Some algebraic universal semigroup compactifications

PII. S0161171201004380 http://ijmms.hindawi.com © Hindawi Publishing Corp.

SOME ALGEBRAIC UNIVERSAL SEMIGROUP

COMPACTIFICATIONS

H. R. EBRAHIMI-VISHKI

(Received 20 November 1999)

Abstract.Universal compactifications of semitopological semigroups with respect to the properties satisfying the varieties of semigroups and groups are studied through two func-tion algebras.

2000 Mathematics Subject Classification. 22A20, 43A60, 54H15.

1. Introduction. In the main approach to semigroup compactification, whose con-siderations goes back at least to the pioneering paper of de Leeuw and Glicksberg [2], the spectra of someC∗_{-algebras of functions on a semitopological semigroup are} em-ployed to construct certain universal compactifications. For instance, Junghenn in his elaborate study of distal functions characterized the universal group compactifica-tion, [7]. The main goal of the present work is to construct two function algebrasᏲV andᏲᐂwhose corresponding compactifications are universal with respect to the

prop-erties satisfying a varietyVof semigroups and a varietyᐂof groups (the structures of which in terms of subdirect products are given in [1, Section 3.3]).

2. Preliminaries. Here we highlight some required notions and notations from Berglund et al. [1], which is our ground rule. OnᏯ(S), theC∗_{-algebra of all} contin-uous bounded complex-valued functions on a semitopological semigroupS, the left and right translationsLs andRs are defined so that,(Lsf )(t)=f (st)=(Rtf )(s). A unitalC∗_{-subalgebraᏲofᏯ(S)which is a left translation invariant (i.e.,Lsf∈Ᏺfor} alls∈S andf∈Ᏺ) is calledm-admissible if the functions(Tµf )(s)=µ(Lsf )lies inᏲfor allf∈Ᏺand µ∈SᏲ _{(equal to the spectrum ofᏲ). Then the pair(ε,S}Ᏺ_)is

a (semigroup) compactification (calledᏲ-compactification) ofS, in whichSᏲ _is

fur-nished with the Gelfand topology and the multiplicationµν=µ◦Tν, andε:S→SᏲ_is

the evaluation mapping.

Them-admissible algebras of the left multiplicatively continuous, (weakly) [strongly] almost periodic, and strongly minimal distal functions, are denoted byᏸᏹᏯ,(ᐃᏭᏼ) [᏿Ꮽᏼ]Ꮽᏼ, andᏳᏼ, respectively; see [1,7].

3. The main results. From now on,FAis a fixed free semigroup on the countable al-phabetA, on which the identities ofVare expressed, andp(=mi=1ai)=q(=nj=1bj) is a fixed arbitrary formal identity ofV; see [6].

lim α

   m

i=2 lim α Rθα(ai)  _f 

_θαa1=lim

α

   n

j=2 lim α Rθα(bj)  _f 

_{θαb1, (3.1)}

lim_α    m

i=2

lim_α Rθα(ai)

  _lim

α Rsαf



_θαa1=lim

α

   n

j=2

lim_α Rθα(bj)

  _lim

α Rsαf



_θαb1,

(3.2) 

m

i=1

lim_α Rθα(ai)

 _f=

 n

j=1

lim_α Rθα(bj)



_{f , (3.3)}

 m

i=1

lim_α Rθα(ai)

  _lim α Rsαf =  n

j=1

lim_α Rθα(bj)

  _lim

α Rsαf

, (3.4)

are valid for all identitiesmi=1ai=j=n 1bjofV, and all netsθα(of mappings fromA intoS), for which the involved pointwise limits exist. (Note that, the notationis being used for the iterated multiplications in different semigroups, and one should realize thatmi=2limαRθα(ai)plays the role of identity operator wheneverm=1; similarly for n=1.)

The next lemma presentsᏲV in terms ofSᏸᏹᏯ_.

Lemma3.2. A functionf ∈ᏸᏹᏯbelongs toᏲV if and only if for every homomor-phismΦ:FA→SᏸᏹᏯ_{, each identityp=qofV, and everyµ∈S}ᏸᏹᏯ_{,Φ(p)(f )=Φ(q)(f ),}

(Φ(p)µ)(f )=(Φ(q)µ)(f ),TΦ(p)f=TΦ(q)f, andT(Φ(p)µ)f=T(Φ(q)µ)f.

Proof. For the necessity, it is enough to show thatΦ(p)(g)=Φ(q)(g)andTΦ(p)g=

TΦ(q)g, for allg∈Xf∪{f}, whereXfis the pointwise closure ofRSfinᏯ(S). There exists a netθα:A→Ssuch that, for eachiandj, limαε(θα(ai))=Φ(ai)and limαε(θα(bj))

=Φ(bj). Now using (3.1) and (3.2), we have

Φ(p)(g)=Φa1  m

i=2 TΦ(ai)

 _g

=lim_α    m

i=2

lim_α Rθα(ai)

 _g



_θαa1

=lim_α    n

j=2

lim_α Rθα(bj)

 _g



_θαb1

=Φ(q)(g).

(3.5)

A similar argument, using (3.3) and (3.4), shows thatTΦ(p)g=TΦ(q)g. Conversely, for any netθα, of mappings fromAintoS, consider the mappingθ:A→SᏸᏹᏯ _defined

by θ(a)=limαε(θα(a)). Using the crucial property of free semigroups,θ can be extended to a homomorphismΦ:FA→SᏸᏹᏯ_{. Therefore, for everyg∈Xf∪{f},}

 m

i=1

lim_α Rθα(ai)

 _g=

 m

i=1 TΦ(ai)



_{g=TΦ(p)g=TΦ(q)g=}  n

j=1

lim_α Rθα(bj)



which is equivalent to (3.3) and (3.4), taken together. A similar argument can be used to obtain the other required identity, which is equivalent to (3.1) and (3.2) taken together.

Now we have the next result which describes the main property ofFV.

Theorem3.3. ᏲV ism-admissible andᏲV-compactification is universal with respect to the property satisfyingV.

Proof. UsingLemma 3.2, them-admissibility ofᏲV may be readily verified. Again Lemma 3.2implies that, for each homomorphismΦ:FA→SᏲV _{andf∈ᏲV,Φ(p)(f )=}

Φ(q)(f ); which meansSᏲV _{∈V. It is enough to show that for every other} compacti-fication(ψ,X)ofS, withX∈V,ψ∗_{(Ꮿ(X))⊆ᏲV(S). Letπ:S}ᏸᏹᏯ_{→Xbe the}

homo-morphism for whichπ◦ε=ψ. This implies thatψ∗_{(Ꮿ(X))⊆ᏸᏹᏯ(S). Furthermore,} ifh∈Ꮿ(X), then for each homomorphismΦ:FA→SᏸᏹᏯ_{and eachµ∈S}ᏸᏹᏯ_,

Φ(p)µψ∗_{(h)=h(π◦Φ)(p)π(µ)=h(π◦Φ)(q)π(µ)=Φ(q)µψ}∗_{(h). (3.7)} Similar arguments show that

Φ(p)ψ∗_(h)=Φ(q)ψ∗_(h), TΦ(p)ψ∗(h)=TΦ(q)ψ∗(h),

T(Φ(p)µ)ψ∗(h)=T(Φ(q)µ)ψ∗(h).

(3.8)

NowLemma 3.2implies thatψ∗_{(h)∈ᏲV(S), as claimed.}

LetᏲVcconsist of thosef∈Ꮿ(S)such thatf (Φ(p))=f (Φ(q)),f (Φ(p)s)=f (Φ(q)s), f (sΦ(p))=f (sΦ(q)), andf (sΦ(p)t)=f (sΦ(q)t), for alls,t∈S, for every homomor-phismΦ:FA→S, and all identitiesp=qofV. Trivially,ᏲV⊆ᏲVc (with the equality holding in the compact setting). Due to joint continuity of the multiplication ofSᏭᏼ_,

we get the next simplification ofᏲV∩Ꮽᏼ.

Proposition3.4. ᏲV∩Ꮽᏼ=ᏲVc∩Ꮽᏼ.

Proof. According toLemma 3.2it is enough to show that iff∈ᏲVc∩Ꮽᏼ, then for each homomorphismΦ:FA→SᏭᏼ_{, and allµ∈S}Ꮽᏼ_{,Φ(p)(f )=Φ(q)(f ),(Φ(p)µ)(f )=}

(Φ(q)µ)(f ),TΦ(p)f=TΦ(q)f, andT(Φ(p)µ)f=T(Φ(q)µ)f. Imitating the methods of the proof of Lemma 3.2, let Φα : FA → S be that net (of homomorphisms) for which limαε(Φα(ai))=Φ(ai) and limαε(Φα(bj))=Φ(bj); and alsotα be a net inS such that limαε(tα)=µ. Now, for alls∈S;

TΦ(p)µf(s)=    m

i=1 Φai

 _µ

 _Lsf

=    m

i=1

lim_α εΦαai

  _lim

α ε

tα

 _Lsf

=lim_α εΦα(p)tαLsf=lim_α fsΦα(p)tα

=lim_α fsΦα(q)tα=TΦ(q)µf(s).

(3.9)

Remarks. (a) The present results are a generalization of what we have described in [4], for the familiar varieties of abelian semigroups,AB, bands,BD, and semilattices, SL.

ByDefinition 3.1,ᏲABconsists of thosef∈ᏸᏹᏯsuch that the identities lim_α lim_α Rsαftα=lim_α lim_α Rtαfsα,

lim_α lim_α Rsα lim_α Ruαftα=lim_α lim_α Rtα lim_α Ruαfsα,

lim

α Rsα limα Rtαf

=lim

α Rtα limα Rsαf

,

lim_α Rsα lim_α Rtα lim_α Ruαf=lim_α Rtα lim_α Rsα lim_α Ruαf

(3.10)

hold for all netssα,tα, anduαinS, for which the limits exist. SinceSᏲAB _is semitopo-logical, one realizes that the latter seemingly complicated limit process is preparatory, and can be condensed so that it presentsᏲABin the simple form

f∈ᐃᏭᏼ:f (st)=f (ts), andf (stu)=f (sut)∀s,t,u∈S. (3.11) In other words,ᏲAB=ᏲABc∩ᐃᏭᏼ.

Again,Definition 3.1implies thatᏲBDis the set of allf∈ᏸᏹᏯ, for which lim_α lim_α Rsαfsα=lim_α fsα,

lim_α lim_α Rsα lim_α Rtαfsα=lim_α lim_α Rtαfsα,

lim

α Rsα limα Rsαf

=lim

α Rsαf ,

lim_α Rsα lim_α Rsα lim_α Rtαf=lim_α Rsα lim_α Rtαf;

(3.12)

for all netssαandtαinS, for which the limits exist. In contrast to the situation for ᏲAB, in generalᏲBD⊆ᐃᏭᏼ(e.g., a direct verification shows that the characteristic function of even numbers onN, with its maximum multiplication, lies inᏲBDbut not inᐃᏭᏼ). Furthermore, as it is seen by easy examples (e.g., consider[0,1]×Nunder its rectangular multiplication)ᏲBD≠ᏲBDc, in general. It would be desirable to investigate the equality ofᏲBD=ᏲBDc∩ᏸᏹᏯ, the left and right introversion ofᏲBD, the relation betweenᏲBDand the space similarly defined in terms of the left translations, and to characterize the topological center ofSᏲBD_{. We believe that there are close connections} among these problems.

By the joint continuity theorem of Lawson [8],ᏲSL⊆Ꮽᏼand soᏲSL=ᏲSLc∩Ꮽᏼ. It would be desirable to examineᏲV for the variety of simple semigroups.

(b) It might be readily verified that the conditions (3.3) and (3.4) in the definition of ᏲV, taken together, are equivalent to the fact that the enveloping semigroup(S,Xf∪ {f}), of the natural flow(S,Xf∪{f}), lies inV; and so the latter is satisfied whenever f ∈ᏲV. A natural question that arises is whether the converse is also true. As a helpful answer, one can verify that; a functionf∈Ᏻᏼlies inᏲVif and only if(S,Xf) lies inV; that is,ᏲV∩Ᏻᏼ= {f∈Ᏻᏼ:(S,Xf)∈V}.

Now, for a varietyᐂof groups, defined by the set of lawsΩ(see [9]), we defineᏲᐂ(S)

It is easy to verify that, a functionf∈Ᏻᏼlies inᏲᐂif and only ifTωµf=Tµf for

allµ∈SᏳᏼ_{and all values inS}Ᏻᏼ_{of the lawsω∈Ω; which is also equivalent to the fact}

that(S,Xf)∈ᐂ. Using these, one may obtain the next result, which is the group version ofTheorem 3.3.

Theorem3.5. Ᏺᐂism-admissible andᏲᐂ-compactification is universal with respect

to the property satisfyingᐂ.

It should be mentioned thatᏲᐂis not inᐃᏭᏼin general (e.g., for the variety of all

groups we getᏳᏼwhich is not always inᐃᏭᏼ). Using the (joint continuity) theorem of Ellis [5], we have

Ᏺᐂ∩ᐃᏭᏼ=Ᏺᐂ∩Ꮽᏼ=Ᏺᐂ∩᏿Ꮽᏼ. (3.13)

More precisely, each side of (3.13) (and hence Ᏺᐂ, when S is compact) consists of

thosef∈᏿Ꮽᏼsuch that,(mi=1(Rsi)li)(Rsf )=Rsf, for all laws

m

i=1xiliinΩ, and all s1,s2,...,sm, andsinS. (For instance, for the variety of abelian groups,Ᏺᐂconsists

of thosef ∈᏿Ꮽᏼ such thatf (stu)=f (sut)for all s,tand uin S, [3]; which, of course, is equal toᏲAB∩Ᏻᏼ, see the previous remarks (b)). Hence, for every element of ᐂ, each side of (3.13) is equal to ᏿Ꮽᏼ, and so for a compact element S of ᐂ, Ᏺᐂ(S)=Ꮿ(S).

As we have shown in [3], for the variety of nilpotent groups, in the definition ofᏲᐂ,

limαRsαf can be replaced byf. However this seems not to be possible in general.

Acknowledgement. This work was in part supported by a grant from IPM.

References

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H. R. Ebrahimi-Vishki: Faculty of Mathematics, University of Mashhad, P.O. Box

91775-1159, Mashhad, Iran

Current address: Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran