Stone-Gelfand duality for metrically complete lattice-ordered groups

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arXiv:2210.15341v3 [math.FA] 25 Nov 2024

Stone-Gelfand duality for metrically complete lattice-ordered

groups

Marco Abbadinia, Vincenzo Marrab,∗, Luca Spadac

aSchool of Computer Science, University of Birmingham, B15 2TT Birmingham, United Kingdom

bDipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, via Cesare Saldini 50,

20133 Milano, Italy

cDipartimento di Matematica, Università degli Studi di Salerno, Piazza Renato Caccioppoli, 2, 84084 Fisciano

(SA), Italy

Abstract

We extend Yosida’s 1941 version of Stone-Gelfand duality to metrically complete unital lattice-

ordered groups that are no longer required to be real vector spaces. This calls for a generalised

notion of compact Hausdorﬀ space whose points carry an arithmetic character to be preserved

by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant

of) a metrically complete additive subgroup of the real numbers containing 1—namely, either

nZfor an integer n= 1,2,..., or the whole of R. The main result needed to establish the ex-

tended duality theorem is a substantial generalisation of Urysohn’s Lemma to such “arithmetic”

compact Hausdorﬀ spaces. The original duality is obtained by considering the full subcategory

of spaces every point of which is assigned the entire group of real numbers. In the Introduction

we indicate motivations from and connections with the theory of dimension groups.

Keywords: Stone-Gelfand duality, Lattice-ordered group, Compact Hausdorﬀ space, Normal

space, Urysohn’s Lemma, Tychonoﬀ cube.

2020 MSC: Primary: 06F20. Secondary: 54A05, 54C30.

1. Introduction

Let us write KH for the category of compact Hausdorﬀ spaces and continuous maps. In the

Thirties and Forties of the last century a number of mathematicians realised that the opposite

category KHop can be represented, to within an equivalence, in several useful ways that variously

relate to known mathematical structures of relevance in functional analysis. It is common to

refer to this conglomerate of results by the umbrella term Stone-Gelfand Duality. To provide a

historically accurate picture of these theorems and of their mutual relationships would require

far more space than we can aﬀord here. Let us mention in passing that the authors involved

include at least Gelfand and Kolmogorov [11], Gelfand and Neumark [12], the Krein brothers

[19], Kakutani [17], Yosida [26], and Stone [24]; and that the mathematical structures used

in these representation theorems include at least rings of continuous functions, commutative

C∗-algebras, Kakutani’s (M)-spaces, vector lattices (alias Riesz spaces), and divisible Abelian

lattice-groups. In the present paper the more relevant structures are the ones used by Yosida,

as follows.

∗Corresponding author

Email addresses: marco.abbadini.uni@gmail.it (Marco Abbadini), vincenzo.marra@unimi.it

(Vincenzo Marra), lspada@unisa.it (Luca Spada)

Preprint submitted to arXiv

Let us write Vfor the category whose objects are unital vector lattices—that is, lattice-

ordered real vector spaces equipped with a distinguished unit—and whose morphisms are the

linear lattice homomorphisms preserving the distinguished units. We recall that an element 1

of the vector lattice Vis a (strong order)unit if for any v∈Vthere is a positive integer p

such that v6p. (In the latter expression, pstands for p·1, and we shall use such shorthands

without further warning.) Then Vcan be equipped with a pseudometric induced by its unit 1

by setting, for v, w ∈V,

kv−wk:= inf {λ∈R|λ>0 and λ>|v−w|},(*)

where |x|:=x∨(−x) is the absolute value of x∈V. This pseudometric is a metric just when

the vector lattice is Archimedean, i.e., if for all x, y ∈V, whenever px 6yfor all positive

integers p, then x60. A unital vector lattice Vis then metrically complete if the pseudometric

induced by the unit 1 is a metric, and Vis (Cauchy) complete in this metric. Thus, for instance,

the pseudometric induced on the vector lattice C(X) of real-valued continuous functions on a

compact Hausdorﬀ space Xby the unit given by the function X→Rconstantly equal to 1 is

the uniform metric, and C(X) is of course metrically complete.

Now, there is an adjunction between Vand KHop that takes a unital vector lattice Vto

its maximal spectral space, that is, its set of maximal (vector-lattice) ideals topologised by

the hull-kernel topology; and a compact Hausdorﬀ space Xto the unital vector lattice C(X).

The adjunction descends to an equivalence between the full subcategory of Von the metrically

complete vector lattices, and KHop. This is the version of Stone-Gelfand Duality obtained by

Yosida.

From a ﬁrst perspective, in this paper we pursue the extension of Yosida’s version of Stone-

Gelfand Duality obtained by relaxing the hypothesis that Vbe a real linear space, allowing Vto

be instead a lattice-ordered Abelian group. Let us recall that a lattice-ordered group (ℓ-group, for

short) is a group endowed with a lattice structure such that the group operation distributes over

both lattice operations; for background, please see [1]. The notions of unit and of pseudometric

induced by the unit are deﬁned for Abelian ℓ-groups as in the vector lattice case—in order to

use only the Z-module structure of the underlying Abelian group in deﬁning the pseudometric,

replace the right-hand side of (*) by

inf ßp

q∈Q|p>0, q > 0,and p>q|v−w|™.(†)

Archimedean unital ℓ-groups are deﬁned as in the vector-lattice case; it is a standard result

that Archimedean ℓ-groups are Abelian [1, Théorème 11.1.3], and it is again true that the

pseudometric is a metric precisely when the Archimedean condition holds. A unital ℓ-group is

metrically complete if it is Archimedean and complete in the metric induced by its unit. We

abbreviate ‘unital lattice-ordered Archimedean (hence Abelian) group complete in the metric

induced by its unit’ by metrically complete ℓ-group. We thus have the category Gof unital

Abelian ℓ-groups and lattice-group homomorphisms preserving the distinguished unit; and a

full subcategory Gon those objects which are metrically complete.

Then the question arises, what should replace the category KH in order to regain a dual

equivalence with G.

To obtain some insight, consider the unit interval [0,1] ⊆Rand the ℓ-group ∇of continuous

piecewise-aﬃne functions [0,1] →Rsuch that each aﬃne piece a(x):=z1x+z2has integer

coeﬃcients z1, z2∈Z, with unit the function [0,1] →Rconstantly equal to 1. Then ∇is a

subobject (i.e., a unital sublattice subgroup, known as unital ℓ-subgroup) in Gof C([0,1]). The

ℓ-group ∇is not metrically complete. To compute its completion, let ﬁrst Ax:={z1x+z2|

z1, z2∈Z}be the set of values attained by elements of ∇at the point x∈[0,1]. Next

observe that Axis a dense subgroup of Rif xis an irrational number, but it is the discrete

subgroup 1

dZ:={z

d|z∈Z}if xis a rational number of denominator d. It follows that the

metric completion of ∇is the subobject ∇of C([0,1]) consisting of those functions that, at each

rational point of [0,1] of denominator d, are constrained to take values in the subgroup 1

dZ.

The maximal spectral space of C([0,1])—that is, its Stone-Gelfand-Yosida dual space—is

just [0,1]. However, this is also the maximal spectral space of both ∇and ∇, because the

functions in ∇separate points. In order to discern C([0,1]) from ∇, their maximal spectral

space [0,1] can be enriched by information that speciﬁes, for each x∈[0,1], which subgroup of

Rthe functions in the dual ℓ-group can take values in. This can be summarised by the map

den: [0,1] →Nwhich assigns to a rational number x∈[0,1] its denominator d∈N, meaning

that the set of possible values of functions at xis 1

dZ; and to an irrational number x∈[0,1] the

value 0, meaning that, at x, the values of the functions can range in the whole group R. Here, the

choice of 0 is natural rather than conventional: it will transpire that a morphism, say, ∇ → ∇,

contravariantly induces a continuous map f: [0,1] →[0,1] which decreases denominators in the

sense that it carries denominators to their divisors, i.e., for x∈[0,1], den f(x) divides den x;

thus, there is no constraint on where such a map may send a point of denominator 0.

What characteristic properties does the assignment den: [0,1] →Nsatisfy? We just pointed

out that the dual continuous maps decrease denominators, and so the divisibility order of Nis

relevant; further inspection conﬁrms that den : [0,1] →Nis continuous when its codomain Nis

equipped with the upper topology for its divisibility order—the topology whose closed sets are

precisely the ﬁnite downsets, along with Nitself. (Please see Section 4 for more details on the

upper topology.)

Abstracting from this example with the beneﬁt of considerable hindsight, we deﬁne an arith-

metic space (a-space, for short) to be a topological space equipped with a denominator map, a

function to Nthat is continuous with respect to the upper topology induced by the divisibility

order of the natural numbers. Morphisms between a-spaces are continuous maps that decrease

denominators with respect to the divisibility order; we call them a-maps. Fundamental examples

of a-maps are (restrictions of) aﬃne maps between ﬁnite-dimensional real vector spaces which

descend to aﬃne maps between the Z-submodules of lattice (i.e., integer-coordinate) points with

respect to chosen bases.

Now, it is clear that not every a-space can arise as the dual object of a metrically complete ℓ-

group; at the very least, the underlying space ought to be compact and Hausdorﬀ. In Deﬁnition

Deﬁnition 4.1 we are able to identify the needed subclass of arithmetic spaces, which we call

arithmetically normal; as we shall prove, they provide the appropriate substitute for KH. In

other words, we prove the following generalisation of Yosida duality: The category of metrically

complete ℓ-groups is dually equivalent to the category of arithmetically normal spaces and a-

maps between them.

Before proceeding, we should emphasise that the category of a-spaces admits a more con-

ceptual presentation through lax comma categories; we provide some details on this intriguing

connection in Remark 2.3 below.

From a second perspective, our main result is motivated by, and relates to, the theory

of approximately ﬁnite-dimensional (AF) C∗-algebras [7] and their ordered K0groups, known

as dimension groups [13]. In the remarks that follow, C∗-algebras and dimension groups are

always assumed to be unital. Elliott [9] showed dimension groups to be complete isomorphism

invariants of AF C∗-algebras. Elliott’s result generalised previous well-known work by Glimm

(1960) and Dixmier (1967), in turn rooted in the classical Murray-von Neumann approach to

the classiﬁcation of factors. Incidentally, we may now point out the signiﬁcant conceptual

diﬀerences between the two perspectives we are discussing in this Introduction. Let Xbe a

compact Hausdorﬀ space. In the ﬁrst perspective, elements of C(X) arise as observables of

a classical—i.e., a commutative C∗-algebra—system. In the second perspective, they arise as

Murray-von Neumann “dimensions” of a suﬃciently tame—i.e., an AF C∗-algebra with lattice-

ordered K0—non-classical system.

Eﬀros, Handelman, and Shen proved ([8, Theorem 2.2], see also [13, Chapter 3]) that dimen-

sion groups are exactly the directed, isolated—also known as “unperforated”—unital partially

ordered Abelian groups satisfying the Riesz interpolation property; any unital Abelian ℓ-group

is such. Goodearl and Handelman oﬀered in [14] a systematic investigation of completions of a

unital directed partially ordered Abelian groups with respect to the pseudometric induced on

it by a state, i.e., a normalised positive group homomorphism to the reals. Within the theory

they developed, Goodearl and Handelman obtained the following representation theorem for

metrically complete unital Abelian ℓ-groups.1

Theorem 1.1 (Goodearl and Handelman, [14, Theorem 5.5]).Let Xbe a compact Hausdorﬀ

space. For each x∈X, choose a subgroup Axof R, so that Ax:=Ror Ax:=1

nZfor some

positive integer n. Set

B:={p∈C(X)|p(x)∈Axfor each x∈X},

and give Bthe order inherited from C(X). Then Bis an Archimedean metrically complete

lattice-ordered Abelian group with unit. Conversely, any such group is isomorphic to one of this

form.

We retain the notation of Theorem 1.1 in the remarks that follow. The maximal spectrum Yof

Bis always a continuous image of X, but Xand Ymay not be homeomorphic because Bmay

fail to separate the points of X. For instance, take X= [0,1] and Ax=Zfor each x∈[0,1].

Then Bis the set of constant functions from [0,1] to Z; thus, Bis Zand Yis a singleton.

Moreover, diﬀerent assignments {Ax}x∈Xon a space Xmay give rise to the same ℓ-group B.

Consider for instance the one-point compactiﬁcation of the discrete space N, with the following

two assignments. The ﬁrst assignment is constantly Z; it yields an ℓ-group that we call again

B. The second assignment is everywhere Zexcept at the accumulation point, where it is 1

2Z; it

yields an ℓ-group that we call B′. It is clear that B⊆B′. The converse inclusion holds because

the continuity of any f∈B′forces the value of fat the accumulation point to belong to Z.

Theorem 1.1 provides a “complete structural description of all” [14, p. 862] objects in G; the

preceding paragraph points out that the relationship between Xand Bin that description is

loose. In order to tighten it, we begin observing that the assignment x7→ Axmakes Xinto an a-

space via the function ζ:X→Ndeﬁned by x7→ 0 if Axis Rand x7→ nif Axis 1

nZ. We further

observe that for any maximal ideal mof Bthe quotient B/mhas exactly one unital isomorphism

to either Ror 1

nZ, for a uniquely determined natural number n. Hence, the maximal spectrum

Yof Bbecomes an a-space via the function ζ′:Y→Ndeﬁned by m7→ 0 if B/mis isomorphic

to R, and m7→ nif B/mis isomorphic to 1

nZ. To restate a part of our main results in relation to

Theorem 1.1: (X, ζ) is naturally isomorphic as an a-space to (Y, ζ′) exactly when it is a normal

a-space (Theorem 6.11). This is a consequence of the fact that arithmetically normal spaces

satisfy the arithmetic version of Urysohn’s Lemma given by Theorem 6.7.

To sum up, the generalisation of Yosida duality that we obtain here strengthens and extends

Theorem 1.1—without assuming it—to a duality for the category G.

1As a terminological aside, we remark that Goodearl and Handelman called “norm-complete” those unital

Archimedean ℓ-groups which are complete in the metric (†) induced by their unit. Here, we quote their theorem

using the terminology we introduced above, which adopts “metrically complete” rather than “norm-complete”.

Apart from this and from minor diﬀerences in notation, we quote verbatim.

Either perspective sketched above promptly suggests much further research—for the sake

of brevity, we do not elaborate. We do announce one further result driven by considerations

of a rather diﬀerent nature from the ones motivating this paper: To within an equivalence of

categories, Gmay be axiomatised by ﬁnitely many equations in an inﬁnitary algebraic language

with primitive operations of at most countably inﬁnite arity; for classical Stone-Gelfand Duality,

the corresponding result is the main one in [20]. The announced result will appear as a separate

contribution.

We now outline the structure of the paper with the aim of emphasising the proof strategy

of our main result, Theorem 7.10. The proof proceeds in a number of steps going from a more

general dual adjunction to the duality of Theorem 7.10. The ﬁrst steps, readily ﬂowing from

general theory [6, 21], are collected in Appendix A.

The starting point is a dual adjunction between the categories Algτand Topa(Proposi-

tion A.4), where Topais the category of a-spaces, and Algτis the category of all structures in

the algebraic signature τ:={+,∧,∨,−,0,1}of unital lattice-groups. This ﬁrst dual adjunction

is natural in that it is induced by the dualising object R.

The next step is to consider the full subcategory Gof Algτon the unital Abelian ℓ-groups,

and the full subcategory Kaof Topaon the compact a-spaces. We prove in Theorem A.5 that the

ﬁrst dual adjunction descends to a second one between these full subcategories. We note here

that this second adjunction, unlike the ﬁrst, is not natural with respect to the underlying-set

functors on Gand Ka—Ris not compact, and thus not an object of Ka. It is the naturality

of the ﬁrst adjunction that makes it a convenient starting point for the proof of the second,

non-natural adjunction.

Readers not interested in the details of these ﬁrst two adjunctions can safely ignore Ap-

pendix A and begin directly from Section 2. Here, we set the stage with the deﬁnition of

a-space and recall as Proposition 2.9 the dual adjunction between Gand Kaconstructed in

Appendix A.

The body of the paper is then devoted to characterising the ﬁxed subcategories of the dual

adjunction between Gand Ka, i.e., the full subcategories on those objects at which the component

of the (co)unit is an isomorphism.

Work begins in earnest in Section 3, where we prove that the compact a-spaces ﬁxed by the

adjunction are exactly the compact subspaces of RI(for some set I), on which the denominator

function is deﬁned canonically (Theorem 3.2) via the standard notion of denominator of a point

of RI. The next aim is to obtain an intrinsic characterisation of such a-spaces as precisely the

normal a-spaces in the sense of Deﬁnition 4.1. In Section 4 we prove that the compact subspaces

of RIindeed are normal a-spaces. The converse implication—that any normal a-space embeds

as a subobject of RI—requires the material in Sections 5 and 6. The key result is a substantial

generalisation of Urysohn’s Lemma in which denominators are taken into account (Theorem 6.7).

The two implications are then assembled into the intrinsic characterisation in Theorem 6.11 at

the end of Section 6.

Concerning the objects of Gﬁxed by the adjunction, these are characterised in Theorem 7.9

of Section 7. They are exactly the metrically complete ℓ-groups. To achieve this result we use

a strengthening of the Stone-Weierstrass Theorem for ℓ-groups (Theorem 7.7).

Finally, we state and prove our main result Theorem 7.10, and in Corollary 7.11 we detail

how it specialises to Yosida duality.

2. Arithmetic spaces

We consider N:={0,1,2,...}with its divisibility order. For n∈Nwe write div nfor the set

of natural numbers that divide n. Recall that any (pre)ordered set (X, 6) can be equipped with

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arXiv:2210.15341v3[math.FA]25Nov2024Stone-Gelfanddualityformetricallycompletelattice-orderedgroupsMarcoAbbadinia,VincenzoMarrab,∗,LucaSpadacaSchoolofComputerScience,UniversityofBirmingham,B152TTBirmingham,UnitedKingdombDipartimentodiMatematica“FederigoEnriques”,UniversitàdegliStudidiMilano,viaCesare...

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Stone-Gelfand duality for metrically complete lattice-ordered groups

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