Recapturing semigroup compactifications of a group from those of its closed normal subgroups

© Hindawi Publishing Corp.

RECAPTURING SEMIGROUP COMPACTIFICATIONS OF A GROUP

FROM THOSE OF ITS CLOSED NORMAL SUBGROUPS

M. R. MIRI and M. A. POURABDOLLAH

(Received 8 January 1998 and in revised form 20 June 1998)

Abstract.We know that ifSis a subsemigroup of a semitopological semigroupT, andᏲ stands for one of the spacesᏭᏼ,ᐃᏭᏼ,᏿Ꮽᏼ,ᏰorᏸᏯ, and(,TᏲ_{)denotes the canonical} Ᏺ-compactification ofT, whereT has the property thatᏲ(S)=Ᏺ(T )|s, then(|s,(S))is

anᏲ-compactification ofS. In this paper, we try to show the converse of this problem whenTis a locally compact group andSis a closed normal subgroup ofT. In this way we construct various semigroup compactifications ofTfrom the same type compactifications ofS.

Keywords and phrases. Semigroup (ᏼ-) compactification, conjugation invariance. 2000 Mathematics Subject Classification. 43A60.

1. Introduction. For notation and terminology we follow Berglund et al. [2], as much as possible. Thus a topological semigroup is a semigroupSthat is a Hausdorff topological space, the multiplication (s,t)→st:S×S →S being continuous. S is called a semitopological semigroup if the multiplication is separately continuous, i.e., the mapsλs:t→standρs:t→tsfromS intoSare continuous for eachs∈S. ForS

to be right topological only, the mapsρsare required to be continuous. LetGdenote

a locally compact group, andNis a closed normal subgroup ofG. A semigroup com-pactification ofGis a pair(ϕ,X), whereXis a compact right topological semigroup with identity 1, andϕ:G→X is a continuous homomorphism withϕ(G)=X, and ϕ(G)⊂Λ(X)= {x∈X|λx:X→X is continuous}; Λ(X)is called the topological

center ofX. When there is no risk of confusion we often refer to(ϕ,X), or even toX, as a compactification ofG.

A homomorphism from a compactification(ψ,X)ofSto a compactification(ϕ,Y ) of S is a continuous function θ :X →Y such thatθ◦ψ=ϕ. Then, Y is called a factor ofX, andX is an extension ofY. A compactification with a given propertyP (such as that of being a semitopological semigroup or a topological group) is called a P-compactification. A universalP-compactification ofSis aP-compactification which is an extension of everyP-compactification ofS(see [1, 2, 3]).

TheC∗_{-algebra of all bounded continuous complex-valued functions onGis denoted}

byᏯ(G)with left and right translation operators,Ls andRs, defined for alls∈Gby

Lsf=f◦λs and Rsf =f◦ρs. IfᏭis aC∗-subalgebra ofᏯ(G)containing the

con-stant functions, we denote byGᏭ_{the spectrum ofᏭfurnished with Gelfand topology}

(i.e., the weak*-topology induced fromᏭ∗_{); the natural map:G→G}Ꮽ _{is defined by} (s)f=f (s). WhenᏭis left translation invariant (i.e.,Lsf∈Ꮽfor alls∈Gandf∈Ꮽ)

we can define an action ofGonGᏭ_{by(s,ν)→(s)ν, where((s)ν)(f )=ν(L}

translation invariance andν(s)are analogously defined (see [5, 7]).

A left translation invariantC∗_{-subalgebra of Ꮿ(G) containing the constant}

func-tions is called leftm-introverted if the functions →(νf )(s)=v(Lsf ) is in Ꮽ for

allf ∈Ꮽand ν∈GᏭ_{; in this situation the product of µ,ν∈G}Ꮽ _{can be defined by} (µν)(f )=µ(νf ). This makes(,GᏭ_{)a semigroup compactification ofG. The spaces}

of almost periodic, weakly almost periodic, left continuous and distal functions, which are denoted byᏭᏼ,ᐃᏭᏼ,ᏸᏯ, andᏰ, respectively, are leftm-introverted. We refer the reader to [2, 5] for the one-to-one correspondence between compactifications ofGand leftm-introvertedC∗_{-subalgebras ofᏯ(G), and also for a discussion of propertiesP}

of compactifications and associated universal mapping properties.

2. Main results. LetGbe a locally compact group with a closed normal subgroup N, and let(ϕ,X)be a compactification ofN. Let∼be the equivalence relation onG×X with equivalence classes(sr−1_{,ϕ(r )x)|r∈N. Thus}

(s,x)∼(t,y)if and only ift−1_s∈Nandϕt−1_{sx=y. (2.1)}

π:G×X→(G×X)/∼will denote the quotient map. Clearlyπis one-to-one on{e}×X, so we can identifyX {e}×Xwithπ({e}×X). It is important that(G×X)/∼is locally compact and Hausdorff. In this connection we have the following lemmas, which are stated in [6].

Lemma2.1. (i) The graph of∼is closed.

(ii) π:(G×X)→(G×X)/∼is an open mapping.

(iii) LetKbe a compact subset ofGand letL=KN, thenπ(K×X)=π(L×X).

This lemma has the following easy consequences.

Lemma2.2. The quotient space(G×X)/∼is locally compact and Hausdorff. Lemma2.3. IfG=KNfor some compact subsetKofG, then(G×X)/∼is compact.

Letµ:G→(G×X)be defined byµ(s)=(s,1), where 1 is the identity ofX. Then, π◦µ:G→(G×X)/∼is continuous as a composition of two continuous functions, andπ◦µ(G)=π(G×ϕ(N)), since for each(s,ϕ(r ))∈G×ϕ(N),(s,ϕ(r ))∼(sr ,1), andπ◦µ(sr )=π(sr ,1)=π(s,ϕ(s)). Furthermore, ifϕis a homeomorphism ofN intoX, thenπ◦µis also a homeomorphism.

We now defineσs(r )=s−1r sfors∈Gandr∈N, it is obvious thatσs:N→Nis a

surjective homomorphism for eachs∈G.

Definition2.4. Aᏼ-compactification(ϕ,X)ofNis said to be a conjugation in-variantᏼ-compactification ofN if(ϕ◦σs,X)is a ᏼ-compactification ofNfor each

s∈G. When we write ᏼ-compactification instead of P-compactification, this means that we want to emphasize its conjugation invariance, see Corollary 2.7.

compactification ofNwhich may not be aᏼ-compactification ofN, thus(ψ,X)can fail to be aᏼ-conjugation invariant compactification ofN. On the other hand, if(ψ,X)is a ᏼ-conjugation invariant compactification ofN, i.e.,(ψ◦σs,X)is aᏼ-compactification

ofNfor eachs∈G, it is not always true thatσshas an extension fromXtoX. Lemma2.5. LetG be a locally compact group,N a closed normal subgroup, and

(ϕ,X)a conjugation invariant universalᏼ-compactifications ofN, then eachσs can be extended continuously to a mapping fromXtoX.

Proof. By conjugation invariance of(ϕ,X),(ϕ◦σs,X)is aᏼ-compactification of

N, and by universality of(ϕ,X)there exists a continuous homomorphismν:X→X such thatϕ◦σs=ν◦ϕfor eachs∈N. Thisνis the continuous function extending

σs.

It is obvious that if(ϕ,X)is a conjugation invariant universalᏼ-compactification of N, then eachσs determines a continuous transformation ofX, for which we use the

same notationσs.

Corollary2.6. LetNbe contained in the center ofG, then each compactification

(ϕ,X)ofNis conjugation invariant.

Corollary2.7. Let(,Nᏼ_{)denote a universalᏼ-compactification ofNand letᏼbe} a purely algebraic property, then(,Nᏼ_{)is a conjugation invariantᏼ-compactification} ofN.

Notice our deviation from the usual notation.

Corollary2.8. Let (ϕ,X) be an Ᏺ-compactification of N, where Ᏺ stands for either of the spacesᏭᏼandᐃᏭᏼ, then(ϕ,X) is a conjugation invariant universal

Ᏺ-compactification ofN.

Lemma2.9. Let(ϕ,X)be a conjugation invariantᏼ-compactification ofN, then for eachs∈G,σsis a continuous automorphism ofX.

Proof. σsis a homeomorphism ofXontoX(sinceσs(N)=Nandσsσs−1=I, the identity mapping). Now, we show thatσsis a homomorphism. Obviously,

σs(xy)=σs(x)σs(y) for eachx,y∈ϕ(N). (2.2)

SinceXis a right topological semigroup withϕ(N)⊂Λ(X), we conclude that (2.2) holds for eachx∈ϕ(N),y∈X. Then it follows that (2.2) holds for allx,y∈X, as required.

IfNis a closed subgroup ofG, andXis a conjugation invariantᏼ-compactification ofN, then we can define a semidirect product structure onG×X by(s,x)(t,y)= (st,σt(x)y), whereσtis the conjugation map.

Lemma2.10. LetGbe a locally compact group with a closed normal subgroupN, and let(ϕ,X)be a conjugation invariantᏼ-compactification ofN, thenG×Xis a right topological semigroup. Furthermore, the map

is continuous, and the equivalence relation∼is a congruence onG×X.

Proof. The continuity is an easy conclusion of Ellis theorem. Now, we show that

∼ is a congruence. Suppose (s,x)∼(t,y) and (u,z) ∈G×X, then t−1_{s∈ N and}

ϕ(t−1_{s)x=y, so(s,x)(u,z)=(su,σ}

u(x)z)and(t,y)(u,z)=(tu,σu(y)z).

On the other hand,(su,σu(x)z)∼(tu,σu(y)z)since(tu)−1su=u−1t−1su∈N

and

ϕ(tu)−1_suσ

u(x)z=σu(y)z, (2.4)

thus

(s,x)(u,z)∼(t,y)(u,z). (2.5)

Similarly

(u,z)(s,x)∼(u,z)(t,y). (2.6)

The following theorem is an easy consequence of the previous corollaries and lemmas.

Theorem2.11. LetG be a locally compact group with a closed normal subgroup

N, and let(ϕ,X)be a conjugation invariant compactification ofN. Then(G×X)/∼is a locally compact right topological semigroup, and a compactification ofG, provided thatG=KNfor some compact subsetKofG.

Theorem2.12. The compactification(π◦µ,(G×X)/∼)ofGdescribed in the pre-vious theorem has the following universal property; let(ϕ,Y )be a semigroup compact-ification ofGsuch thatϕ|Nextends to a continuous homomorphismφ:X→Y in such a way that for eachs∈Gandx∈X,

φσs(x)=ϕ(s−1)φ(x)ϕ(s), (2.7)

then there exists a (unique) continuous homomorphismθ:(G×X)/∼→Y such that

θ◦π◦µ=ϕ.

Proof. We defineθ0:G×X→Y byθ0(s,x)=ϕ(s)φ(x), thenθ0is a continuous

homomorphism which is constant on∼-classes ofG×X. Now we takeθ=θ0◦π. Theorem2.13. LetN be a closed normal subgroup ofG withG=KN for some compact subsetK ofG. Suppose thatᏼ is a property of compactifications such that

(ϕ|N,ϕ(N))is aᏼ-compactification ofNwhenever(ϕ,ϕ(G))is aᏼ-compactification ofG. Suppose that(,Nᏼ_{)is a conjugation invariant universalᏼ-compactification ofN.} If(G×Nᏼ_{)/∼has the propertyᏼ, then(G×N}ᏼ_{)/∼is the universalᏼ-compactification} ofG.

Proof. We show that(G×Nᏼ_{)/∼ is the universalᏼ-compactification ofG. Let} (ϕ,X)be aᏼ-compactification ofGsuch that (ϕ|N,ϕ(N))is aᏼ-compactification

ofN, by the universal property ofNᏼ_{there exists a continuous homomorphismφ:} Nᏼ_{→Xsuch thatφ◦=ϕ|}

s, and we haveφ(σs(x))=ϕ(s−1)φ(x)ϕ(s)for alls∈G

their connection. Indeed, for fixeds∈G, both sides represent homomorphisms ofNᏼ

intoX, both sides are continuous inx, and coincide on the dense subspaceN. Now the mapϕ×φ:(G×Nᏼ_{)→Xdefined by(ϕ×φ)(s,x)=ϕ(s)φ(x)is continuous and}

a homomorphism, since

(ϕ×φ)(s,x)(t,y)=(ϕ×φ)st,σt(x)y=ϕ(st)φσt(x)y =ϕ(s)ϕ(t)φσt(x)φ(y)=ϕ(s)φ(x)ϕ(t)φ(y) =ϕ×φ(s,x)ϕ×φ(t,y).

(2.8)

Alsoϕ×φis constant on∼-classes, thus the quotient ofϕ×φgives a continuous homomorphism from(G×Nᏼ_)/∼toX.

Corollary2.14. LetNbe a closed normal subgroup ofGwithG=KNfor some compact subsetKofG, then

(i) (G×NᏸᏯ_{)/∼is the universalᏸᏯ-compactification ofG.}

(ii) (G×NᏰ_{)/∼is the universalᏰ-compactification ofG.}

Proof. (i) Since (G×NᏸᏯ_{)/ ∼ is a compactification of G, by Theorem 2.13,} (G×NᏸᏯ_{)/∼is the universalᏸᏯ-compactification ofG.}

(ii) SinceNᏰ_{is a group,(G×N}Ᏸ_{)/∼, the quotient by a congruence of a semidirect}

product of groups is also a group, thus by Theorem 2.13(G×NᏰ_{)/∼is the universal} Ᏸ-compactification ofG.

In some situations, we want to be able to conclude that the right topological semi-group(G×X)/∼of Theorem 2.13 is also left topological. The following lemma can be helpful in this connection.

Lemma2.15. LetGbe a locally compact group with a closed normal subgroupN

and let X be a universal conjugation invariant compactification of N. Suppose that

G=KN for some compact subsetKofGands→σs(x):G×X→Xis continuous for allx∈X. Then(G×X)/∼is semitopological.

Proof. Since(s,x)→σs(x):G×X→Xis a group action, it is continuous by Ellis

theorem, thusG×X is semitopological semigroup and also(G×X)/∼=π(G×X).

Corollary2.16. LetGbe a locally compact group with a closed normal subgroup

N,G=KNfor some compact subsetKofGand suppose thats→σs(x):G→NᐃᏭᏼ is continuous for all x∈NᐃᏭᏼ_{, then(G×N}ᐃᏭᏼ_{)/∼ is the universal semitopological} semigroup compactification ofG.

Proof. SinceNᐃᏭᏼ_{is a semitopological semigroup, by Lemma 2.15,(G×N}ᐃᏭᏼ_)/∼

is semitopological semigroup. Thus by Theorem 2.13,(G×NᐃᏭᏼ_{)/∼is the universal}

semitopological semigroup compactification ofG. A similar argument yields the following corollary.

Corollary2.17. LetGbe a locally compact group with a closed normal subgroup

continuous for allx∈NᏭᏼ_{, then(G×N}Ꮽᏼ_{)/∼is the universal topological semigroup} compactification ofG.

Lemma2.18. LetNbe a closed normal subgroup ofGwithG=KNfor some compact subsetK ofG. LetᏲandᏳbe leftm-introverted subalgebras ofᏸᏯ(N)andᏸᏯ(G), respectively. ThenNᏲ_{is a conjugation invariantᏲ-compactification ofNif and only if} Ᏻ|N=Ᏺand(G×NᏲ)/∼is theᏳ-compactification ofG.

Proof. LetᏳ|N=Ᏺ, we defineσs(x)(f )fors∈G, x∈NᏲandf∈Ᏺbyσs(x)(f )=

x(g◦σs|N), whereg∈Ᏻ,g|N=f. Since every such extensiongyields ag◦σsagreeing

withf◦σs onN,σs(x)is well defined. SoNᏲis a conjugation invariantᏲ-

compacti-fication ofN.

Conversely, since the quotient mapπ :G×NᏲ_→(G×NᏲ_{)/∼is injective on the}

compact set NᏲ _{{e} ×N}Ᏺ_{, it gives a topological isomorphism of N}Ᏺ _{into G}Ᏻ (G×NᏲ_)/∼.

Corollary2.19. LetGbe a compact group with a closed normal subgroupN, then

(i) (G×NᐃᏭᏼ_)/∼GᐃᏭᏼ_.

(ii) (G×NᏭᏼ_)/∼GᏭᏼ_.

Corollary2.20. LetNbe a closed normal subgroup of a locally compact groupG

contained in the center ofG, then

G×NᐃᏭᏼ_∼GᐃᏭᏼ_{. (2.9)}

The next example shows that the continuity of s→ σs(x)in Corollary 2.14 and

Lemma 2.15 is an essential condition.

Example2.21. LetG=C×Tbe the Euclidean group of the plane with(z,w)(z1,w1) =(z+wz1,ww1) and N=C× {1}, then N is a closed normal subgroup ofG and

Ꮽᏼ(G)|Nis a proper subset ofᏭᏼ(N)[4, 8], so by Lemma 2.15(G×CᏭᏼ)/∼is not the

universalᏭᏼcompactification ofG.CᏭᏼ_{is a conjugation invariant compactification}

ofN, so the continuity ofs→σs must fail to hold Lemma 2.15. From [4, 8], we can

similarly conclude that(G×CᐃᏭᏼ_{)/∼is not the universalᐃᏭᏼ-compactification of} Gand that the continuity ofs→σs, as required by Corollary 2.14, also fails to hold.

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Miri: Department of Mathematics, University of Birjand, Birjand, Iran

Pourabdollah: Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran