IJMMS 26:6 (2001) 353–357
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 function algebras.
2000 Mathematics Subject Classification. 22A20, 43A60, 54H15.
1. Introduction. In the main approach to semigroup compactification, whose considerations goes back at least to the pioneering paper of de Leeuw and Glicksberg [2],
the spectra of some C ∗ -algebras of functions on a semitopological semigroup are employed to construct certain universal compactifications. For instance, Junghenn in his
elaborate study of distal functions characterized the universal group compactification, [7]. The main goal of the present work is to construct two function algebras ᏲV
and Ᏺᐂ whose corresponding compactifications are universal with respect to the properties satisfying a variety V of semigroups and a variety ᐂ of 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. On Ꮿ(S), the C ∗ -algebra of all continuous bounded complex-valued functions on a semitopological semigroup S, the left
and right translations Ls and Rs are defined so that, (Ls f )(t) = f (st) = (Rt f )(s). A
unital C ∗ -subalgebra Ᏺ of Ꮿ(S) which is a left translation invariant (i.e., Ls f ∈ Ᏺ for
all s ∈ S and f ∈ Ᏺ) is called m-admissible if the function s (Tµ f )(s) = µ(Ls f ) lies
in Ᏺ for all f ∈ Ᏺ and µ ∈ S Ᏺ (equal to the spectrum of Ᏺ). Then the pair (ε, S Ᏺ ) is
a (semigroup) compactification (called Ᏺ-compactification) of S, in which S Ᏺ is furnished with the Gelfand topology and the multiplication µν = µ ◦ Tν , and ε : S → S Ᏺ is
the evaluation mapping.
The m-admissible algebras of the left multiplicatively continuous, (weakly) [strongly]
almost periodic, and strongly minimal distal functions, are denoted by ᏸᏹᏯ, (ᐃᏭᏼ)
[᏿Ꮽᏼ]Ꮽᏼ, and Ᏻᏼ, respectively; see [1, 7].
3. The main results. From now on, FA is a fixed free semigroup on the countable alm
n
phabet A, on which the identities of V are expressed, and p(= i=1 ai ) = q(= j=1 bj )
is a fixed arbitrary formal identity of V ; see [6].
Definition 3.1. We define ᏲV (S) as the set of those f ∈ ᏸᏹᏯ(S) such that the
identities
354
H. R. EBRAHIMI-VISHKI

lim 
α
i=2

lim 
α


m
i=1


m
m
i=1
m
i=2
 
lim Rθα (ai ) f  θα a1
α

= lim 
α
n
j=2
 
lim Rθα (bj ) f  θα b1 ,
α
(3.1)




n
lim Rθα (ai) lim Rsα f  θα a1 = lim  lim Rθα (bj) lim Rsα f  θα b1 ,

α
α


lim Rθα (ai ) f = 
α
α
n
j=1
j=2
α
α
(3.2)

lim Rθα (bj ) f ,
(3.3)
α


n


lim Rθα (ai )
lim Rθα (bj )  lim Rsα f ,
lim Rsα f =

α
α
j=1
α
(3.4)
α
n
m
are valid for all identities i=1 ai = j=1 bj of V , and all nets θα (of mappings from A
into S), for which the involved pointwise limits exist. (Note that, the notation is being
used for the iterated multiplications in different semigroups, and one should realize
m
that i=2 limα Rθα (ai ) plays the role of identity operator whenever m = 1; similarly for
n = 1.)
The next lemma presents ᏲV in terms of S ᏸᏹᏯ .
Lemma 3.2. A function f ∈ ᏸᏹᏯ belongs to ᏲV if and only if for every homomorphism Φ : FA → S ᏸᏹᏯ , each identity p = q of V , and every µ ∈ S ᏸᏹᏯ , Φ(p)(f ) = Φ(q)(f ),
(Φ(p)µ)(f ) = (Φ(q)µ)(f ), TΦ(p) f = TΦ(q) f , and T(Φ(p)µ) f = T(Φ(q)µ) f .
Proof. For the necessity, it is enough to show that Φ(p)(g) = Φ(q)(g) and TΦ(p) g =
TΦ(q) g, for all g∈ Xf ∪{f }, where Xf is the pointwise closure of RS f in Ꮿ(S). There exists
a net θα : A → S such that, for each i and j, limα ε(θα (ai )) = Φ(ai ) and limα ε(θα (bj ))
= Φ(bj ). Now using (3.1) and (3.2), we have


m
TΦ(ai ) g
Φ(p)(g) = Φ a1 

= lim 
α

i=2
m
i=2
 
lim Rθα (ai ) g  θα a1
α
(3.5)
 
n


= lim
lim Rθα (bj ) g  θα b1
α
j=2
α
= Φ(q)(g).
A similar argument, using (3.3) and (3.4), shows that TΦ(p) g = TΦ(q) g. Conversely, for
any net θα , of mappings from A into S, consider the mapping θ : A → S ᏸᏹᏯ defined
by θ(a) = limα ε(θα (a)). Using the crucial property of free semigroups, θ can be
extended to a homomorphism Φ : FA → S ᏸᏹᏯ . Therefore, for every g ∈ Xf ∪ {f },






m
m
n

lim Rθα (ai ) g = 
TΦ(ai ) g = TΦ(p) g = TΦ(q) g = 
lim Rθα (bj ) g;
(3.6)
i=1
α
i=1
j=1
α
SOME ALGEBRAIC UNIVERSAL SEMIGROUP COMPACTIFICATIONS
355
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 of FV .
Theorem 3.3. ᏲV is m-admissible and ᏲV -compactification is universal with respect
to the property satisfying V .
Proof. Using Lemma 3.2, the m-admissibility of ᏲV may be readily verified. Again
Lemma 3.2 implies that, for each homomorphism Φ : FA → S ᏲV and f ∈ ᏲV , Φ(p)(f ) =
Φ(q)(f ); which means S ᏲV ∈ V . It is enough to show that for every other compactification (ψ, X) of S, with X ∈ V , ψ∗ (Ꮿ(X)) ⊆ ᏲV (S). Let π : S ᏸᏹᏯ → X be the homomorphism for which π ◦ ε = ψ. This implies that ψ∗ (Ꮿ(X)) ⊆ ᏸᏹᏯ(S). Furthermore,
if h ∈ Ꮿ(X), then for each homomorphism Φ : FA → S ᏸᏹᏯ and each µ ∈ S ᏸᏹᏯ ,
Φ(p)µ ψ∗ (h) = h (π ◦ Φ)(p)π (µ) = h (π ◦ Φ)(q)π (µ) = Φ(q)µ ψ∗ (h) . (3.7)
Similar arguments show that
Φ(p) ψ∗ (h) = Φ(q) ψ∗ (h) ,
TΦ(p) ψ∗ (h) = TΦ(q) ψ∗ (h),
∗
(3.8)
∗
T(Φ(p)µ) ψ (h) = T(Φ(q)µ) ψ (h).
Now Lemma 3.2 implies that ψ∗ (h) ∈ ᏲV (S), as claimed.
Let ᏲVc consist of those f ∈ Ꮿ(S) such that f (Φ(p)) = f (Φ(q)), f (Φ(p)s) = f (Φ(q)s),
f (sΦ(p)) = f (sΦ(q)), and f (sΦ(p)t) = f (sΦ(q)t), for all s, t ∈ S, for every homomorphism Φ : FA → S, and all identities p = q of V . Trivially, ᏲV ⊆ ᏲVc (with the equality
holding in the compact setting). Due to joint continuity of the multiplication of S Ꮽᏼ ,
we get the next simplification of ᏲV ∩ Ꮽᏼ.
Proposition 3.4. ᏲV ∩ Ꮽᏼ = ᏲVc ∩ Ꮽᏼ.
Proof. According to Lemma 3.2 it is enough to show that if f ∈ ᏲVc ∩ Ꮽᏼ, then for
each homomorphism Φ : FA → S Ꮽᏼ , and all µ ∈ S Ꮽᏼ , Φ(p)(f ) = Φ(q)(f ), (Φ(p)µ)(f ) =
(Φ(q)µ)(f ), TΦ(p) f = TΦ(q) f , and T(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
limα ε(Φα (ai )) = Φ(ai ) and limα ε(Φα (bj )) = Φ(bj ); and also tα be a net in S such
that limα ε(tα ) = µ. Now, for all s ∈ S;

 
m


Φ ai µ  Ls f
TΦ(p)µ f (s) =

= 
i=1
m
i=1
lim ε Φα
α



ai
lim ε tα  Ls f
α
= lim ε Φα (p)tα Ls f = lim f sΦα (p)tα
α
α
= lim f sΦα (q)tα = TΦ(q)µ f (s).
α
Similar arguments may apply to the other required equalities.
(3.9)
356
H. R. EBRAHIMI-VISHKI
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.
By Definition 3.1, ᏲAB consists of those f ∈ ᏸᏹᏯ such that the identities
lim lim Rsα f tα = lim lim Rtα f sα ,
α
α
α
α
lim lim Rsα lim Ruα f
tα = lim lim Rtα lim Ruα f
sα ,
α
α
α
α
α
α
(3.10)
lim Rsα lim Rtα f = lim Rtα lim Rsα f ,
α
α
α
α
lim Rsα lim Rtα lim Ruα f
= lim Rtα lim Rsα lim Ruα f
α
α
α
α
α
α
hold for all nets sα , tα , and uα in S, for which the limits exist. Since S ᏲAB is semitopological, one realizes that the latter seemingly complicated limit process is preparatory,
and can be condensed so that it presents ᏲAB in the simple form
(3.11)
f ∈ ᐃᏭᏼ : f (st) = f (ts), and f (stu) = f (sut)∀s, t, u ∈ S .
In other words, ᏲAB = ᏲAB c ∩ ᐃᏭᏼ.
Again, Definition 3.1 implies that ᏲBD is the set of all f ∈ ᏸᏹᏯ, for which
lim lim Rsα f sα = lim f sα ,
α
α
α
lim lim Rsα lim Rtα f
sα = lim lim Rtα f sα ,
α
α
α
α
α
lim Rsα lim Rsα f = lim Rsα f ,
α
α
α
lim Rsα lim Rsα lim Rtα f
= lim Rsα lim Rtα f ;
α
α
α
α
(3.12)
α
for all nets sα and tα in S, for which the limits exist. In contrast to the situation for
ᏲAB , in general ᏲBD ⊆ ᐃᏭᏼ (e.g., a direct verification shows that the characteristic
function of even numbers on N, with its maximum multiplication, lies in ᏲBD but not
in ᐃᏭᏼ). Furthermore, as it is seen by easy examples (e.g., consider [0, 1]×N under its
rectangular multiplication) ᏲBD ≠ ᏲBDc , in general. It would be desirable to investigate
the equality of ᏲBD = ᏲBDc ∩ ᏸᏹᏯ, the left and right introversion of ᏲBD , the relation
between ᏲBD and the space similarly defined in terms of the left translations, and to
characterize the topological center of S ᏲBD . We believe that there are close connections
among these problems.
By the joint continuity theorem of Lawson [8], ᏲSL ⊆ Ꮽᏼ and so ᏲSL = ᏲSLc ∩ Ꮽᏼ.
It would be desirable to examine ᏲV 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 in V ; 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 function f ∈ Ᏻᏼ lies in ᏲV if and only if (S, Xf )
lies in V ; that is, ᏲV ∩ Ᏻᏼ = {f ∈ Ᏻᏼ : (S, Xf ) ∈ V }.
Now, for a variety ᐂ of groups, defined by the set of laws Ω (see [9]), we define Ᏺᐂ (S)
m
as the set of all f ∈ Ᏻᏼ(S) such that ( i=1 (limα Rsiα )li )(limα Rsα f ) = limα Rsα f , for
m
all laws i=1 xi li in Ω, and all nets siα and sα in S for which the required limits exists.
SOME ALGEBRAIC UNIVERSAL SEMIGROUP COMPACTIFICATIONS
357
It is easy to verify that, a function f ∈ Ᏻᏼ lies in Ᏺᐂ if and only if Tωµ f = Tµ f for
all µ ∈ S Ᏻᏼ and all values in S Ᏻᏼ of the laws ω ∈ 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 of Theorem 3.3.
Theorem 3.5. Ᏺᐂ is m-admissible and Ᏺᐂ -compactification is universal with respect
to the property satisfying ᐂ.
It should be mentioned that Ᏺᐂ is not in ᐃᏭᏼ in general (e.g., for the variety of all
groups we get Ᏻᏼ which is not always in ᐃᏭᏼ). 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
m
m
those f ∈ ᏿Ꮽᏼ such that, ( i=1 (Rsi )li )(Rs f ) = Rs f , for all laws i=1 xi li in Ω, and all
s1 , s2 , . . . , sm , and s in S. (For instance, for the variety of abelian groups, Ᏺᐂ consists
of those f ∈ ᏿Ꮽᏼ such that f (stu) = f (sut) for all s, t and u in S, [3]; which, of
course, is equal to ᏲAB ∩ 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 of Ᏺᐂ ,
limα Rsα f can be replaced by f . However this seems not to be possible in general.
Acknowledgement. This work was in part supported by a grant from IPM.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
J. F. Berglund, H. D. Junghenn, and P. Milnes, Analysis on Semigroups: Function
Spaces, Compactifications, Representations, John Wiley and Sons, New York, 1989.
MR 91b:43001. Zbl 727.22001.
K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math.
105 (1961), 63–97. MR 24#A1632. Zbl 104.05501.
H. R. Ebrahimi-Vishki and M. A. Pourabdollah, The universal nilpotent group compactification of a semigroup, Proc. Amer. Math. Soc. 125 (1997), 2171–2174. MR 97i:43002.
, The universal semilattice compactification of a semigroup, Int. J. Math. Math. Sci.
22 (1999), no. 1, 85–89. MR 2000h:22004. Zbl 916.22004.
R. Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957), 119–125.
MR 19,561b. Zbl 079.16602.
J. M. Howie, An Introduction to Semigroup Theory, no. 7, Academic Press, London, 1976,
L.M.S. Monographs. MR 57#6235. Zbl 355.20056.
H. D. Junghenn, Distal compactifications of semigroups, Trans. Amer. Math. Soc. 274 (1982),
no. 1, 379–397. MR 84h:43017. Zbl 514.22003.
J. D. Lawson, Joint continuity in semitopological semigroups, Illinois J. Math. 18 (1974),
275–285. MR 49#454. Zbl 278.22002.
D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80,
Springer-Verlag, New York, 1982. MR 84k:20001. Zbl 483.20001.
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
E-mail address: vishki@math.um.ac.ir