorphic to Y/∼.Moreover,∼= {∼F : F ∈F}. Thus, up to homeomorphism, X is the direct limit of {YF ,πFG}.

Corollary 4.1. Each C∗-algebra A is an inverse limit of clean C∗-algebras, each of which embeds into the Dedekind algebra D(A).

Proof. Let A be a C∗-algebra. By Gelfand-Neumark-Stone duality, A is isomorphic to C(XA). As XA is compact Hausdorﬀ, by Theorem 4.2, XA is the direct limit of a direct family {Xi,ϕij } of Stone spaces, each of which is a quotient space of the Gleason cover of XA. By Corollary 3.2, the Gleason cover of XA is XD(A).As each Xi is a quotient of XD(A),eachC(Xi)embedsinD(A). Since each Xi is a ∗ Stone space, each C(Xi) is a clean C -algebra. Therefore, {C(Xi), ϕij } is an inverse ∗ family of clean C -algebras, and C(XA) is the inverse limit of {C(Xi), ϕij }.

Theorem 4.3. A ∈ ba is a C∗-algebra if and only if A is an inverse limit of clean C∗-algebras.

Proof. If A ∈ ba is a C∗-algebra, then by Corollary 4.1, A is an inverse limit of clean C∗-algebras. Conversely, if A is an inverse limit of clean C∗-algebras, then by Proposition 4.1, A is a C∗-algebra.

Appendix. An Algebraic Proof of Gleason’s Theorem The proof of Theorem 3.3 relies heavily on Gleason’s theorem.

Gleason’s Theorem.

(1) The projective objects in KHaus are precisely the extremally disconnected compact Hausdorﬀ spaces. (2) Each compact Hausdorﬀ space X has a projective cover Y .

Since by Gelfand-Neumark-Stone duality, the category KHaus of compact Hausdorﬀ spaces is dual to the category C∗Alg of C∗-algebras, part one of Glea- son’s theorem gives that the injective objects in C∗Alg are precisely the C∗-algebras C(X), where X ∈ ED.In[24], Wright gave a direct proof of this statement using functional analysis, and obtained part one of Gleason’s theorem as a corollary. Also,

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Gelfand-Neumark-Stone duality and part two of Gleason’s theorem give that injective hulls exist in C∗Alg.In[11, 12], Gonshor used this insight to obtain several results about injective hulls in C∗Alg. In keeping with the spirit of the present paper, as well as [4], we discuss a diﬀerent proof of Gleason’s theorem that emphasizes algebraic ideas. In fact, we prove the following stronger algebraic statement in the larger category ba :

Gleason’s Theorem (Algebraic Version).

(1) The injective objects in ba are precisely the Dedekind algebras. (2) If A ∈ ba , then D(A) is the injective hull of A in ba .

This of course was proved already in Theorem 3.3 and Corollary 3.2, but the proof given there relied on Gleason’s theorem, which is precisely what we wish to give an alternative proof for in this section.

Proofof(1):The left to right implication is easy to prove. Indeed, if A is an injective object in ba , then there is α : D(A) → A with α ◦ δA =idA.Since δA is essential, α is monic, and hence an isomorphism. Thus, A is a Dedekind algebra. For the right to left implication, let A be a Dedekind algebra. The key step is to show that for each monomorphism α : A → B in ba , there is a morphism β : B → A in ba with β ◦ α =idA.ByZorn’slemma,thereisan-ideal J of B maximal with respect to the condition α−1(J) = 0. Therefore, the canonical map α : A → B/J is an essential monomorphism. Thus, by Theorem 3.2,eachelement of B/J is a meet of elements from α(A). Since essential monomorphisms preserve existing meets and α(A) is meet-dense in B/J,wehaveα is onto, and hence an isomorphism. Now, setting β =(α)−1 ◦ π,whereπ : B → B/J is the canonical map, −1 −1 we obtain β ◦ α =(α) ◦ π ◦ α =(α) ◦ α =idA, as desired. We are ready to prove that A is an injective object in ba .Letα : B → C ∗ and β : B → A be morphisms in ba ,withα monic. Then α : XC → XB and ∗ ∗ β : XA → XC are continuous, with α onto as α is monic. Let X be the pullback of this diagram in the category KHaus of compact Hausdorﬀ spaces and continuous maps. Therefore, there exist continuous maps ϕ : X → XC and η : X → XA such that α∗◦ϕ = β∗ ◦η.Asα∗ is onto, so is η. This yields ϕ◦α∗ = η◦β∗,whereη is monic as η is onto. Since η◦ φA is monic, our key step in the previous paragraph produces a morphism γ : C(X) → A in ba with γ ◦ η◦ φA =idA.Set = γ ◦ ϕ◦ φC : C → A. Then

∗ ◦ α = γ ◦ ϕ ◦ φC ◦ α = γ ◦ ϕ ◦ α ◦ φB

∗ = γ ◦ η ◦ β ◦ φB = γ ◦ η ◦ φA ◦ β

=idA ◦ β = β.

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This proves A is an injective object in ba . α / B FF C FF FF FF FFφB FFφC FF FF F# F# αf∗ / β C(XB ) C(XC )

f∗ AWFF β ϕe FF FFφA FF F# ηe / C(XA) C(X)

Proof of (2): Note that D(A) is injective by (1), and, for each A ∈ ba ,the canonical embedding δA : A → D(A) is essential (Theorem 3.2). Thus, D(A)isthe injective hull of A.

We conclude by indicating how Gleason’s theorem, as stated at the beginning of the appendix, can be deduced as a corollary of the algebraic version. By (1), the injectives in ba are the Dedekind algebras. Since Dedekind algebras are C∗- algebras and the reﬂector A → C(XA) preserves monomorphisms, it follows that the injectives in C∗Alg are precisely the Dedekind algebras. Gelfand-Neumark-Stone duality and the Stone–Nakano Theorem then imply that the projectives in KHaus are precisely the extremally disconnected spaces in KHaus.Next,letX ∈ KHaus. By (2), D(C(X)) is the injective hull of C(X). By (1), Y = XD(C(X)) is projective and, by Theorem 3.2, the dual map Y → X is irreducible. So, Y is the projective cover of X.

Acknowledgments We are very grateful to the referee whose suggestions have greatly improved the proofs and exposition of the main results of the paper. We also thank Warren McGovern for his suggestions and pointers to the literature.

References [1] J. Ad´amek, H. Herrlich, J. Rosick´y and W. Tholen, Injective hulls are not natural, Algebra Universalis 48 (2002) 379–388. [2] B. Banaschewski and A. W. Hager, Injectivity of Archimedean -groups with order unit, Algebra Universalis 62 (2009) 113–123. [3] G. Bezhanishvili, Varieties of monadic Heyting algebras. II. Duality theory, Studia Logica 62 (1999) 21–48.

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