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DEDEKIND COMPLETIONS of BOUNDED ARCHIMEDEAN C December 6, 2012 10:1 WSPC/S0219-4988 171-JAA 1250139 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; essen- tial 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 (commuta- tive 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 1250139-1 December 6, 2012 10:1 WSPC/S0219-4988 171-JAA 1250139 G. Bezhanishvili, P. J. Morandi & B. Olberding 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 comple- tion 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 con- tinuous 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 1250139-2 December 6, 2012 10:1 WSPC/S0219-4988 171-JAA 1250139 Dedekind Completions of Bounded Archimedean -Algebras 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 inter- section 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 repre- sentation 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 1250139-3 December 6, 2012 10:1 WSPC/S0219-4988 171-JAA 1250139 G. Bezhanishvili, P. J. Morandi & B. Olberding 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.
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