almost periodicity of functions on pseudocompact groupoids

206 Pages·2016·1.45 MB·English

ROMAIJ.,v.11,no.2(2015),1–11
ALMOST PERIODICITY OF FUNCTIONS ON
PSEUDOCOMPACT GROUPOIDS
Mitrofan M. Choban, Dorin I. Pavel
Department of Mathematics, Tiraspol State University, Chi¸sin˘au, Republic of Moldova
[email protected],[email protected]
Abstract In this article the weakly almost periodic functions and almost periodic functions on
a quasigroup are studied. It is proved that the almost periodic compactification is a
groupoidcompactificationanditisgeneratedbythefamilyofweaklyalmostperiodic
functions.
WorkpresentedasinvitedlectureatCAIM2015,September17-20,2015,
University“S¸tefancelMare”ofSuceava,Romania.
Keywords:almostperiodicfunction,weaklyalmostperiodicfunction,compactification.
2010MSC:54H10,06B30,54C35,20N15.
1. INTRODUCTION
The aim of the present article is to study the compactifications of topological
groupoids. Any space is considered to be Tychonoff and non-empty. We use the
terminologyfrom[12].
LetG be a non-empty topological space. A continuous mapping ω : G2 G is
→
called a binary operation on G. Let ω : G2 G be a binary operation on a space
→
G. Then the pair (G,ϕ) is called a topological groupoid. Subgroupoids, homomor-
phisms, isomorphisms and Cartesian products of topological groupoids are defined
intraditionalway[1,13,8].
Apair(E,ϕ)isaalgebraicalgeneralizedcompactificationoranag-compactification
of a topological groupoid G if E is a compact topological groupoid, ϕ : G E is
→
a continuous homomorphism and the set ϕ(G) is dense in E. If (Z,ϕ) and (Y,ψ)
are a-compactifications of G, then (Z,ϕ) (Y,ψ) if and only if there exists a con-
≤
tinuous homomorphism g : Y Z such that ϕ = g ψ. If (Y,ϕ) and (Z,ψ)
→ ◦
are ag-compactifications of G, (Y,ϕ) (Z,ψ) and (Z,ψ) (Y,ϕ), then the ag-
≤ ≤
compactifications (Y,ϕ), (Z,ψ) are called equivalent and there exists a unique topo-
logical isomorphism g : Y Z such that ψ = g ϕ. We identify the equivalent
→ ◦
ag-compactifications. In this case the class of all ag-compactifications of the space
G isaset. IfϕisanembeddingofG in E,then E iscalledana-compactification. In
thiscaseweassumethatGisasubspaceof E andϕ(x) = xforeach x G.
∈
IfGisatopologicalgroupoid,thenCom (G)isthesetofallag-compactifications
a
ofthetopologicalgroupoidG.
Thefollowingpropertiesareobvious.
1
2 Mitrofan M. Choban, Dorin I. Pavel
Property1.1. ThesetCom (G)isacompletelatticeforeverytopologicalgroupoid
a
G andforeverynon-emptysubset L Com (X)thereexistthemaximalelement L
a
⊆ ∨
andtheminimalelement L.
∧
Property1.2. Inthelatticeofallag-compactificationsofthetopologicalgroupoid
G thereexiststhemaximalag-compactification(β G,β ).
a G
Property1.3. Inthelatticeofallag-compactificationsofthetopologicalgroupoid
Gthereexiststheminimalag-compactification(µ G,µ ),µ Gisasingletongroupoid.
a G a
FixatopologicalspaceG. LetC(G)bethespaceofreal-valuedcontinuousfunc-
tionsonthespaceG inthetopologyofuniformconvergence. ThetopologyonC(G)
is generated by the metric d(f,g) = sup f(x) g(x) : x G . Let C (G) be the
◦
{| − | ∈ }
subspace of all bounded continuous functions. Then C (G) is a Banach space with
◦
the norm f = sup f(x) : x G . For some f,g C(G) it is possible that
k k {| | ∈ } ∈
d(f,g) = . We have C(G) = C (G) if and only if the space G is pseudocompact.
◦
∞
If C(G) , C (G), then C(G) is a linear space, but is not a topological linear space.
◦
ThespaceC(G)isatopologicalringrelativelytotheoperations f +gand f g. For
any number λ R the correspondence t (f) = λf is a continuous mapping o·fC(G)
λ
∈
intoC(G). For λ , 0, the correspondence t : C(G) C(G) is a linear mapping, a
λ
→
homeomorphismandagrouphomomorphism.
Compactificationsofthespacescanbeproducedinavarietyofways. Onewayis
byusesubspacesofthespaceC (G).
◦
Let F C (G) be a non-empty subspace. Consider the mapping e : G RF,
◦ F
where e (⊆x) = (f(x) : f F). Denote by b G the closure of the set e (G)→in RF.
F F F
∈
Then(b G,e )isageneralizedcompactification(briefly,ag-compactification)ofG
F F
andβG = β GistheStone-C˘echmaximalcompactificationofG(see[12]).
C (G)
◦
Let (E,ϕ) be a g-compactification of a topological space G. If C (G) = f ϕ :
E
{ ◦
f C(E) , then C (G) is the maximal subalgebra F of the Banach algebra C (G)
E ◦
∈ }
suchthat(E,ϕ)=(b G,e ). Inparticular, (E,ϕ)=(b G,e ). DenoteCa(G)
F F CE(G) CE(G)
=C (G).
β G
a
Question A. Let G be a topological quasigroup and F C (G). Under which
◦
⊆
conditions(b G,e )isagroupoidag-compactificationofG ande : G b G isa
F F F F
→
homomorphism?
Question B. Let G be a topological quasigroup and f C(G). Under which
∈
conditions f Ca(G)?
∈
Lemma 1. Let (X,d) be a complete metric space. For a non-empty subset L of the
spaceX thefollowingassertionsareequivalent:
1. TheclosureH =cl LofthesetLinX isacompactsubsetofX.
X
2. Foe every ǫ > 0 there exists a finite subset S(ǫ) of X such that d(x,S(ǫ)) =
inf d((x,y) : y S(ǫ) ǫ foreach x L.
{ ∈ } ≤ ∈
Proof. Follows immediately from Theorem 4.3.29 from [12], which affirm that a
metrizable space Y is compact if and only if on Y there exists a metric ρ which is
bothtotallyboundedandcomplete.
Almost periodicity of functions on pseudocompact groupoids 3
2. P-ALMOST PERIODIC FUNCTIONS ON
SPACES
Fix a topological space G. Denote by Π(G) the set of all continuous mappings
ϕ : G G. Relatively to the operation of composition ϕ ψ, where (ϕ ψ)(x) =
→ ◦ ◦
ϕ(ψ(x)) for ψ,ψ Π(G) and x G, the set Π(G) is a semigroup with identity e ,
G
∈ ∈
wheree (x) = xforeach x G. WesaythatΠ(G)isthesemigroupofallcontinuous
G
∈
translations ofG. If f C(G) and ϕ Π(G), then f = f ϕ (f (x) = f(ϕ(x)) for
ϕ ϕ
∈ ∈ ◦
any x G). Evidently, f C(G).
ϕ
∈ ∈
Fixanon-emptysubsetP Π(G). WesaythatPisasetofcontinuoustranslations
⊆
of G. The set P is called a transitive set of translations of G if for any two points
x,y Gthereexistsϕ Psuchthatϕ(x) = y. Obviously,thesetΠ(G)istransitive.
∈ ∈
For any function f C(G) we put P(f) = f : ϕ P . If f C (G), then
ϕ ◦
∈ { ∈ } ∈
P(f) C (G).
◦
⊆
Definition 2.1. A function f C(G) is called a P-periodic function of a spaceG if
theclosureP¯(f)ofthesetP(f∈)inthespaceC(G)isacompactset.
Denote by P-ap(G) the subspace of all P-almost periodic functions of a spaceG.
If the set P is finite, then P-ap(G) = C(G). The distinct spaces of almost periodic
functionswasstudiedin[5,14,4,6,8,9,11].
Theorem2.1. Let PbeasetofcontinuoustranslationsofG. Then P-ap(G)hasthe
followingproperties:
1. P-ap(G)isalinearsubspaceofthelinearspaceC(G).
2. P-ap(G)isatopologicalsubringofthetopologicalringC(G).
3. P-ap(G)isaclosedsubspaceofthecompletemetricspaceC(G). Inparticular,
P-ap(G)isacompletemetricspace.
4. If f C(G)isaconstantfunction,then f P-ap(G).
∈ ∈
5. If f P-ap(G), thenforany x G thereexistsanumber c(f,x) > 0suchthat
∈ ∈
f(ϕ(x)) c(f,x)foranyϕ P.
| | ≤ ∈
6. If f P-ap(G),ψ Π(G)andg(x) = f(ψ(x))foreachx G,theng P-ap(G).
∈ ∈ ∈ ∈
Inparticular, f P-ap(G)forall f P-ap(G)andψ Π(G).
ψ
∈ ∈ ∈
Proof. Fix f,g P-ap(G). Since P¯(f), P¯(g), P¯(f) P¯(f)+P¯(g)and P¯(f) P¯(g)are
compactsubsets∈of P-ap(G)and P¯(f)=P¯( −f),P¯(f +g) P¯(f)+P¯(g),P¯·(f g)
P¯(f) P¯(g),then f, f +g, f g −P-ap(G).−Hence P-ap(G⊆)isatopologicalsu·brin⊆g
· − · ∈
ofthetopologicalringC(G).
If f P-ap(G) and λ R, then correspondence t (f) = λf is a continuous
λ
mapping∈ofC(G)intoC(G)∈andP¯(λf)=t (P¯(f)). Henceλf P-ap(G)andP-ap(G)
λ
∈
isalinearsubspaceofthelinearspaceC(G).
Let f P-ap(G) : n N and f = lim f . It is well known that f C(G).
n n n
Fixǫ >{0. ∈Thereexistn ∈Nan}dafinitesubs→et∞S ofC(G)suchthat: ∈
∈
- f (x) f(x) ǫ/3foreach x G;
n
| − | ≤ ∈
4 Mitrofan M. Choban, Dorin I. Pavel
-d(g,S) ǫ/3foreachg P(f ).
n
≤ ∈
Fix ϕ P. For a given ǫ > 0 there exists g S such that g(x) f (ϕ(x))
n
∈ ∈ | − ≤
ǫ/3+ǫ/3foreach x G. Then g(x) f(ϕ(x) g(x) f (ϕ(x) + f (ϕ(x)) f(ϕ(x)
n n
∈ | − |≤ | − | | − |
< ǫ/3+ǫ/3+ǫ/3 = ǫ. Hence d(h,S) ǫ for each h P(f). By virtue of Lemma
≤ ∈
1, f P-ap(G). Hence, P-ap(G) is a closed subspace of the complete metric space
∈
C(G).
Theassertion4isobvious.
Assume that f C(G), b G and the set f(ϕ(x)) : ϕ P is unbounded.
Then there exists a∈sequence ϕ∈ P : n N{ such that f(∈ϕ (b})) 2 + f(b)
and f(ϕ (b)) 2 + f(ϕ (b{))nf∈or each n∈ N}. We put g| (x)1= f|(ϕ≥(x)). |Then|
d(f,|g ) n+12n f|o≥r each|n nN. |Hence P(f)∈is an unboundend subset onf C(G) and
n
≥ ∈
f < P ap(G). Theassertion5isproved.
−
Fixψ Π(G). ConsiderthemappingΦ :C(G) C(G),whereΦ(h)(x)=h(ψ(x))
∈ −→
for all h C(G) and x G. We have d(Φ(f),Φ(g)) d(f,g) for all f,g C(G).
∈ ∈ ≤ ∈
Fix now f P-ap(G) and put g(x) = f(ψ(x)) for each x G. Let ǫ > 0. Then
∈ ∈
there exists a finite subset S of C(G) such that d(h,S) ǫ for each h P(f). We
≤ ∈
have g (x) = g(ϕ(x)) = f(ϕ(ψ(x))) for each x G and each ϕ P. Assume that
ϕ
∈ ∈
ϕ P, δ > 0, h C(G) and d(f ,h) δ. Since f(ϕ(x) h(x) δ for any x G,
ϕ
∈ ∈ ≤ | − | ≤ ∈
we have f(ϕ(ψ(x)) h(ψ(x)) δ for any x G. Hence, the set Φ(S) is finite and
| − | ≤ ∈
d(h,Φ(S)) ǫ foreachh P(g). ByvirtueofLemma1,theassertion6isproved.
≤ ∈
Corollary 2.1. If P is a transitive set of translations ofG, then any function f P-
∈
ap(G)isboundedandP-ap(G)isaBanachalgebraofcontinuousfunctions.
Theorem 2.2. Let P be a set of continuous translations of G and F be a compact
subset of the complete metric space P-ap(G). Then the closure H of the set P(F) =
P(f) : f F isacompactsubsetofthespaceP-ap(G).
∪{ ∈ }
Proof. Fixǫ > 0. ThereexistsafinitesubsetS ofFsuchthatd(h,S ) ǫ/2foreach
1 1
≤
h F. Foreach f F thereexistsafinitesubsetS ofP(f)suchthatd(h,S ) ǫ/2
f f
∈ ∈ ≤
for each h P(f). We put S = S : f S . Fix h F and ϕ P. There
f 1
∈ ∪{ ∈ } ∈ ∈
exists f S such that d(f,h) ǫ/2. In continuation, there exists g S such that
1 f
∈ ≤ ∈
d(f ,g) ǫ/2. Since d(h , f ) d(h, f), we have d(h ,g) d(h , f )+d(f ,g) .
ϕ ϕ ϕ ϕ ϕ ϕ ϕ
≤ ≤ ≤ ≤
Henced(h,S) ǫ foreachh P(F). Theproofiscomplete.
≤ ∈
3. GROUPOIDS
Let (G, ) be a groupoid. For each a G we put L (x) = a x and R (x) = x a
a a
· ∈ · ·
for each x G. The mappings L ,R are called the left and right multiplication
a a
∈
operators or the left and right actions. A groupoid (G, ) with topology is called a
·
semitopologicalgroupoidiftheactions L ,R :G G arecontinuous. Let(G, )be
a a
→ ·
agroupoid. WeputL = L : a G andR = R : a G .
G a G a
{ ∈ } { ∈ }
DenotebySL theminimalsetofcontinuoustranslationsofGwithproperties:
G
-L SL ;
G G
⊂
Almost periodicity of functions on pseudocompact groupoids 5
-ifϕ,ψ SL ,thenϕ ψ SL .
G G
∈ ◦ ∈
DenotebySR theminimalsetofcontinuoustranslationsofGwithproperties:
G
-R SR ;
G G
⊂
-ifϕ,ψ SR ,thenϕ ψ SR .
G G
∈ ◦ ∈
NowweputS =SL SR .
G G G
∪
Foranyfunction f : G Rweput L (f)= f : ϕ SL(G) ,R (f)= f : ϕ
G ϕ G ϕ
→ { ∈ } { ∈
SR(G) andS (f)= f : ϕ S(G) =L (f) R (f).
G ϕ G G
} { ∈ } ∪
Let(G, )beasemitopologicalgroupoid. If f C(G),thenS (f) C(G).
G
· ∈ ⊆
Definition 3.1. Let (G, ) be a semitopological groupoid. A function f C(G) is
· ∈
called:
- a left almost periodic function of G if the closure L¯ (f) of the set L (f) in the
G G
spaceC(G)isacompactset;
-arightalmostperiodicfunctionofG iftheclosureR¯ (f)ofthesetR (f)inthe
G G
spaceC(G)isacompactset;
- a bi-almost periodic function of G if the closure S¯ (f) of the set S (f) in the
G G
spaceC(G)isacompactset.
Denote by L-ap(G) the subspace of all left almost periodic continuous functions,
by R-ap(G) the subspace of all right almost periodic continuous functions and by
SB-ap(G) the subspace of all bi-almost periodic continuous functions of the semi-
topologicalgroupoidG. Bydefinitions,wehaveSB-ap(G)=L-ap(G) R-ap(G).
∩
Forthespaces L-ap(G),R-ap(G)andSB-ap(G)aretrueallassertionsofthesec-
ondSection.
If G is a compact semitopological semigroup, then L-ap(G) = R-ap(G) = SB-
ap(G)=C(G)(see[14]).
A pseudometric d is an invariant pseudometric on a topological groupoid (G, ) if
·
d isacontinuouspseudometricandd(xy,uv) d(x,u)+d(y,v)forall x,y,u,v G.
≤ ∈
Thepseudometricd iscontinuousonGifthesets B(x,d,r)= y G : d(x,y) < r
{ ∈ }
areopeninGforall x Gandr > 0.
∈
If f SB-ap(G),thend (x,y)= sup f(ψ(x)) f(ψ(y)) : ψ S(G) isainvariant
f
∈ {| − | ∈ }
pseudometriconG. Ifthefunction f SB-ap(G)isbounded,thenthepseudometric
∈
d istotallybounded.
f
Example3.1. Let(A, )bethemultiplicativesemigroupofrationalswiththetopology
·
generated by the open base (u,v) : u,v A,u < v . The space A is a metrizable
{ ∈ }
topological semigroup. Denote by B= x A : x 0 as a subsemigroup of A. Let
{ ∈ ≥ }
d be an invariant pseudometric on B. Assume that 0 a < b and d(a,b) = 4k > 0.
≤
Since g(t) = d(0,t) is a continuous function on G and g(0) = 0, there exists λ > 0
suchthatg(t) < kforallt λ. Thend(u,v) 2kprovided0 u v λ. Thereexist
≤ ≤ ≤ ≤ ≤
three numbers p,q,r Bsuch that 0 p < q λ, r > 0, a = rp and b = rq. Then
∈ ≤ ≤
4k = d(a,b)=d(rp,rq) d(r,r)+d(p,q) 0+2k = 2k,acontradiction. Thus:
≤ ≤
-ontopologicalsemigroupsAand Banyinvariantpseudometricd istrivial
(d(x,y) = 0forall x,y);
6 Mitrofan M. Choban, Dorin I. Pavel
-ontopologicalsemigroupsAand Banyalmostperiodicfunctionisconstant.
Now we put S = L R and S = ϕ ψ : ϕ S ,ψ S for
(G,1) G G (G,n+1) (G,1) (G,n)
eachn N. ∪ { ◦ ∈ ∈ }
∈
Definition 3.2. Let (G, ) be a semitopological groupoid. A function f C(G) is
calledaweaklyalmostp·eriodicfunctionofG iftheclosureS¯ (f)ofthese∈tS (f)=
n n
f(ϕ(x)) : ϕ S inthespaceC(G)isacompactsetforeachn N.
(G,n)
{ ∈ } ∈
Definition 3.3. Let (G, ) be a semitopological groupoid. A function f C(G) is
calledanalmostperiodic· functionofGiftheclosureS(¯f)ofthesetS(f)=∈f(ϕ(x)) :
ϕ S ,n N inthespaceC(G)isacompactsetforeachn N. {
(G,n)
∈ ∈ } ∈
DenotebyS-wap(G)thesubspaceofallweaklyalmostperiodiccontinuousfunc-
tionsofthesemitopologicalgroupoidGandS-wap (G)=S-wap(G) C (G). Denote
◦ ◦
∩
byS-ap(G)thesubspaceofallalmostperiodiccontinuousfunctionsofthesemitopo-
logicalgroupoidGandS-ap (G)=S-ap(G) C (G).
◦ ◦
∩
IfGisasemitopologicalsemigroup,thenS-wap(G)=S-ap(G)andS(f)=S (f)
2
foranyfunction f C(G).
∈
Theorem3.1. IfGisacompacttopologicalgroupoid,thenS-wap(G)=C(G).
Proof. Fix f C(X).
∈
Consider the mappings Φ : G C(G) and Ψ : G C(G), where Φ(y)(x) =
−→ −→
f(L (x)) and Ψ(y)(x) = f(R (x)) for all x,y G. Since G is a compact space, the
y y
∈
mappingsΦandΨarecontinuous. HencethesetS (f)=Φ(G) Ψ(G)isacompact
1
∪
set.
Fix n 2. Fix an n-collection ξ = (i ,i ,...,i ), where i L,R for any j n, and
1 2 n j
≥ { } ≤
apointa = (a ,a ,...,a ) Gn. Weputϕ = L fori = L,ϕ =R fori = R
1 2 n ∈ (a,i) aj j (a,i) aj j
andϕ =ϕ ϕ ... ϕ . ConsiderthemappingΦ :Gn C(G),where
(ξ,a) (a,1)◦ (a,2)◦ ◦ (a,n) ξ −→
Φ (x) = f ϕ for any x Gn. Since G is a compact space, the mapping Φ is
ξ ◦ (ξ,x) ∈ ξ
continuous. HencethesetΦ (G)iscompact. Byconstruction,S (f)= Φ (G) : ξis
ξ n ξ
∪{
ann-collection . Sincewehaveafinitesetofn-collections,thesetS (f)iscompact.
n
}
Theproofiscomplete.
Corollary3.1. IfGisatopologicalgroupoid,thenCa(G)=S-wap (G)and(β G,β )
◦ a G
=(b G,e )forF =S-wap (G).
F F ◦
Proof. Let F = S-wap (G). For each f F the pseudometric d is totally bounded
◦ f
∈
and invariant. The operation : G2 G is uniformly continuous relatively to
· −→
pseudometrics d , f F. Hence for each f F on b G there exists a unique
f F
∈ ∈
continuous pseudometric ρ such that d (x,y) = ρ (e (x),e (y)) for all x,y G.
f f f F F ∈
As was proved [12], on b G there exists a structure of groupoid such that e is a
F F
homomorphism and ρ are invariant pseudometrics. The invariant pseudometrics
f
inducedonb G thegroupoidtopology. Henceb isacompactgroupoid. Theproof
F F
iscomplete.
Almost periodicity of functions on pseudocompact groupoids 7
4. QUASIGROUPS
A groupoid (G, ) is called a division groupoid if for any two points a,b G the
· ∈
equations a x = b and y a = b are solutions. If there exist a unique point x G
· · ∈
suchthata x = b,andauniquepointy Gsuchthaty a = b,then(G, )iscalleda
· ∈ · ·
quasigroup.
Adivisiongroupoid(G, )withtopologyiscalled:
·
-asemitopologicaldivisiongroupoidiftheactionsL ,R :G Garecontinuous;
a a
→
-aparatopologicaldivisiongroupoidiftheoperation :G2 Giscontinuous;
· →
- a topological division groupoid if G is a paratopological division groupoid and
there exist two continuous binary operation l,r : G2 G such that x l(x,y) =
→ ·
r(y,x) y=yforall x,y G.
· ∈
The operation l(x,y) is the left division and the operation r(x,y) is the right divi-
sion.
A division groupoid (G, ) with the divisions l,r is a quasigroup if and only if
·
l(x,x y)=l(r(x,y),x)=r(y x,x)=r(x,l((y,x))=yforall x,y G.
· · ∈
Some properties of paratopological and semitopological groups were studied in
[4, 2, 3]. There exists a semitopological group which is not a paratopological group
(see[4,2]). Thereexistsasimpleexampleofaparatopologicalgroupwhichisnota
topologicalgroup.
Example 4.1. Let A be the additive group of rationals with the topology generated
by the open base [u,v) : u,v A,u < v . The space A is countable and metrizable.
{ ∈ }
ThegroupAisaparatopologicalgroupwhichisnotatopologicalgroup.
Example4.2. Let Abethesetofrationalsand B= (u,v) : u,v A betheadditive
{ ∈ }
groupwiththeoperation(u,v)+(w,t)=(u+w,v+t)andwiththetopologygenerated
bytheopenbase O((u,v),r) = ((u r,u] (v r,v]) ([u,u+r) [v,v+r)) : u,v,r
{ − × − ∪ × ∈
A,r > 0 . ThespaceBiscountableandmetrizable. ThegroupBisasemitopological
}
group which is not a paratopological group. The inverse operation z ( z) is not
→ −
continuous.
Example4.3. Let AbethesetofrationalsandC = (u,v) : u,v A betheadditive
{ ∈ }
groupwiththeoperation(u,v)+(w,t)=(u+w,v+t)andwiththetopologygenerated
bytheopenbase O((u,v),r)= (u,v) (x,y) C : u r < x < u+r, x u < y v <
{ { }∪{ ∈ − −| − | −
x u : (u,v) C,r > 0 . ThespaceC iscountableandmetrizable. ThegroupC is
| − |} ∈ }
asemitopologicalgroupwhichisnotaparatopologicalgroup. Theinverseoperation
z ( z)iscontinuous.
→ −
Thefollowingassertionsareevidently.
Proposition 4.1. Let (G,+) be a division groupoid with the topology T and α,β :
G G betwocontinuousmappingsofthespaceG intoG. ConsideronG thenew
−→
binaryoperation x y = α(x)+β(y). Then:
∗
1. If the actions of the groupoid (G,+) are continuous, then the actions of the
groupoid(G, )arecontinuoustoo.
∗
8 Mitrofan M. Choban, Dorin I. Pavel
2. Iftheoperation+iscontinuous,thentheoperation iscontinuoustoo.
∗
3. (G, )isadivisiongroupoidifandonlyifα(G)=β(G)=G.
∗
4. If(G,+)isatopologicaldivisiongroupoidandα,βarehomeomorphisms,then
(G, )isatopologicaldivisiongroupoidtoo.
∗
5. If(G,+)isatopologicalquasigroup, then(G, )isatopologicalquasigroupif
∗
andonlyifα,βarehomeomorphisms.
5. PERIODIC GROUPOIDS
Let f : G G be a mapping of a space G. We put f1 = f, f2 = f f1, ...,
fn+1 = f fn−,→... and f0(x) = x for each x G. The mapping f is called◦periodic
◦ ∈
if there exists an integer n 1 such that fn = f0. The minimal integer n 1 for
≥ ≥
which fn = f0 iscalledtheperiodof f andisnotedbyπ(f). Acontinuousperiodic
mappingisahomeomorphism.
Definition5.1. Agroupoid(G, )withthetopologyT iscalledasemitopologicalΠ-
·
groupoidifthereexiststwoperiodiccontinuoushomomorphismsλ,µ :G G and
−→
a binary operation + on G such that (G,+) is a semitopological group and x y =
·
λ(x)+µ(y)forall x,y G. Inthiscasewesaythat(+,λ,µ)isastronggroupisotope
∈
ofthesemitopologicalΠ-groupoid(G, ).
·
Definition 5.2. A groupoid (G, ) with the topology T is called a semitopological
·
GT-quasigroup if there exists two homeomorphisms λ,µ : G G and a binary
−→
operation+onGsuchthat(G,+)isasemitopologicalgroupandx y=λ(x)+µ(y)for
·
allx,y G. Inthiscasewesaythat(+,λ,µ)isagroupisotopeofthesemitopological
∈
GT-quasigroup(G, ).
·
AgroupoidGiscalledamedialgroupoidif(xu)(vy)=(xv)(uy)forallx,y,u,v G.
∈
AnytopologicalmedialquasigroupisatopologicalGT-quasigroup(see[1,15]). Any
topological medial quasigroup with a multiply unity is a topological Π-quasigroup
[3]. FromProposition4.1itfollows
Corollary 5.1. Any Π-groupoid is a GT-quasigroup and any semitopological GT-
quasigroupisasemitopologicalquasigroup.
Theorem 5.1. Let P be a non-empty set of properties of topological spaces. The
followingassertionsareequivalent:
1. AnysemitopologicalgroupwithpropertiesPisatopologicalgroup.
2. AnysemitopologicalΠ-groupoidwithpropertiesPisatopologicalquasigroup.
3. Any semitopologicalGT-quasigroup with properties P is a topological quasi-
group.
Proof. Since any semitopological group is a semitopological Π-groupoid, the im-
plication 2 1 is obvious. Any semitopological Π-groupoid is semitopological
→
GT-quasigroup. Hencetheimplication3 2istruetoo.
→
Almost periodicity of functions on pseudocompact groupoids 9
Assume now that any semitopological group with properties P is a topological
group. Fix a semitopological GT-quasigroup (G, ) with properties P and a group
·
isotope(+,λ,µ)ofthesemitopologicalGT-quasigroup(G, ). Then(G,+)isasemi-
topological group with properties P. Hence (G,+) is a topo·logical group. Then, by
virtueofProposition4.1,(G, )isatopologicalquasigroup.
·
In[4,2,3]therearegivendistinctpropertiesunderwhichasemitopologicalgroup
isatopologicalgroup. Inparticular,weobtain:
Corollary 5.2. A pseudocompact (or locally pseudocompact) paratopological GT-
quasigroupisatopologicalquasigroup.
Corollary5.3. Ifapseudocompact(orlocallypseudocompact)paratopologicalquasi-
groupGismedial,thenGisatopologicalquasigroup.
6. ALMOST PERIODIC FUNCTIONS ON
PERIODIC GROUPOIDS
If G is a division groupoid, then SL(G) and SR(G) are transitive semigroups of
translations. Hence,fromCorollary2.1itfollows
Corollary 6.1. Let G be a division semitopological groupoid. Then any function
f L-ap(G) R-ap(G) is bounded and L-ap(G), R-ap(G), S-ap(G) are Banach
∈ ∪
algebrasofcontinuousfunctions.
Corollary 6.2. Let G be a semitopological quasigroup. Then any function f L-
∈
ap(G) R-ap(G) is bounded and L-ap(G), R-ap(G), S-ap(G) are Banach algebras
∪
ofcontinuousfunctions.
Theorem6.1. Let(G, )beapseudocompactparatopologicalΠ-groupoid. Then:
·
1. L-ap(G)=R-ap(G)=SB-ap(G)=C(G).
2. (G, )isapseudocompacttopologicalΠ-quasigroup.
3. (β·G,β ) is a compact topological Π-groupoid and β G is the Stone-Cˇech
a G a
compactificationβGofthespaceG.
Proof. Denote by (+,λ,µ) a strong group isotope of the topological Π-groupoid
(G, ). Assume that n,m 1 and λn(x) = µm(x) = x for each x G. By definition,
· ≥ ∈
themappingsλ,µ :G Garetopologicalisomorphismsand x y=λ(x)+µ(y)for
−→ ·
all x,y G. Hence x+y = λ 1(x) µ 1(y) for all x,y G. By virtue of Proposition
− −
∈ · ∈
4.1 and Corollary 5.2, (G, ) is a topological quasigroup and (G,+) is a topological
·
group.
By virtue of W.W. Comfort and K.A. Ross theorem (see [4],Corollary 6.6.3 and
Theorem6.6.4),onheStone-CˇechcompactificationβG ofthespaceG thereexistsa
structureofatopologicalgroupsuchthatGisasubgroupofβG. Thenthereexisttwo
homeomorphismsην : βG βG suchthatλ = ηG andµ = νG. Obviously, ηn(x)
−→ | |
=νm(x)= xforeach x βG. Thusthemappingsη,νareperiodictoo. Theoperation
∈
10 Mitrofan M. Choban, Dorin I. Pavel
x y = η(x)+ν(y), x,y βG, forms a structure of a topological Π-groupoid on βG
· ∈
such that G is a subquasigroup of the compact topological quasigroup βG. Hence,
withoutlossofgenerality,wecanassumethatG = βGisacompactquasigroup.
WeputSL (G)= L = L : a G ,SR (G)=R = L : a G andSL (G)=
1 G a 1 G a i+
{ ∈ } { ∈ }
ϕ ψ : ϕ L ,ψ SL(G) ,SR (G)= ϕ ψ : ϕ R ,ψ SR(G) foreachi 1.
G i i+ G i
{Ob◦viously∈SL(G)=∈ SL(G} ) : i N an{dS◦R(G)=∈ SR(∈G) : i N} ≥
i i
Fixk N. Leta =∪({a ,a ,...,a∈) }Gk. Weputτ =∪L{ L ... ∈L .}ThenL (G)=
∈ 1 2 k ∈ a a1◦ a2◦ ◦ ak k
τ : a Gk . Nowwefix f C(G)anda Gk. Then f(L (x))= f(a x)= f(η(a )+
{ a ∈ } ∈ ∈ a1 1· 1
ν(x)), f(L (L (x)))= f(η(a )+ν(η(a )+ν(x)))= f(η(a )+ν(η(a )+ν2(x))),..., f(τ (x)
a1 a2 1 2 1 2 a
= f(η(a )+ν(η(a )+...+ν(η(a )+ν(x))...))= f(η(a )+ν(η(a ))+...+νk 1(η(a ))+νk(x)).
1 2 k 1 2 − k
By construction, SL (f) = f(τ (x)) : a Gk = f(b + νk(x)) : b G . Hence
k a
SL (f)=SL (f)andL ({f)= f : ϕ S∈L(G}) ={ SL (f) : k N ∈= }SL (f) :
k+m k G ϕ k k
{ ∈ } ∪{ ∈ } ∪{
k 1,2,...,m .
∈ { }}
Since the correspondence a f(τ (x)), a Gk, is a continuous mapping of a
a
→ ∈
compact space Gk into C(G), the set SL (f) is compact. Hence the set L (f) is
k G
compactastheunionofafinitefamilyofcompactsets. Therefore L-ap(G)=C(G).
In the similar way we establish that R-ap(G) = C(G). Thus L-ap(G) = R-ap(G) =
SB-ap(G)=C(G). Theproofiscomplete.
Example 6.1. LetG be the compact space of complex number z with z = 1. Rela-
| |
tively to the multiplicative operation and inverse operation 1 the space G is a
−
{·} { }
compactcommutativegroupwiththeunite1. Letg : G G beahomeomorphism
−→
and x y = x g(y) for all x,y G. Then (G, ) is a topological GT-quasigroup.
∗ · ∈ ∗
Denote by P(g) the translations of the topological quasigroup (G,ω ). Obviously,
g
g P(g).
∈
In[4]wasconstructedsuchhomeomorphismg forwhichonlyconstantfunctions
0
are continuous almost periodic on (G,ω ). Since G is a compact space, then, by
g0
virtueofTheorem3.1,wehaveS-wap(G)=C(G).
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