IJMMS 2003:51, 3277–3280
PII. S0161171203202052
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
REDUCTIVE COMPACTIFICATIONS OF
SEMITOPOLOGICAL SEMIGROUPS
ABDOLMAJID FATTAHI, MOHAMAD ALI POURABDOLLAH,
and ABBAS SAHLEH
Received 10 February 2002
We consider the enveloping semigroup of a flow generated by the action of a semitopological semigroup on any of its semigroup compactifications and explore the
possibility of its being one of the known semigroup compactifications again. In this
way, we introduce the notion of E-algebra, and show that this notion is closely related to the reductivity of the semigroup compactification involved. Moreover, the
structure of the universal E Ᏺ-compactification is also given.
2000 Mathematics Subject Classification: 22A20, 43A60.
1. Introduction. A semigroup S is called right reductive if a = b for each
a, b ∈ S, since at = bt for every t ∈ S. For example, all right cancellative semigroups and semigroups with a right identity are right reductive.
From now on, S will be a semitopological semigroup, unless otherwise is stipulated. By a semigroup compactification of S we mean a pair (ψ, X), where X is
a compact Hausdorff right topological semigroup, and ψ : S → X is a continuous homomorphism with dense image such that, for each s ∈ S, the mapping
x→ψ(s)x : X→X is continuous. The C ∗ -algebra of all bounded complex-valued
continuous functions on S will be denoted by Ꮿ(S). For Ꮿ(S), the left and right
translations, Ls and Rt , are defined for each s, t ∈ S by (Ls f )(t) = f (st) =
(Rt f )(s), f ∈ Ꮿ(S). The subset Ᏺ of Ꮿ(S) is said to be left translation invariant if for all s ∈ S, Ls Ᏺ ⊆ Ᏺ. A left translation invariant unital C ∗ -subalgebra
Ᏺ of Ꮿ(S) is called m-admissible if the function s→Tµ f (s) = µ(Ls f ) is in Ᏺ
for all f ∈ Ᏺ and µ ∈ S Ᏺ (where S Ᏺ is the spectrum of Ᏺ). Then the product
of µ, ν ∈ S Ᏺ can be defined by µν = µ ◦ Tν and the Gelfand topology on S Ᏺ
makes (, S Ᏺ ) a semigroup compactification (called the Ᏺ-compactification) of
S, where : S→S Ᏺ is the evaluation mapping.
Some m-admissible subalgebras of Ꮿ(S), that we will need, are left multiplicatively continuous functions ᏸᏹᏯ, distal functions Ᏸ, minimal distal functions ᏹᏰ, and strongly distal functions ᏿Ᏸ. We also write Ᏻᏼ for ᏹᏰ ∩ ᏿Ᏸ;
and we define ᏸᐆ := {f ∈ Ꮿ(S); f (st) = f (s) for all s, t ∈ S}. For a discussion of the universal property of the corresponding compactifications of these
function algebras see [1, 2].
3278
ABDOLMAJID FATTAHI ET AL.
2. Reductive compactifications and E-algebras. Let (ψ, X) be a compactification of S, then the mapping σ : S × X→X, defined by σ (s, x) = ψ(s)x, is
separately continuous and so (S, X, σ ) is a flow. If ΣX denotes the enveloping semigroup of the flow (S, X, σ ) (i.e., the pointwise closure of semigroup
{σ (s, ·) : s ∈ S} in X X ) and the mapping σX : S→ΣX is defined by σX (s) = σ (s, ·)
for all s ∈ S, then (σX , ΣX ) is a compactification of S (see [1, Proposition 1.6.5]).
One can easily verify that ΣX = {λx : x ∈ X}, where λx (y) = xy for each
y ∈ X. If we define the mapping θ : X → ΣX by θ(x) = λx , then θ is a continuous
homomorphism with the property that θ ◦ ψ = σX . So (σX , ΣX ) is a factor of
(ψ, X), that is (ψ, X) ≥ (σX , ΣX ). By definition, θ is one-to-one if and only if X
is right reductive. So we get the next proposition, which is an extension of the
Lawson’s result [3, Lemma 2.4(ii)].
Proposition 2.1. Let (ψ, X) be a compactification of S. Then (σX , ΣX ) (ψ, X) if and only if X is right reductive.
A compactification (ψ, X) is called reductive if X is right reductive. For example, the ᏹᏰ-, Ᏻᏼ-, and ᏸᐆ-compactifications are reductive.
An m-admissible subalgebra Ᏺ of Ꮿ(S) is called an E-algebra if there is a
compactification (ψ, X) such that (σX , ΣX ) (, S Ᏺ ). In this setting (ψ, X) is
called an E Ᏺ-compactification of S. Trivially for every reductive compactification (ψ, X), ψ∗ (Ꮿ(X)) is an E-algebra. But the converse is not, in general,
true. For instance, for any compactification (ψ, X), σX∗ (Ꮿ(ΣX )) is an E-algebra;
however, it is possible that ΣX would be nonreductive, as the next example
shows.
Example 2.2. Let S = {a, b, c, d} be the semigroup with the following multiplication table:
a
b
c
d
a
a
a
a
a
b
a
a
a
c
c
a
a
a
a
d
a
c
a
b
Then for the identity compactification (i, X) of S, ΣX is not right reductive; in
fact, λa ≠ λb , however, λat = λbt for every t ∈ S.
Lemma 2.3. If (ψ, X) is a compactification satisfying X 2 = X, then the compactification (σX , ΣX ) is reductive.
Proof. Since X 2 = X, for each x1 , x2 ∈ X, from λx1 λy =λx2 λy for every
λy ∈ ΣX , it follows that λx1 =λx2 . So ΣX is right reductive.
Corollary 2.4. Let sS (or Ss) be dense in S, for some s ∈ S, then for every
compactification (ψ, X) of S, it follows that X 2 = X and so (σX , ΣX ) is reductive.
REDUCTIVE COMPACTIFICATIONS OF SEMITOPOLOGICAL . . .
3279
Now, we are going to construct the universal E Ᏺ-compactification of S. For
this end we need the following lemma.
Lemma 2.5. Let Ᏺ be an m-admissible subalgebra of Ꮿ(S). Then Tν f ∈
for all f ∈ Ᏺ and ν ∈ S ᏸᏹᏯ .
σS∗Ᏺ (Ꮿ(ΣS Ᏺ ))
Proof. Since ΣS Ᏺ = {λµ : µ ∈ S Ᏺ }, we can define g : ΣS Ᏺ →C by g(λµ ) =
µ(Tν f ), where C denotes the complex numbers. Since the mapping λµ → µν :
ΣS Ᏺ → S Ᏺ is p-weak∗ continuous, g is a bounded continuous function and it is
easy to see that σS∗Ᏺ (g) = Tν (f ). Therefore, Tν f ∈ σS∗Ᏺ (Ꮿ(ΣS Ᏺ )) for all ν ∈ S Ᏺ .
If ν̃ ∈ S ᏸᏹᏯ and ν is the restriction of ν̃ to Ᏺ, then Tν̃ f = Tν f for all f ∈ Ᏺ. So
the conclusion follows.
Proposition 2.6. Let Ᏺ be an E-algebra. Then
GᏲ := f ∈ ᏸᏹᏯ : Tν f ∈ Ᏺ ∀ν ∈ S ᏸᏹᏯ
(2.1)
is an m-admissible subalgebra of Ꮿ(S) and (, S GᏲ ) is the universal E Ᏺ-compactification of S.
Proof. It is easy to verify that GᏲ is an m-admissible subalgebra of Ꮿ(S)
containing Ᏺ. By definition of GᏲ we can define the mapping θ : S Ᏺ → ΣS GᏲ
by θ(µ) = λµ̃ , where µ̃ is an extension of µ to S GᏲ . Clearly, θ is continuous
and θ ◦ = σS GᏲ . Thus (, S Ᏺ ) ≥ (σS GᏲ , ΣS GᏲ ). On the other hand, since Ᏺ is
an E-algebra, there exists a compactification (φ, Y ) of S such that (σY , ΣY ) (, S Ᏺ ) and Ᏺ = σY∗ (Ꮿ(ΣY )). By Lemma 2.5, we have Tν f ∈ σY∗ (Ꮿ(ΣY )), for
each ν ∈ S ᏸᏹᏯ and each f ∈ φ∗ (Ꮿ(Y )). This means that φ∗ (Ꮿ(Y )) ⊂ GᏲ
and so, by [1, Propositiion 1.6.7], (σY , ΣY ) ≤ (σS GᏲ , ΣS GᏲ ). Therefore, (, S Ᏺ ) (σS GᏲ , ΣS GᏲ ) and (, S GᏲ ) is an E Ᏺ-compactification of S. Finally, if (ψ, X) is
an E Ᏺ-compactification of S and f ∈ ψ∗ (Ꮿ(X)), then by Lemma 2.5, Tµ f ∈
σX∗ (Ꮿ(ΣX )) = Ᏺ for all µ ∈ S ᏸᏹᏯ . So ψ∗ (Ꮿ(X)) ⊂ GᏲ and (ψ, X) ≤ (, S GᏲ ).
Examples 2.7. (a) We have GᏹᏰ = Ᏸ. To see this, if f ∈ GᏹᏰ , then for all
µ, ν, η ∈ S ᏸᏹᏯ with η2 = η, we have µην(f ) = µη(Tν f ) = µ(Tν f ) = µν(f ). So
f ∈ Ᏸ. Also if f ∈ Ᏸ, then for all µ, ν, η ∈ S ᏸᏹᏯ with η2 = η, we have µη(Tν f ) =
µην(f ) = µν(f ) = µ(Tν f ). That is, Tν f ∈ ᏹᏰ for all ν ∈ S ᏸᏹᏯ and so f ∈ GᏹᏰ
(see also [4, Lemma 2.2]).
(b) By a similar proof, we can show that GᏳᏼ = ᏿Ᏸ (see [4, Lemma 2.2 and
Theorem 2.6]).
(c) Let ᏾ := {f ∈ ᏸᏹᏯ(S) : f (r st) = f (r t) for r , s, t ∈ S}. Clearly, ᏾ is an
m-admissible subalgebra of Ꮿ(S). If f ∈ ᏾ and ν ∈ S ᏸᏹᏯ , then for each
r , s, t ∈ S we have Lr t f (s) = f (r ts) = f (r s) = Lr f (s). So Tν f (r t) =
ν(Lr t f ) = ν(Lr f ) = Tν f (r ). That is, Tν f ∈ ᏸᐆ. On the other hand,
if f ∈ Gᏸᐆ , then f (r st) = (T(t) f )(r s) = (T(t) f )(r ) = f (r t) and so
f ∈ ᏾. Therefore, Gᏸᐆ = ᏾.
3280
ABDOLMAJID FATTAHI ET AL.
Acknowledgment. In the end, we would like to thank Dr H. R. EbrahimiVishki for his valuable comments.
References
[1]
[2]
[3]
[4]
J. F. Berglund, H. D. Junghenn, and P. Milnes, Analysis on Semigroups, Function
Spaces, Compactifications, Representations, Canadian Mathematical Society
Series of Monographs and Advanced Texts, John Wiley & Sons, New York,
1989.
H. D. Junghenn, Distal compactifications of semigroups, Trans. Amer. Math. Soc.
274 (1982), no. 1, 379–397.
J. D. Lawson, Flows and compactifications, J. London Math. Soc. (2) 46 (1992), no. 2,
349–363.
M. A. Pourabdollah and A. Sahleh, Flows and universal compactifications, J. Sci.
Islam. Repub. Iran 8 (1997), no. 4, 269–273.
Abdolmajid Fattahi: Department of Mathematics, Faculty of Science, Razi University,
Kermanshah, Iran
E-mail address: abfattahi@yahoo.ca
Mohamad Ali Pourabdollah: Faculty of Mathematics, Ferdowsi University of Mashhad,
P.O. Box 91775-1159, Mashhad, Iran
E-mail address: pourabd@math.um.ac.ir
Abbas Sahleh: Department of Mathematics, Guilan University, Rasht, Iran
E-mail address: sahlehj@kd.gu.ac.ir