OF PRODUCTS OF TWO G-SPACES
NATELLA ANTONYAN
Received 5 September 2005; Revised 27 December 2005; Accepted 8 January 2006
LetGbe any Hausdorfftopological group and letβGXdenote the maximalG-compactifi- cation of aG-TychonoffspaceX. We prove that ifXandY are twoG-Tychonoffspaces such that the productX×Y is pseudocompact, thenβG(X×Y)=βGX×βGX.
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1. Introduction
LetGbe any Hausdorfftopological group and letβGXdenote the maximalG-compac- tification of aG-TychonoffspaceX (i.e., a TychonoffG-space possessing aG-compac- tification). Recall that a completely regular Hausdorfftopological space is called pseudo- compact if every continuous function f :X→Ris bounded.
In this paper, we prove that if X and Y are twoG-Tychonoffspaces such that the productX×Yis pseudocompact, thenβG(X×Y)=βGX×βGX(seeTheorem 2.2). This is aG-equivariant version of the well-known result of Glicksberg [16], which for Ga locally compact group was proved earlier by de Vries in [10]. Note that even in the case of a locally compact acting groupG, our proof is shorter than that of [10, Theorem 4.1].
It follows fromProposition 2.7that the equalityβG(X×Y)=βGX×βGXdoes not imply, in general, the pseudocompactness ofX×Y even ifXandY both are infinite (cf. [16, Theorem 1]).
Theorem 2.10says that if a pseudocompact groupGacts continuously on a pseudo- compact spaceX, thenβGX=βX.
Let us introduce some terminology we will use in the paper.
Throughout the paper, all topological spaces are assumed to be Tychonoff(i.e., com- pletely regular and Hausdorff). The letter “G” will always denote a Hausdorff(and hence, completely regular) topological group unless otherwise stated.
For the basic ideas and facts of the theory ofG-spaces or topological transformation groups, we refer the reader to [5,7,11]. However, we recall below some more special notions and facts we need in the paper.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 93218, Pages1–9
DOI10.1155/IJMMS/2006/93218
By aG-space we mean a TychonoffspaceX endowed with a continuous actionG× X→X of a topological groupG. A continuous map ofG-spaces f :X→Y is called a G-map or an equivariant map if f(gx)=g f(x) for allx∈Xandg∈G.
IfX is aG-space andSa subset ofX, thenG(S) denotes theG-saturation ofS, that is,G(S)= {gs|g∈G, s∈S}. In particular,G(x) denotes theG-orbit{gx∈X|g∈G} ofx. IfG(S)=S, thenSis said to be an invariant set. The orbit space endowed with the quotient topology is denoted byX/G.
For a closed subgroupH⊂G, byG/H we will denote theG-space of cosets{gH|g∈ G}under the action induced by left translations.
On any product ofG-spaces we always consider the diagonal action ofG.
AG-compactification of aG-spaceXis a pair (b,bX), whereb:X→bXis aG-homeo- morphic embedding into a compactG-spacebXsuch that the imageb(X) is dense inbX.
UsuallybX alone is a sufficient denotation. We will say that twoG-compactifications b1X and b2X are equivalent if there exists a G-homeomorphism f :b1X→b2X such that f(b1(x))=b2(x) for allx∈X. Clearly, the equivalence ofG-compactifications is an equivalence relation in the class of allG-compactifications ofX. We will identify equiva- lentG-compactifications; any class of equivalentG-compactifications will be denoted by the same symbolbX, wherebXis anyG-compactification from this equivalence class. An order relation in the family of allG-compactifications is defined as follows:b1Xb2X if there exists aG-map f :b2X→b1Xsuch that f b2=b1. It is easy to see thatb1X and b2X are equivalent if and only if b1Xb2X andb2Xb1X. We will writeb1X=b2X wheneverb1Xandb2X are equivalentG-compactifications. In a standard way, one can show that each nonempty family ofG-compactifications ofX has a least upper bound with respect to the order. In particular, if aG-spaceXhas aG-compactification, then there exists a largestG-compactificationβGXwith respect to the order;βGX is called the maximalG-compactification ofX.
A continuous real-valued function f :X→Ron aG-spaceXis said to beG-uniform if for anyε >0, there exists a neighborhoodU of the identity element in Gsuch that
|f(gx)−f(x)|< εfor allx∈X,g∈U.
AG-spaceX is said to beG-Tychonoffif for any closed setA⊂X and any pointx∈ X\A, there exists aG-uniform function f :X→[0, 1] such thatf(x)=0 andA⊂ f−1(1).
It is evident that each continuous function on a compactG-space isG-uniform, and hence every compactG-space isG-Tychonoff. Since an invariant subspace of aG-Tych- onoff space is again G-Tychonoff, we see that if aG-space has a G-compactification, then it isG-Tychonoff. The converse is also true (see, e.g., [1,2]). Thus, aG-space is G-Tychonoffif and only if it admits aG-compactification, and in particular, a maximal G-compactification. In [8,9], it was proved that ifGis a locally compact group, then ev- ery TychonoffG-space isG-Tychonoff. The local compactness ofGis essential here (see [18]).
Given a spaceZ, we will denote by C(Z,R) the space of all continuous real-valued functions f :Z→Requipped with the compact-open topology (see, e.g., [13, Chapter 12, Section 1]). A subsetK⊂C(Z,R) is called equicontinuous at a pointz0∈Zif for any ε >0, there exists a neighborhoodOofz0∈Zsuch that|f(z)−f(z0)|< εfor allz∈O
and f ∈K. IfK is equicontinuous at each pointz0∈Z, then we will say that it is an equicontinuous set.
If additionallyZ is a G-space for a groupG, then one can define the following (in general not continuous) action ofGonC(Z,R):
(gψ)(z)=ψg−1z, ψ∈C(Z,R),z∈Z,g∈G. (1.1)
IfGis locally compact, then this action is continuous, otherwise it may be discontinuous (see, e.g., [7, Chapter I, Section 2.1]). However, the following result is true.
Lemma 1.1. LetZbe aG-space andKan invariant equicontinuous subset ofC(Z,R). Then the closureKis also an invariant set and the restriction of the action (1.1) toG×Kis con- tinuous.
Proof. For everyg∈G, define the mapg∗:C(Z,R)→C(Z,R) by settingg∗(ψ)=gψ, wheregψis defined as in (1.1). First we show thatg∗is a continuous map.
Indeed, letCbe a compact set inZ,Uan open set inR, andM(C,U)= {ψ∈C(Z,R)| ψ(C)⊂U}. Since all the sets of the formM(C,U) constitute a subbase of the compact- open topology ofC(Z,R) andg∗−1(M(C,U))=M(g−1C,U), we infer thatg∗is continu- ous.
Now chooseϕ∈K andh∈Garbitrary. One needs to show thathϕ∈K. LetV be a neighborhood ofgϕ. Since the above-defined maph∗is continuous, the seth−∗1(V)= h−1V is a neighborhood ofϕ. Consequently,h−1(V)∩K= ∅, which is equivalent to V∩hK= ∅. ButhK=KbecauseK is invariant. Hence,V∩K= ∅, as required. Thus, the proof that the closureKis an invariant subset is complete.
Next we observe that the closure of an equicontinuous set is again equicontinuous [17, Chapter 7, Theorem 14]; soKis an equicontinuous invariant subset ofC(Z,R).
Now the continuity of the restriction of the action (1.1) toG×Kfollows easily from the continuity of the evaluation mapω:K×Z→Rdefined byω(ψ,z)=ψ(z),ψ∈K, z∈Z (see, e.g., [17, Chapter 7, Theorem 15]). We refer the reader to [2, Lemma 2] for
more details.
We will need this lemma in the proof ofTheorem 2.2.
In what follows, we will need also the following two characterizations of the maximal G-compactificationβGXestablished in [8] (see also [4]).
Proposition 1.2. LetGbe a group andXaG-Tychonoffspace. Then the following hold.
(1) EachG-mapf :X→Bto a compactG-space has a uniqueG-extensionF:βGX→B.
(2) LetbXbe aG-compactification ofXsuch that everyG-map f :X→Bto a compact G-space has aG-extensionF:bX→B. ThenbXis equivalent toβGX.
Proposition 1.3. LetGbe a group andXaG-Tychonoffspace. Then the following hold.
(1) Each boundedG-uniform function f :X→Rpossesses a unique continuous exten- sionF:βGX→R.
(2) IfbXis aG-compactification such that each boundedG-uniform functionf :X→R admits a continuous extensionF:bX→R, thenbXis equivalent toβGX.
2. Main results
Lemma 2.1. LetGbe any group,XaG-space, andAa denseG-subset ofX. Assume that f :X→Ris a continuous map such that the restriction f|A:A→RisG-uniform. Then f isG-uniform as well.
Proof. Define the map f:X→C(G,R) by setting f(x)(g)= f(gx),x∈X,g∈G. The continuity of ffollows from the fact that the compact-open topology is proper (see [14, Theorem 3.4.1]).
It is easy to see that theG-uniformness of f is just equivalent to the equicontinuity of the image f(X) inC(G,R). Since the restriction f|AisG-uniform, we infer that the set f(A) is equicontinuous. But closure of an equicontinuous set is again equicontinuous [17, Chapter 7, Theorem 14]; so f(A) is equicontinuous. By continuity of f, f(X)⊂ f(A), yielding that f(X) is also equicontinuous. Hence, f isG-uniform.
Theorem 2.2. LetGbe any group and letXandY beG-Tychonoffspaces such thatX×Y is pseudocompact. ThenβG(X×Y)=βGX×βGY.
Proof. According toProposition 1.3, it suffices to prove that every boundedG-uniform function f :X×Y→Rhas a continuous extensionF:βGX×βGY→R.
The idea is first to extend f to a boundedG-uniform functionϕ:βGX×Y→R, and then to extend in a similar wayϕto obtain the desired extensionF. In the nonequivariant case, this is due to Todd [21].
Define the mapf:X→C(G×Y,R) by setting
f(x)(g,y)=f(gx,g y) ∀x∈X, (g,y)∈G×Y. (2.1) Continuity of ffollows from the fact that the compact-open topology is proper (see [13, Theorem 3.1]).
Claim 2.3. The image f(X) is an equicontinuous set inC(G×Y,R).
Proof of the claim. Letε >0 and (g0,y0)∈G×Y. We have to show that there exist neigh- borhoodsUofg0andVofy0such that
f(x)(g,y)−f(x)g0,y0< ε ∀x∈X,g∈U, y∈V. (2.2) Since f is aG-uniform function, one can choose a neighborhoodUof the unity inG such that
f(tx,ty)−f(x,y)<ε
3 ∀(x,y)∈X×Y,t∈U. (2.3) Then
f(x)(g,y)−f(x)g0,y0=f(gx,g y)−fg0x,g0y0
≤f(gx,g y)−fgx,g0y0+fgx,g0y0
−fgx,g y0 +fgx,g y0
−fg0x,g0y0.
(2.4)
It follows from (2.3) that for allx∈Xandg∈Ug0, we have fgx,g y0
−fg0x,g0y0< ε
3. (2.5)
It is known that the formula ϕ(y)=sup
x∈X
f(x,y)−fx,g0y0, y∈Y, (2.6)
defines a continuous functionϕ:Y→R(see [15, Lemma 1.3]).
Sinceϕ(g0y0)=0, we conclude that there is a neighborhoodV ofg0y0inY such that ϕ(v)< ε/3 for allv∈V. Hence, one has
f(x,v)−fx,g0y0<ε
3 ∀v∈V,x∈X. (2.7)
By continuity of the action onY, there exist neighborhoodsOandW ofg0 and y0, respectively, such thatOW⊂V andO⊂Ug0. Consequently, ifg∈Oandy∈W, then g y∈V andg y0∈V. Hence, (2.7) yields for allx∈X
f(gx,g y)−fgx,g0y0<ε
3, fgx,g y0
−fgx,g0y0< ε
3. (2.8) Now, (2.4), (2.5), and (2.8) imply for allg∈Ug0andy∈Wthat
f(x)(g,y)−f(x)g0,y0< ε 3+ε
3+ ε
3=ε, (2.9)
as required. Thus, f(X) is indeed an equicontinuous set, and the proof of the claim is
complete.
Now we continue with the proof ofTheorem 2.2. ConsiderG×Y as aG-space en- dowed with the actionh∗(g,y)=(gh−1,hy). Then the induced action (1.1) becomes the following action:
(hψ)(g,y)=ψgh,h−1y ∀ψ∈C(G×Y,R), g,h∈G,y∈Y. (2.10) We claim that fis algebraically equivariant, that is,h f(x)= f(hx) for allx∈Xand h∈G. Indeed, if (g,y)∈G×Y, then we have
h f(x)(g,y)=f(x)gh,h−1y=f(ghx,g y)=f(hx)(g,y)=
h f(x)(g,y), (2.11) which means thath f(x)=f(hx).
Consequently, f(X) is an invariant subset of C(G×Y,R). ByLemma 1.1 and the above claim, the closureT= f(X) also is an invariant subset ofC(G×Y,R), and the restriction of the action (2.10) toG×Tis continuous.
Further, since f(X) is a bounded subset ofC(G×Y,R), it follows from the Arzela- Ascoli theorem [13, Theorem 6.4] thatTis compact.
Thus,Tis a compactG-space. Next, since f:X→Tis aG-map, byProposition 1.2, fextends to aG-mapF:βGX→T⊂C(G×Y,R).
Define the mapφ:βGX×Y→Rby the formulaφ(z,y)=F(z)(e,y), where (z,y)∈ βGX×Y andeis the unity ofG. Clearly,φis bounded.
Since the evaluation mapω:T×(G×Y)→Rdefined byω(ψ,t)=ψ(t),ψ∈T,t∈ G×Y, is continuous (see, e.g., [17, Chapter 7, Theorem 15]), we infer that φis also continuous.
If (x,y)∈X×Y, thenφ(x,y)=F(x)(e,y)=f(x)(e,y)= f(x,y), showing thatφex- tends f. Since f isG-uniform, it follows fromLemma 2.1thatφisG-uniform.
Since the product of a pseudocompact space and a compact space is pseudocompact (see, e.g., [14, Corollary 3.10.27]),βGX×Y is a pseudocompactG-space. Consequently, by the same way, one can prove that the boundedG-uniform functionφ:βGX×Y→R extends to a continuous functionF:βGX×βGY→R, which is the desired extension of
f. This completes the proof.
Remark 2.4. ForGa locally compact group,Theorem 2.2was proved earlier by de Vries in [10] in a different way. IfG, as a topological space, is ak-space (i.e., a quotient image of a locally compact space) andXis a pseudocompactG-space, thenβGX=βX(see [10, Lemma 5.5]). Hence,Theorem 2.2follows in this case directly from the classical result of Glicksberg [16] (this is just [10, Corollary 5.7]).
In the following lemma, we just list two known important cases when the product of two pseudocompact spaces is pseudocompact.
Lemma 2.5. The productX×Yof two spaces is pseudocompact, if at least one of the follow- ing conditions is fulfilled:
(1)Xis a pseudocompactk-space andY is a pseudocompact space;
(2)Xis a pseudocompact topological group andY is a pseudocompact space.
Proof. For the first statement, see, for example, [14, Theorem 3.10.26]. The second one is
proved in [20, Corollary 2.14].
Corollary 2.6. LetGbe any group,H a closed subgroup ofGsuch thatG/H is compact, and letXbe a pseudocompactG-Tychonoffspace. ThenβG(G/H×X)=G/H×βGX.
The following simple result shows that the converse ofTheorem 2.2is not true even if XandY both are infinite (cf. [16, Theorem 1]).
Proposition 2.7. LetGbe any group,Ha closed subgroup ofGsuch thatG/His compact, and letXbe a Tychonoffspace endowed with the trivial action ofG. ThenβG(G/H×X)= G/H×βX.
Proof. Evidently,G/H×βX is a G-compactification ofG/H×X. Hence, according to Proposition 1.3, it suffices to prove that every boundedG-uniform function f :G/H× X→Rhas a continuous extensionF:G/H×βX→R.
Define a function f:X→C(G/H,R) by f(x)(t)= f(t,x), where (t,x)∈G/H×X.
Then fis continuous, and it follows from theG-uniformness of f that the image f(X) is an equicontinuous set inC(G/H,R). Besides, the set f(X)(t0)= {f(x)(t0)|x∈X}is bounded for allt0∈G/H. Consequently, by the Arzela-Ascoli theorem [13, Theorem 6.4], f(X) has a compact closure f(X) inC(G/H,R). Hence, fhas a continuous extension
F:βX→ f(X)⊂C(G/H,R). DefineF:G/H×βX→RbyF(t,z)= f(z)(t). The com- pactness ofG/H insures thatFis continuous (see, e.g., [14, Theorem 3.4.3]). It remains
only to observe thatFextends f.
Recall that aG-spaceXis called free if for everyx∈X, the equalitygx=ximplies that g=e, the unity ofG.
Below, we will need the following well-known result.
Lemma 2.8. LetGbe a compact group andXa freeG-space. Then (G×X)/GisG-homeo- morphic toX, whereGacts on the orbit space (G×X)/Gaccording to the ruleh∗G(g,x)= G(gh−1,x).
Proof. The desiredG-homeomorphism f : (G×X)/G→Xis defined as follows:
fG(g,x)=g−1x ∀(g,x)∈G×X, (2.12) whereG(g,x) stands for theG-orbit of the pair (g,x).
It is easy to verify that f is continuous and bijective. The closedness of f follows from that of the mapG×X→X, (g,x)→g−1x(see [5, Chapter I, Theorem 1.2]).
If the action ofGonXis not trivial, thenProposition 2.7is no longer true. Namely, we have the following proposition.
Proposition 2.9. LetGbe an infinite, compact, metrizable group andXa finite-dimension- al, paracompact, noncompact, freeG-space. ThenβG(G×X)=G×βGX.
Proof. Suppose the contrary, thatβG(G×X)=G×βGX. Passing to the orbit spaces, we have
G×βGX
G =
βG(G×X)
G . (2.13)
Using the formula (βGZ)/G=β(Z/G) (see [4, Corollary 4.10]), we get βG(G×X)
G =βG×X G
. (2.14)
Hence,
G×βGX
G =βG×X G
. (2.15)
It is known that a finite-dimensional, paracompact, freeG-space has a freeG-compac- tification and in this caseβGXis also a freeG-space (see [3, Proposition 3.7]). Conse- quently, by virtue ofLemma 2.8, one has that (G×X)/G=Xand (G×βGX)/G=βGX. In sum, we getβX=βGX, which implies that each bounded continuous function f :X→R isG-uniform. However, this is not true.
Indeed, sinceX is paracompact and noncompact, it is not countably compact [14, Theorem 3.10.3]. Hence, there exists a locally finite, disjoint, countable family{U1,U2,...} of open subsets ofX. SinceGis infinite, one can choose a countable base{O1,O2,...}of neighborhoods of the unity inG. For eachn≥1, choose a pointxn∈Unarbitrary. Then,
by continuity of theG-action atxn∈X, there exists an elementgn∈Onsuch thatgnis different from the unity ofGandgnxn∈Un,n=1, 2,.... SinceXis a freeG-space, we see thatgnxn=xn,n≥1.
Now, let fn:X→[0, 1] be a continuous function such that fn(xn)=1, fn(gnxn)=0 and fn(X\Un)= {0}. Define f(x)=∞
n=1fn(x),x∈X. Since{U1,U2,...}is disjoint and locally finite, f is a well-defined, continuous, bounded functionX→R. Hence, it should be alsoG-uniform, which yields a neighborhoodQof the unity inGsuch that|f(gx)− f(x)|<1/2 for allx∈Xandg∈Q. We choosen≥1 so large thatOn⊂Q. This implies thatgn∈Q, and hence 1= |f(gnxn)−f(xn)|<1/2, a contradiction.
In general, if the acting groupGis not discrete, an actionG×X→X cannot be ex- tended (continuously) to an actionG×βX→βX; the natural rotation-action of the circle group on the planeR2provides a counterexample (see [19, Section 1.5]). However, the following result holds true.
Theorem 2.10. LetGbe a pseudocompact group andXa pseudocompactG-space. ThenX isG-TychonoffandβGX=βX.
Proof. The actionα:G×X→X uniquely extends to a continuous mapϕ:β(G×X)→ βX. ByLemma 2.5(2), the productG×X is pseudocompact, and hence, according to Glicksberg’s theorem [16],β(G×X)=βG×βX. Thus,ϕcan be treated as a continuous map ofβG×βX inβX which extendsα. But remember thatβG is a topological group containingGas a dense subgroup (see, e.g., [6, Theorem 4.1(f)]).
Further, the fact thatαsatisfies the two algebraic conditions of action implies easily that the mapϕ:βG×βX→βX satisfies these conditions as well. Thus,ϕis an action, and henceβX is a βG-space. In particular,βX is aG-space. Consequently,βX is a G- compactification ofX, and henceXis aG-Tychonoffspace. It is also clear thatβXis the maximalG-compactification ofX, that is,βGX=βX, as required.
Remark 2.11. It is worth to mention that there exists a pseudocompact group whose underlying topological space is not ak-space (see, e.g., [12,20]).
Acknowledgments
The author was supported by the Grants U42573-F from CONACYT and IN-105803 from PAPIIT, Universidad Nacional Aut ´onoma de M´exico (UNAM). We are thankful to the referee for useful comments.
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Natella Antonyan: Departamento de Matem´aticas, Divisi ´on de Inginier´ıa y Arcitectura, Instituto Tecnol ´ogico y de Estudios Superiores de Monterrey, 14380 M´exico,
Distrito Federal, M´exico
E-mail address:[email protected]
