Internat. J. Math. & Math. Sci.
VOL. 18 NO. 3 (1995) 497-500
497
QUASIMINIMAL DISTAL FUNCTION SPACE AND
ITS SEMIGROUP COMPACTIFICATION
R. D. PANDIAN
Department of Mathematics
North Central College
Naperville, IL 60566
(Received July 27, 1993 and in revised form February 12, 1994)
ABSTRACT.
Quasiminimal distal function on a semitopological semigroup is
introduced.
The concept of right topological semigroup compactification is
employed to study the algebra of quasiminimal distal functions.
The universal
mapping property of the quasiminimal distal compactification is obtained.
KEY WORDS AND PHRASES.
Quasiminimal distal flows and distal functions, right
topological semigroup compactification, universal mapping property.
1980 AMS SUBJECT CLASSIFICATION CODES. Primary 43A60, 22A20 secondary 43A07
In a recent paper, we generalized the notion of distal flows and distal functions
on an
arbitrary semitopological semigroup S and studied their function spaces
[3].
The study was an extension of Junghenn’s work on distal functions and their
semigroup compactifications [2].
is the study
by Berglund
et
The impetus for the author’s work in this area
al. in which the authors have introduced several
kinds of function spaces which are admissible
C*-subalgebras
of C(S) and studied
these function spaces through semigroup compactifications [i].
Interestingly,
these compactifications possess certain universal mapping properties, universal
in the sense that the compactifications are maximal with respect to these
properties.
This approach of studying function spaces through semigroup
compactifications enables one to take advantage of the universal mapping
properties the compactifications enjoy.
For example, in the case of distal
functions on a semigroup S, Junghenn has effectively used the universal mapping
properties to show that the space of weakly almost periodic distal functions on S
is the direct sum of the
algebra of strongly almost periodic functions and two
ideals of weakly almost "flight" functions on S
[2].
In this paper, we
introduce the quasiminimal distal function space QMD(S), and show that this space
is an admissible
C*osubalgebra
of C(S).
As a consequence, we obtain the
QMD(S)-compactification of S which is the spectrum of QMD(S).
We then
characterize this semigroup compactification in terms of the universal mapping
property the compactification possesses.
i.
PRELIMINARIES
Throughout our work, S denotes a semitopological semigroup with Hausdorff
topology and C(S) denotes the C*-algebra of all bounded complex-valued continuous
functions on S.
A subspace F of C(S) is translation invariant if
LsF=F
and
RsF=F
R. D. PANDIAN
498
where the operators h
f(ts) for s,
S and f
t
on C(S) are defined
Tx.F
We define
C(S).
f(st) and Rsf(t)
by Lsf(t)
homomorphisms on F.
A
C*-subalgebra F of C(S)
is called admissible if it is
TxF=F
translation invariant, contains the constant functions, and
spectrum of F.
x
x(L f) where
B by Txf(.
the spectrum of F which is the space of nonzero continuous complex
F and x
f
Rs
and
A pair (X,) is called
X is a continuous
of S if X is a compact right topological semigroup and a’S
homomorphism with dense image such that the mapping x
subalgebra of C(S) and
compactification.
a
X is continuous
a(s)x’X
It is called an F-compactification of S if F is an admissible
S.
for each s
for every
right topological Gompactification
*
We call X the phase .space of the
F.
C(X)
The set of all multiplicative means on F, denoted by MM(F), is
w-compact right topological semigroup with binary operation defined in MM(F)
f(s)), then e(S)
xoTy. If eS MM(F) is the evaluation map (e(s)(f)
by xy
For a
is dense in MM(F) and the pair (MM(F), e) is an F-compactification of S.
fixed F, all Focompactifications are isomorphic both algebraically and
following properties (for details see [I. 1.4]).
T
TxoTy
b)
[e(s)x](f)
x
For s
B has the
X,
F and x, y
S, f
[x e(s)](f)
c)
(Lsf)
Tx’F
The map
(MM(F), e), the canonical F-compactification of S.
a)
We call
Thus, there is a unique F-compactification of S.
topologically.
(Rsf).
x
A right
topological compactification (X,) of S is said to be maximal (universal) with
respect to a property P if (X,) has the property P and if (Y,) is a right
topological compactification of S with property P, then
%o
that
for some continuous homomorphism %’XY.
subalgebra of C(S) is LMC(S)
x
MM C(S).
If
C(S)
s
x
factors
in the sense
A very useful admissible
(Lsf)
is continuous for
every
It is the largest admissible subalgebra of C(S) and the hMC(S)-
compactification of S is maximal with respect to the property that it is a right
topological compactification of S [i].
XX is a continuous
If X is a compact topological space and ’S
X is continuous for each s
homomorphism such that (s)’X
(S, X, ) is called a flo____w.
For convenience, we write sx for (s)(X).
is called distal (respectively, quasidistal) if, whenever
lim
s
x
lim
for every s
S).
s y
S, then the triple
y (respectively, sx
for some net (s) in S, then x
A function f
The flow
x,y,X such that
sy
IMC(S) is called a distal (respectively, a
quasidistal) function if the flow (S, Z, ), where
the topology of pointwise convergence on C(S) and
Z
is the closure of
(s)’flz
Rsf
in
is distal
a quasidistal).
It is called a minimal function if g
Z implies
Junghenn has shown that the set of all minimal distal functions, MD(S),
(respectively,
f
Z.
is an admissible
subalgebra of C(S) and that MD(S)ocompactification (Y,) is
maximal with respect to the property that Y is left simple
proved analogous results for quasidistal functions, QD(S).
[2].
The author has
For more on these
functions and related theorems, the reader is referred to [3,4].
2.
QUASIMINIMAL DISTAL FUNCTIONS
DEFINITION 2-1.
A function of f
LMC(S) is quasiminimal if for every
g
Z, there exists x X such that Tuf Tuxg for every u
phase space of the canonical LMC(S)-compactification (X,).
We denote the set of all quasiminimal functions by QM(S).
X where X is the
QUASIMINIMAL DISTAL FUNCTION SPACE AND ITS SEMIGROUP COMPACTIFICATION
PROPOSITION 2"2
PROOF.
Tuf
{Txf
Tyf
TuxTyf Tuxyf
X}
x
X.
some y
Let g
X.
compactification of S.
f,
QM(S).
LMC(S) is said to be quasiminimal
We write QMO(S) for QM(S)nQD(S).
LMC(S) and (X,a) denote the canonical LMC(S)-
Let f
PROPOSITION 2 4.
Txg
TuxTyf
Tuxyf
Tuf
for
Tyf
which implies that g
X such that
A function of
QM(S)nQD(S).
distal if f
X
x
Then by definition, f
DEFINITION" 2"3.
every v
{Txf
Zf
For this y, there exists x
for every
X such that
X.
for every
Sufficiency"
X.
Let y
By definition, there exists x
Zf
X.
for every u
Tuf
Assume f is quasiminimal.
Necessity"
X, there
QM(S) if and only if for each y
Then f
X (depending on y and f) such that Tuxyf
exists x
Then
LMC(S) and (X,a) denote the canonical LMC(S)-
Let f
compactification of S.
499
ve(f) for
QD(S) and v(f)
QMD(S) if and only if f
f
E(x)
e
Before we present the proof of this proposition, we state
Remark 2-5.
here without proof the author’s characterization of quasidistal functions QD(S)
A function f
,
QD(S) if and only if uev(f)
LMC(S) is quasidistal, that is f
7
uv(f) for every u
X and
v
E(X) where X is the phase space of the
e
LMC(S)-compactification of S.
PROOF.
X 2" then v
Let v
v
(2v=f)
and
2v2xef
-
the same result holds for v
Let y
which implies exeye
e.
vuexey(f) (the last
vue(f)
X.
X.
two equalities
We recall that X has
QM(S).
X such that (exe)(eye)
by 2.5)
Since
was
arbitrary, Tuxyf
QMD(S), v
QMD(S) is one admissible subalgebra of C(S).
Z
v(Lsf)
and e
E(X).
For s
which implies that
v(s)(f) (Remark 2.5)
translation invariant.
S, ve(Lsf)
Lsf
QMD(S).
(s)ve(f) --(s)v(f)
Now ve(Rsf
LMC(S).
Trivially, v(1)
QMD(S) contains 1 and hence contains all constant functions.
proof by showing that
x
Tzf
QMD(S) for every z
MM QMD(S) be the restriction map.
continuous homomorphism onto MM QMD(S).
For s
S, Tf(s)
Let
vem(s)(f)
QMD(S). Thus QMD(S) is
v(Rsf) which proves that R_f
That QMD(S) is an algebra follows from the fact that X
the set of all multiplicative means on
and
Tuf.
QM(S)
Linearity of the space QMD(S) is immediate from Proposition 2.4.
(Prop. 2.4)
e
X, V(Tuxf
vuexy(f)
vuxy(f)
vuexeye(f) (by hypothesis)
Let (X,) denote the canonical LMC(S)-compactification of S.
PROOF.
f
That
ve(f).
For any v
v(Tuf).
Therefore by Proposition 2.2, f
THEOREM 2-6.
vlve(f)
E (K(X)), eXe is a maximal subgroup of X
There exists x
Let u
vu(f) (by hypothesis)
vlvxe (f)
vlv. (f)
follows from the right topological property of X.
the smallest ideal K(X) and for each e
with identity e [5].
X (2.2).
Therefore, v 1(Tv2xef)
(by 2-2)
It suffices to prove that f
Sufficiency.
X.
E(X)
Let e
for every u
Tuf
Now, v(f)
vlv.(f)
v(2v=f)
v(vxf
Tuxef
by (2.2).
2v2i
which implies that vlv2xe (f)
vv2exe(f) (by 2"1)
QD(S)QM(S).
X such that
exists x
vlv
QMD(S)
Assume f
Necessity.
By Proposition i, there
w(Lf)
z(Lsf)
MM QMD(S).
Tf(s).
Thus
Tf
So
We complete this
Let z
xlQMD(S ).
I.e., D(x)
There exists w
ve(1).
MM QMD(S)
Then
is a
X such that D(w)
Tzf.
is
Let v
Z,
z.
u
X,
500
R. D. PANDIAN
E(x).
and e
vu(Twf)
Then
vu(Tzf ).
(since f
veu(Tzf
Tzf
Thus
QD(S))
Now ve(Tf)
QD(S).
QD(S))
vuw(f) (since f
vw(f)
vew(f)
ve(Twf
Therefore, by Proposition 2"4,
v(Tf).
THEOREM 2"7.
veuw(f)
veu(Twf
QMD(S).
Tzf
QMD(S)-compactification (Y,) of S maximal with respect to
the property that y2 is left simple.
Let (X,a) denote this LMC(S)-compactification of S and let e’X
PROOF.
denote the restriction map.
e
Then
is a continuous homomorphism of X onto Y.
first prove that y2 is left simple.
x
X
,
E(X) such that O(x)
d
E(2).
y2 and e
Let u
u and 0(d)
ue
QMD(S), ue(f)
u.
simple.
O(x)0(d)(f)
e(xd)(f)
Since Y is right topological, ue
There exists
xd(f)
x(f)
For
F.
u for all u
Therefore
u(f).
is left
Thus
To prove that (Y,) maximal with respect to this property, we let (Yo,o)
is left
be a right topological compactification of S with the property that
simple.
o
C(Yo)
=
og
g ((=(s)).
X
v
for some g
Since
=
X and e
X
((u)(v)) (since
Further ue(f)
Y
is dense in
X, x(f)
Then uev(f)
is left simple)
g (6(ue))
Yo
=
g ((u)(e))
QMD(S)
[i, III Theorem 2.4].
[2, page 385].
Therefore, there
such that
o.
o
6o
=(s)(o’g
Now =(s)(f)
C(Yo).
E(X).
C(Y o)
LMC(S)
exists a continuous homomorphism 6
u
o
Then it remains to show that
Junghenn has shown that
Then f
We
(We note that @-i (e) is a
e.
compact subsemigroup of X and hence has an independent d.)
f
Y
g((x)) for every x
g ((uev))
g((uv))
u(f).
C(Yo).
g (o(S))
Let
X.
g ((u)6(e)(v))
uv(f).
g ((u))
o"
Let f
g(s)
g
Therefore f
QD(S).
Thus f
QMD(S) and
the proof is complete.
Remark 2-8.
Theorem 2"7 raises a question.
Is there an admissible
subalgebra F of C(S) such that the Fnocompactification (Y,) of S is maximal with
respect to the property that
yn
is left simple?
for generalized distal functions
[3].
We proved an analogous result
We conjecture that the notion of minimal
distal functions could be generalized and that the nth order minimal distal
function space MDn(S)-compactification (Y,) is maximal with respect to the
property that
Yn+
is left simple.
REFERENCES
i.
BERGLUND, J., JUNGHENN, H. and MILNES, P. Compact Right Topological
Semigroups and Generalizations of Almost Periodicity, Lecture Notes in
Mathematics 663, Springer-Verlag, New York, 1978.
2.
JUNGHENN, H.D.
3.
PANDIAN, R.D.
Distal Compactifications of Semigroups, Trans. Amer.
Math. Soc. 274 (1982) 379-397.
Semigroup Compactifications by Generalized Distal
Functions and a Fixed Point Theorem, Internat. J. Math. & Math Sci. Vol 14
No. 2 (1991) 253-260.
4.
PANDIAN, R.D.
Dynamical Properties of Distal Function Spaces and Fixed
Point Theorems, Semigroup Forum 31 (1985) 207-218.
5.
RUPPERT, W. Rechtstopologische Halbgruppen, J
(1973) 123-133.
Reine. Angew. Math. 261