Invariance of compactness for the Bohr topology

Topology and its Applications 111 (2001) 161–173

Invariance of compactness for the Bohr topology✩

Salvador Hernández∗, Sergio Macario

a Departamento de Matemáticas, Universidad Jaume I, Campus de Penyeta Roja, 12071-Castellón, Spain

Received 5 October 1998; received in revised form 20 May 1999

Abstract

We define the g-extension of a topological Abelian groupG as the set of all characters onG suchthat the restriction to every equicontinuous subset ofG is continuous with respect to the pointwiseconvergence topology. A g-group is a topological Abelian group(G, τ) such that its g-extensioncoincides with its completion. The Bohr topology of a topological group(G, τ) is the topologythat the group inherits as a subset of its Bohr compactification. A topological group(G, τ) respects apropertyP if the subsetsA ofG that satisfy the propertyP are exactly the same for the Bohr topologyand for the original topology of the group [Trigos-Arrieta, J. Pure Appl. Algebra 70 (1991) 199].All groups here are assumed to be Abelian. We prove that every complete g-group when endowedwith its Bohr topology is aµ-space. As a consequence, we obtain that for a complete g-groupthe properties of respecting functionally boundedness, pseudocompactness, countable compactnessand compactness are all equivalent and a characterization of this property is also provided. Finally,we extend a theorem of Rosenthal about the existence of sequences equivalent to the`1-basis. Weprove that for aCech-complete g-group the property of respecting compactness is equivalent to theexistence of conveniently placed sequences equivalent to the`1-basis. 2001 Elsevier Science B.V.All rights reserved.

Keywords:Bohr topology;Cech-complete group; g-group; Respects compactness;C-embedded;C∗-embedded

AMS classification:22A05; 46A03; 54C45

1. Introduction

The following question is considered in this paper: Characterize the topological Abeliangroups whose Bohr compact subsets are compact in the original topology of the group.

✩ Research partially supported by Spanish DGES, grant number PB96-1075, and Fundació Caixa Castelló, grantnumber P1B98-24.* Corresponding author.E-mail addresses:[email protected] (S. Hernández), [email protected] (S. Macario).

0166-8641/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0166-8641(99)00196-0

162 S. Hernández, S. Macario / Topology and its Applications 111 (2001) 161–173

Using the terminology introduced by Trigos-Arrieta in [22] we are concerned with thedetermination of the topological Abelian groups which respect compactness. The firstresult along this line is due to Glicksberg [12] who proved the preservation of compactnessfor locally compact Abelian groups. After Glicksberg’s result, there have been severalapproaches to the topic (see [11] to find basic references). Nevertheless, it is not known ofany general criterion to distinguish the groups having this property. In fact, although therehave been found several conditions sufficient to guarantee the invariance of compactnessfor wide families of groups, it seems to be harder to provide necessary conditions.Hence the goal here is to find properties necessary and sufficient for the preservation ofcompactness from the Bohr topology to the original topology of the group. Our strategyhas been to extend and unify the duality methods that hold for LCA groups and locallyconvex vector spaces to several maximally almost periodic topological Abelian groups. Ofcourse, several difficulties arise in this procedure that must be considered. For example,the fundamental Theorem of Grothendieck [13] characterizing the completion of a locallyconvex vector space does not have an easy translation in general, even for locally quasi-convex groups (see [2]). Thus, in order to apply successfully the duality methods tothis wider context, we introduce the concept of “g-group” that allows us to handle avariant of Grothendieck’s completeness theorem. We prove that every complete g-groupis a topologicalµ-space in its Bohr topology and this result is applied to characterizethe invariance of compactness by the existence of special discrete andC-embeddedsubsets. Moreover we show how, for complete g-groups the properties of respectingcompactness, countable compactness, pseudocompactness and functionally boundednessare all equivalent.

In the last section we consider the question of finding sequences equivalent to the`1-basis and we extend Rosenthal’s classical result for Banach spaces to different familiesof topological Abelian groups. The result is used again to characterize the invariance ofcompactness by the existence of sequences equivalent to the`1-basis.

2. Notation and terminology

Let G andH be a pair of topological, not necessarily Abelian, groups (the symbolismdesignating the topology of the group is omitted when there is no possible confusion). Wedenote byHom(G,H) (respectivelyHomc(G,H)) the set of all group homomorphisms(respectively continuous group homomorphisms) ofG into H . If α is a cardinal number,Homα(G,H) designates the set of all group homomorphisms ofG into H that arecontinuous on every subset ofG with cardinality less or equal thanα. For X ⊆Hom(G,H), σ(X) denotes the group topology weakly generated byX on G. WhenHis compact, the group(G,σ(X)) is totally bounded and its completion is a compact groupthat we denote byb(G,σ(X)).

Let (G, τ) denote a maximally almost periodic Abelian group (hereafter referredas MAPA group). That is, a topological Abelian group such that for allx ∈ G, x 6=eG, there exists a continuous characterχ with χ(x) 6= 1 (here, a character means a

S. Hernández, S. Macario / Topology and its Applications 111 (2001) 161–173 163

continuous homomorphism ofG into the one-dimensional torusT). Every MAPA grouphas canonically associated three other topological groups:

(i) (G, τk(G)), dual group of(G, τ). It contains all continuous characters on(G, τ)provided with the compact open topology.

(ii) (G,τk(G)), dual group of(G, τk(G)). It containsG as an algebraic subgroup andit is called the bidual group of(G, τ).

(iii) (bG,bτ), dual group of(G, τd) (whereτd denotes the discrete topology). Thisis a compact group containingG as a dense subgroup and it is called the Bohrcompactification of(G, τ).

A MAPA group topologically isomorphic to its bidual group is said to satisfyPontryagin–van Kampen duality.

On every MAPA group one can consider two canonical topologies besides the originaltopology of the group. They are the topologies inherited from the groups(G,τk(G)) and(bG,bτ). The three topologiesτ , τk(G)|G and bτ |G are all different in general. Thetopologybτ |G is called the Bohr topology of(G, τ). Notice thatbτ |G coincides withσ(G),the topology of pointwise convergence onG. bτ is the finest totally bounded topologyamong the ones coarser thanτ .

Given a topological Abelian group(G, τ) we say that a subsetM of G is quasi-convexif to eachx ∈ G \M there is a characterγ ∈ G such that<(γ (m)) > 0 for all m ∈ Mand<(γ (x)) < 0. The group is said to be locally quasi-convex if it admits a base ofneighborhoods of neutral element consisting of quasi-convex sets (see [3]).

Locally quasi-convex groups form a wide collection containing most of relevant familiesof topological Abelian groups. Among others, locally compact Abelian groups, nucleargroups and locally convex vector spaces considered as additive groups. In the sequel, allgroups are assumed to be locally quasi-convex (hence, Abelian).

Let (G, τ) be a locally quasi-convex group and let(G, τk(G)) denote its topologicaldual group. ForA⊆G andY ⊆Hom(G,T) we define

AFY ={φ ∈ Y : <(φ(x))> 0, for all x ∈A}.

Similarly, if B ⊆Hom(G,T) andX ⊆G we define

BGX ={x ∈X: <(φ(x))> 0, for all φ ∈B}.

3. g-Groups

Given a locally quasi-convex group(G, τ), letE be the set of all equicontinuous subsetsof G, with respect to(G, τ). We shall denote byG the collection of all those characterson G whose restriction to eachE ∈ E areσ(G)-continuous. It is clear thatG is a groupcontainingG as a subgroup.G is endowed with the topologyτ whose neighbourhood baseof the identity consists of sets of the formEF

Gwith E ∈ E . It is easy to see thatτ is a

group topology. Furthermore, it is a locally quasi-convex topology according to the abovedefinition. The fact that every suchE is equicontinuous implies finally thatτ |G = τ .


We say that(G, τ ) is the g-extension of(G, τ). When (G, τ ) coincides with thecompletionG of the group, it is said that(G, τ) is a g-group. We have the following chainsof algebraic subgroups:

G⊆G⊆ G, G⊆ G⊆ G,so, for a complete g-group all these groups coincide and, therefore, it is semi-reflexive.

The class of g-groups contains additive groups of locally convex vector spaces, LCAgroups and nuclear groups among others. Moreover every metrizable complete locallyquasi-convex and reflexive group is a g-group. However Außenhofer [2] has found anexample of a metrizable complete locally quasi-convex group which is not a g-group.

As usual, given a topological space(T , τ ) we denote byd(T , τ ), (respectivelyw(T , τ))or simply d(T ) (respectivelyw(T )) when no confusion is possible, its density number(respectively topological weight). The family of all compact subsets of a topological space(T , τ ) is denoted byK(T , τ ) (K(T ) when no confusion is possible).

Next result is a technical lemma that provides a sufficient condition for the continuity ofcertain group homomorphisms.

Lemma 3.1. Let G andH be topological groups and suppose thatψ ∈ Homα(G,H).Given a setX ⊆Homc(G,H), if there is a subsetA ofX such that|A|6 α andψ belongsto the pointwise closure ofA, as a subset of the productHG; thenψ is σ(X)-continuouson every member ofK(G,σ(X)).

Proof. Define F = A⊥ = {g ∈ G: φ(g) = eH , for all φ ∈ A}. It is clear thatF is aσ(X)-closed subgroup ofG. Now, let K be aσ(X)-compact subset ofG. We denoteby π :G→G/F the canonical quotient mapping associated to the coset spaceG/F . If onG/F we consider the weak topology defined byA, then it is clear thatπ is continuousfrom (G,σ(X)) onto the homogeneous topological space(G/F,σ(A)). Hence,π(K) is acompact subset of(G/F,σ(A)). This implies that

d(π(K)

)6w

(π(K)

)= |A|6 α.Thus, there is a subsetD of K such that|D|6 |A| andπ(D) is dense inπ(K). LetL

be the subgroup generated byD in G. Since|L|6 α, we know thatψ is continuous onL.Furthermore,ψ is also continuous oncl(G,σ (X))L (indeedψ is continuous onL∪{g} for allg ∈ cl(G,σ (X))L and by [6, I.57.5], this implies the continuity ofψ throughoutcl(G,σ (X))L).

Consider now the following diagram

G

ψ

π G/F

ψ

H

whereψ is defined by puttingψ(π(g))=ψ(g).Notice thatψ is well defined because ifπ(g1)= π(g2), for g1, g2 ∈G, theng−1

1 g2 ∈ Fand thereforeφ(g−1

1 g2)= eH for all φ ∈A. Sinceψ is in the closure ofA⊆HG, it followsthatψ(g−1

1 g2)= eH . That is,ψ(g1)= ψ(g2).


In order to prove thatψ is continuous onπ(K ∩ cl(G,σ (X))L), let us see that for allg ∈K ∩ cl(G,σ (X))L and for all net{gδ}δ∈Λ inK ∩ cl(G,σ (X))L such that{π(gδ)}δ∈Λ convergesto π(g), there is a subnet{gm}m∈M with {ψ(π(gm))}m∈M converging toψ(π(g)).

Indeed, suppose that{gδ}δ∈Λ is a net inK ∩ cl(G,σ (X))L such that{π(gδ)}δ∈Λ convergesto π(g) with g also inK ∩ cl(G,σ (X))L. By compactness, there must be a subnet{gm}m∈Mconverging to some pointg′ ∈K∩cl(G,σ (X))L. Clearly, the net{π(gm)}m∈M must convergeto π(g) as a subnet of{π(gδ)}δ∈Λ, but the continuity ofπ implies that it also convergesto π(g′), so g−1g′ ∈ F and, as a consequence,ψ(g) = ψ(g′). The continuity ofψ oncl(G,σ (X))L gives the convergence of{ψ(gm)}m∈M to ψ(g) = ψ(g′). Bearing in mind thedefinition ofψ the desired condition is proved.

Now, the setK ∩ cl(G,σ (X))L is compact in(G,σ(X)) and containsD. Sinceπ(D)is dense inπ(K), it follows thatπ(K ∩ cl(G,σ (X))L) = π(K). Thus, we have shown thecontinuity ofψ onπ(K). The commutativity of the diagram

K

ψ

ππ(K)

ψ

H

implies the continuity ofψ onK what completes the proof.2A subsetA of a topological spaceX is said to be functionally bounded whenf |A

is bounded for everyf ∈ C(X). The topological spaceX is a µ-space when everyfunctionally bounded subset ofX is relatively compact.

Now follows the main result of this note. It extends to complete g-groups previousresults of Valdivia [24] who proved that quasi-complete locally convex vector spaces areµ-spaces in its weak topology, and Trigos-Arrieta [23] who proved the same result forlocally compact Abelian groups provided with its Bohr topology. We present here a moresystematic treatment that unifies both approaches to this question.

Theorem 3.2. Let (G, τ) be a complete g-group. Then the group(G,σ(G)) is aµ-space.

Proof. LetA be a subset ofGwhich isσ(G)-closed and functionally bounded and assume,to begin with, thatA is σ(G)-countably compact. Then for everyσ(G)-compact subsetLof G we may considerA|L as a subset ofCp(L,C) and, sinceA|L is countably compact,Grothendieck’s theorem [14] implies thatA|L is also compact. Hence, ifx ∈ clbGA, itfollows thatx|L ∈A|L andx is σ(G)-continuous onL. Since(G, τ) is a complete g-groupwe obtain thatclbGA⊆G and the relative compactness ofA follows.

Suppose now thatA is not σ(G)-countably compact. Then there is some sequence{an}n<ω ⊆Awith no closure points inG. Letψ ∈ clbG{an}\G. Again, beingG a completeg-group, by Lemma 3.1 there is a countable setY ⊆ G such thatψ is notσ(G)-continuousonY . Define

N = {φ ∈ bG: φ|Y ≡ 1∈ T}.It is clear thatN is aGδ-open closed subgroup ofbG such that(ψ + N) ∩ G = ∅.

Therefore, there is some∈ C(bG) such that 06 `6 1, with ` > 0 onG and`(ψ) = 0.


Thus, the mapping 1/` is continuous on(G,σ(G)) and is not bounded on{an}n<ω becauseψ is a closure point of this sequence. This contradicts our initial assumption on thefunctional boundedness ofA and the proof is done.2

Given a topological group(G, τ) and a propertyP , we say after Trigos-Arrieta [22] that(G, τ) respects the propertyP when(G, τ) and(G,bτ) have the same sets satisfyingP .Along this line the well known theorem of Glicksberg [12] means that locally compactAbelian groups respect compactness.

Trigos-Arrieta considered some compact-like properties (pseudocompactness, countablecompactness, functional boundedness) obtaining that they are respected by locally compactAbelian groups. Later Banaszczyk and Martín-Peinador [5] generalized these results toarbitrary nuclear groups. We show below that for complete g-groups all the propertiesmentioned are equivalent to respecting compactness. A characterization of this lastproperty is also provided.

Theorem 3.3. Let (G, τ) be a complete g-group. The following assertions are equivalent:(a) (G, τ) respects compactness;(b) (G, τ) respects countable compactness;(c) (G, τ) respects pseudocompactness;(d) (G, τ) respects functional boundedness;(e) Every non-totally bounded subsetA ofG has an infinite subsetB which is discrete

andC-embedded in(G,σ(G)).

Proof. Firstly, notice that, since(G, τ) is complete, it follows that is a realcompact space(hence, aµ-space). Thus the groups(G, τ) and(G,σ(G)) areµ-spaces.

(a) (respectively (b) or (c)) implies (d). LetA be a functionally bounded subsetof (G,σ(G)). Then B = cl(G,σ (G))A is compact (respectively countably compact orpseudocompact). Applying (a) (respectively (b) or (c)) we obtain thatB is compact(respectively countably compact or pseudocompact) as a subspace of(G, τ). Now, B isσ(G)-closed, therefore closed also in(G, τ). Since compactness (respectively countablecompactness or pseudocompactness) imply functional boundedness and(G, τ) is a µ-space, it follows thatB is alwaysτ -compact. Thus,A ⊆ B is functionally bounded in(G, τ).

(d) implies (e). LetA be a non-totally bounded subset of(G, τ). Then there exists aneighbourhood of the identityU and a sequence{an}n<ω in A such that

(an +U)∩ (am +U)= ∅, for all n 6=m.Since(G, τ) is a µ-space, it follows that{an}n<ω may not be functionally bounded in(G, τ) and, by (d), the sequence{an}n<ω is not a functionally bounded subset of(G,σ(G)).Letf ∈ C(G,σ(G)) such thatlimn|f (an)| = +∞. If we takeB as a subsequence{ank }k<ωsuch that|f (ank+1)|> |f (ank )|+1, for all k, then we obtain a subset which is discrete andC-embedded in(G,σ(G)).

(e) implies (a) (respectively (b) and (c)). Assume thatA is a compact (respectivelycountably compact or pseudocompact) subset of(G,σ(G)). By (e), A must be totally


bounded in(G, τ). Thus,B = cl(G,τ)A is τ -compact and it follows thatτ |B = σ(G)|B .Therefore,A is compact (respectively countably compact or pseudocompact).2Remark 3.4. We notice that assertions (a), (b) and (c) follow from (e) in general withoutany assumption about the completeness of(G, τ).

4. Sequences equivalent to the1-unit basis

Definition 4.1. Following Bourgain [7] we say that a system{fγ }γ∈Λ of 1-bounded real(or complex) valued functions on a setK isC-equivalent to the1(Λ)-basis provided that

n∑i=1

|aγi |6 C supx∈K

∣∣∣∣∣n∑i=1

aγi fγi (x)

∣∣∣∣∣for all finite scalar sequenceaγ1, aγ2, . . . , aγn .

WhenG is a topological Abelian group we say that{fγ }γ∈Λ ⊆ G is equivalent to the`1(Λ)-basis when there is a compact subsetK ⊆G and a constantC such that{fγ }γ∈Λ isC-equivalent to the1(Λ)-basis on the setK.

In the rest of the paper we shall make use of the following lemma whose proof is implicitin Rosenthal’s classical theorem [20] characterizing the Banach spaces which contain anisomorphic copy of 1. Here on,C denotes the set of all complex numbers andCp(X,C)is the space of all complex-valued continuous functions onX provided with the topologyof pointwise convergence.

Lemma 4.1 (Rosenthal).LetK be a compact Hausdorff space and consider a sequence{fn}n<ω in Cp(K,C) containing no subsequence pointwise convergent inCK . Then,there are an infinite setM ⊆ ω and two disjoint disks inC, I1 and I2, such that for allsubsequences{fn: n ∈ J } of {fn: n ∈M} there isx ∈ K with fn(x) ∈ I1, for all n ∈ J ,andfn(x) ∈ I2 for all n ∈M \ J .

In particular, {fn}n∈J isC-equivalent to the1-basis.

The aim of this section is extend Rosenthal’s lemma characterizing the sequences ina Banach space that contain`1-subsequences. The following technical lemma will beneeded.

Lemma 4.2. Let (X, τ) be a topological space, and let{gn}n<ω ⊆ C(X, τ) such that it isrelatively countably compact inL ⊆ CX. If there is a sequence{Km}m<ω of τ -compactsubsets ofX such that for everyφ andψ in clL{gn}n<ω with

φ|⋃n<ω Kn

=ψ|⋃n<ω Kn

it holds thatφ =ψ onX. Then either{gn}n<ω admits a pointwise Cauchy subsequence orthere is aτ -compact subsetK of X and a subsequence{gnj }j<ω such that it contains nosubsequence pointwise convergent inCK .


Proof. Let us denote byD the set⋃n<ω Kn. Take the compact setK1, by taking the

corresponding restrictions, we may consider{gn}n<ω as a subset ofC(K1, τ ) and, clearly,{gn}n<ω must be relatively countably compact as a subset ofCK1. In general, we mayconsider that{gn}n<ω is a relatively countably compact subset ofCK1∪···∪Kp for all p < ω.

Suppose that{gn}n<ω does not have any pointwise Cauchy subsequence. We claim thatfor somep < ω no subsequence of{gn}n<ω is pointwise convergent inCK1∪···∪Kp .

If not, using an inductive argument, it follows that for everyp < ω we can find asubsequence of{gn}n<ω, say{g(p)n }n<ω, such that

(1) {g(p+1)n }n<ω is a subsequence of{g(p)n }n<ω, p < ω, and

(2) {g(p)n }n<ωconverges pointwise tofp ∈CK1∪···∪Kp , p < ω.It is also verified thatfp+1|K1∪···∪Kp = fp for all p < ω.

Taking the diagonal subsequence{g(n)n }n<ω we obtain a sequence that convergespointwise tofp onK1 ∪ · · · ∪Kp , for all p < ω. Define thenf :D→K asf |K1∪···∪Kp =fp for all p < ω.

It is clear that{g(n)n }n<ω converges pointwise tof onD. On the other hand,{g(n)n }n<ωis a subsequence of{gn}n<ω which is relatively countably compact inL; so there mustbe someg ∈ L that is a closure point of{g(n)n }n<ω. Hence,g(x) is in the closure of{g(n)n (x)}n<ω, for all x ∈ D. Now, since{g(n)n (x)}n<ω also converges tof (x), it followsthatf (x)= g(x) for all x ∈D. By hypothesis, this means thatg is the only closure pointof {g(n)n }n<ω in L. Therefore,{g(n)n }n<ω converges pointwise tog in X. But this contradictsour initial assumption since{g(n)n }n<ω is a Cauchy subsequence of{gn}n<ω.

Thus, we may assume that there isp < ω such that no subsequence of{gn}n<ω ispointwise convergent onCK1∪···∪Kp . We are done takingK =K1 ∪ · · · ∪Kp. 2

Next corollaries provide several classes of spaces where Rosenthal’s lemma holds.

Corollary 4.3. Let (E, τ) be a locally convex space such thatτ is finer than a metrizablevector topology, sayτm. If A= {xn}n<ω is a bounded sequence ofE such that

cl(E′′,σ (E′))(A−A)∩((E, τm)

′)⊥ = {0}then either {xn}n<ω contains a weak Cauchy subsequence or{xn}n<ω contains asubsequence with no subsequence pointwise convergent inCE′ and equivalent to the1-unit basis.

Proof. Sinceτm is metrizable and weaker thanτ , it follows that (E, τm)′ ⊆ (E, τ)′ andthat (E, τm)′ =⋃n<ω Kn beingKn σ(E)-compact for alln < ω. Thus, it is enough toapply Lemmas 4.1 and 4.2 to the topological space(E′, σ (E)) and the sequence ofσ(E)-continuous functions{xn}n<ω. Indeed, since{xn}n<ω is a bounded sequence, it follows thatis a relatively countably compact subset of(E′′, σ (E′)) (see [17, §23.2(4)]). On the otherhand, assume thatφ andψ are inE′′ ∩ clCE′ {xn}n<ω with φ|∪n<ωKn =ψ|∪n<ωKn . Then

φ −ψ ∈ cl(E′′,σ (E′))(A−A)∩((E, τm)

′)⊥ = {0}and, by hypothesis,φ =ψ onE′. 2


As a consequence we obtain new spaces where Rosenthal’s lemma applies: A locallyconvex space(E, τ) is said to be countably weakly compactly generated (cWCG) ifcontains a sequence of absolutely convex weakly compact subsets{Kn}n<ω whose unionis dense inE. Now let (E, τ) be a locally convex vector space such that(E′, σ (E)) iscWCG; by [16],E = (E′, σ (E))′ contains a metrizable topology coarser than the Mackeytopologyτ (E,E′). Taking in account that the bidual spaceE′′ is the same for(E, τ) and(E, τ(E,E′)), Corollary 4.3 applies to obtain that ifA= {xn}n<ω is a bounded sequenceof E such that

cl(E′′,σ (E′))(A−A)∩((E, τm)

′)⊥ = {0}then either {xn}n<ω contains a weak Cauchy subsequence or{xn}n<ω contains asubsequence with no subsequence pointwise convergent inCE′ and equivalent to the1-unit basis.

The same proof of Corollary 4.3 applies to the corollary below

Corollary 4.4. Let (G, τ) be a locally quasi-convex group such thatτ is finer than ametrizable group topology, sayτm. If A= {xn}n<ω is any sequence ofG such that

cl(bG,σ(G))(A−A)∩((G, τm))⊥ = {0}

then either {xn}n<ω contains a Bohr Cauchy subsequence or{xn}n<ω contains asubsequence with no subsequence pointwise convergent inTG and equivalent to the1-unit basis.

Next we apply the results above to obtain several new characterizations for the invarianceof compactness.

Theorem 4.5. Let (G, τ) be a topological g-group and suppose thatτ contains ametrizable group topology, sayτm. The group(G, τ) respects compactness if, and onlyif, for every non-totally bounded subsetA ofG with

cl(bG,σ(G))(A−A)∩((G, τm))⊥ = {0}

there exists a subsetB ⊆A which isC∗-embedded inbG. Moreover,B is equivalent to the`1-unit basis.

Proof. (Necessity) Suppose thatA is not totally bounded in(G, τ) andcl(bG,σ(G))(A −A)∩ ((G, τm))⊥ = {0}.

Then there is a sequence{an}n<ω ⊆A and a neighbourhoodU such that

(an +U)∩ (am +U)= ∅, n 6=m.Since (G, τ) respects compactness, it follows that no subsequence of{anj }j<ω is apointwise Cauchy subsequence. If not, we would obtain that{anj − an2j }j<ω is pointwiseconvergent toeG. Thus, as the group respects compactness, the sequence{anj − an2j }j<ωwould also beτ -convergent what is impossible. Now, regard{an}n<ω as a sequence ofcontinuous functions defined on the group(G, τ ). SinceG admits a metrizable topology


τm which is weaker thanτ , we can arguemutatis mutandisas in Corollary 4.3 above,replacing the topological vector spaces(E, τ)′ and (E, τm)′ by the topological groups(G, τ ) and(G, τm) , to verify that the hypothesis of Lemma 4.2 hold. Then it is enoughto apply again Lemmas 4.2 and 4.1 to complete the proof.

(Sufficiency) Suppose thatA is aσ(G)-compact subset ofG. If A is totally bounded in(G, τ) thenclGA is compact in(G, τ ) andA is aσ(G)-closed subset ofbG. HenceA isσ(G)-closed inG and, as a consequence,A= clGA is compact in(G, τ).

We show now how the assumption thatA is not totally bounded in(G, τ) implies acontradiction. Indeed, ifA is not totally bounded then, for some neighbourhoodU there isa sequence{an}n<ω ⊆A such that

(an +U)∩ (am +U)= ∅, for n 6=m.Since τm is a Hausdorff topology onG, it follows that G ∩ ((G, τm))⊥ = {0} andthen (A − A) ∩ ((G, τm))⊥ = {0}. Moreover, the compactness ofA − A implies thatcl(bG,σ(G))(A − A) ∩ ((G, τm))⊥ = {0} By Corollary 4.4 and taking into account thecompactness ofA we obtain a subsequence, say{bn}n<ω, which converges to some pointof bG. Thus{bn}n<ω, as a subset ofG, is not totally bounded in(G, τ) but no subsequenceof it can beC∗-embedded inbG. This is a contradiction that completes the proof.2Corollary 4.6. Let (G, τ) be a complete g-group and suppose thatτ is a metrizable grouptopology. The following assertions are equivalent

(1) G respects compactness;(2) for every non-totally bounded subsetA of G there is a subsetB ⊆ A discrete and

C-embedded in(G,σ(G)),C∗-embedded inbG and equivalent to the1-unit basis.

Proof. Sufficiency follows from Theorem 3.3.(Necessity) Suppose thatA is not totally bounded in(G, τ). Then there is a sequence{an}n<ω ⊂ A and a neighbourhoodU of the neutral element ofG such that(an + U) ∩(am + U) = ∅, n 6= m. Since(G, τ) is a complete g-group that respects compactness, itfollows from Theorem 3.3 that there exists a subsequence{bn}n<ω of {an}n<ω whichis discrete andC-embedded in(G,σ(G)). Now, apply Theorem 4.5 to the sequence{bn}n<ω being aware of the fact thatτ is already a metrizable topology and, therefore,the annihilator of(G, τ ) in (bG,σ(G)) is just the neutral element ofbG. Hence, we canapply Theorem 4.5 to obtain a subsequence{cn}n<ω of {bn}n<ω which isC∗-embedded inbG and equivalent to thel1-basis. This completes the proof.2Definition 4.2. A topological group(G, τ) is called almost metrizable if for everyx ∈Gthere exists a compact subsetK ⊂G containingx such that the family of all neighborhoodsof K has a countable basis (see [19,18]; Arhangel’skii [1] has used the name of “space ofpointwise countable type”).

The following lemma can be found in [19, Remark 13.19 (c)].


Lemma 4.7. A topological group(G, τ) is almost metrizable if, and only if, there exists acompact subgroupK such that(G/K,τ/K) is metrizable.

It is known that everyCech-complete group is almost metrizable and, reciprocally,complete almost metrizable groups areCech-complete (see [19]). Moreover, if(G, τ) isCech-complete then the group(G/K,τ/K) of lemma above is metrizable and complete.

As a consequence of these facts it is easily verified that the characterization given inCorollary 4.6 extends to almost metrizable groups.

Theorem 4.8. Let (G, τ) be a Cech-complete g-group. The following assertions areequivalent

(1) G respects compactness;(2) for every non-totally bounded subsetA of G there is a subsetB ⊆ A discrete and

C-embedded in(G,σ(G)),C∗-embedded inbG and equivalent to the1-unit basis.

Proof. Firstly, notice that (2) implies (1) is done in Theorem 3.3, (e) implies (a).Conversely, suppose thatG respects compactness. Ifπ :G→ G/K is the canonical

quotient mapping, withK a compact subgroup ofG, thenA⊆G is compact (respectivelytotally bounded) iffπ(A) is compact (respectively totally bounded) inG/K. Moreover,sinceK is compact, it is readily verified thatG/K respects compactness too. Now, ifAis a non-totally bounded subset ofG then, for some neighborhoodU there is a sequence{an}n<ω such that

(an +U)∩ (am +U)= ∅, for n 6=m.EveryCech-complete group is complete (see [19]), therefore Theorem 3.3 can be applied toG obtaining a subsequence{bn}n<ω, of {an}n<ω, which isC-embedded in(G,σ(G)). Onthe other hand, the sequence{π(bn)}n<ω is a non-totally bounded subset ofG/K, which isa metrizable group respecting compactness, so it contains a subsequence{π(cn)}n<ω thatis C∗-embedded inb(G/K) (= bG/K) and equivalent to the1-unit basis. If we take arepresentantcn of every classπ(cn) we obtain{cn}n<ω, a subsequence of{an}n<ω, whichisC∗-embedded inbG and equivalent to the1-unit basis. 2

We have seen how the preservation of compactness is related to the existence of discretesubsets that there areC-embedded in(G,σ(G)) andC∗-embedded inbG. These resultsare connected with the following theorem of van Douwen [10].

Theorem 4.9 (van Douwen).LetG a discrete Abelian group and letA be an infinite subsetofG. Then there is a subsetB ofAwith |B| = |A| such thatB is discrete andC∗-embeddedin bG andC-embedded in(G,σ(G)).

Let us say that a group(G, τ) satisfies van Douwen property when for every non-totallybounded subsetA of G there is a subsetB ⊆ A discrete,C-embedded in(G,σ(G)) and


C∗-embedded inbG. In [11] it is proved that among others every LCA group satisfy vanDouwen property. With the results above we can extend that result in the following way

Theorem 4.10.Every almost metrizable nuclear group satisfies van Douwen property.

Proof. Let (G, τ) be an almost metrizable nuclear group. Then(G, τ) is also nuclear (see[2, Corollary 21.4]) and by [4] respects compactness. So Theorem 4.8 can be applied andwe are done. 2

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Invariance of compactness for the Bohr topology

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