253
Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 253-260
SEMIGROUP COMPACTIFICATIONS BY GENERALIZED DISTAL FUNCTIONS
AND A FIXED POINT THEOREM
R.D. PANDIAN
Department of Mathematics
North Central College
Napervi[le, IL 60566
(Received August 15, 1989)
compactification" which
The notion of "Semigroup
a generalization of
is in a sense,
the classical Bohr (almost periodic) compactlflcation of the usual additive reals R,
been studied by J.F.
has
[2].
al.
Berglund et.
theory of
Their approach to the
semigroup compactification is based on the Gelfand-Naimark theory of commutative C-
algebras,
C*-a igebras, are the semigroup
of
admissible
spectra
H.D. Junghenn’s extensive study of distal functions is from the
where
the
compactifications.
point of view of semigroup compactiflcations [5].
In this paper, extending Junghenn’s
work, we generalize the notion of distal flows and distal functions on an arbitrary
semitopological semigroup S, and show that these function spaces are admissible
C*-
We then characterize their spectra (semigroup compactifications)
In our
subalgebras of C(S).
in terms of the universal mapping properties these compactlficatlons enjoy.
Also,
work, as it is in Junghenn’s, the Ellis semlgroup plays an important role.
relating the existence of left Invariant means on these algebras to the existence of
fixed points of certain affine flows, we prove the related fixed point theorem.
l.
PRELIMINARIES.
Let S be a semitopologlcal semigroup (binary operation separately continuous)
with
a
valued
Hausdorff
continuous
functions
For s
LS
S, define
f(ts) (f e C(S) and t
invariant if
LsF
and C(S)
topology,
c__ F
and
(all
S
Rs
A
S).
(RsF
on
denote the
topologies
C(S)
on
C*-algebra
subspace
F
are
assumed
C(S)
is
left
be
to
f(st)
byLsf(t)
of
all bounded complex
of
Hausdorff).
Rsf(t)
and
(right)
translation
c_ F). It is translation invarlant if it is both left and
A C*-subalgebra F of C(S) is called admissible if it is
right translation invarlant.
translation
i.e.,
of
F
Txf(.)
(the
case, T :F
x
topological
topological
invariant,
x(L(.)f)
space
of
contains
constant functlons,
the
is a member of F whenever f
all
nonzero
continuous
and is
left-m-introverted,
F and x belongs to the spectrum
homorphisms
on
F).
In
F is called the left-m-introversion operator determined by x.
compactificatlon
semigroup
of
(i. e.,
S
is
X
a
is
pair (X,), where
a
compact
X
is
a
semigroup
this
A right
compact
right
with
the
mapping x/xy:X/X continuous for all y
X), and :S/X is a continuous homorphlsm with
If, in
dense image such that for each s e S, the mapping x/(s)x:X+X is continuous.
F where F is an admissible subalgebra of C(S) and
:C(X)/C(S) is
A right
the dual mapping f/fo, then (X,) is called an F-compactiflcation of S.
topological compactlfication X,) of S is said to be maximal with respect to a
addition, a C(X)
R.D. PANDIAN
254
property P if K,) has
S
of
compact if [cat ion
}omo,orphi,n :X+Y such
referred
with
maxima[
result
to
will
.
a
be
with
used
the
to
The
property
frequently
then
specific
[2,
a
is
t.opologica[
a
cont [,uous
by
i.
F-compactficatio,s
are
the
of
[[[
reference
right
exists
there
of X,).
that a CtX) c_ F
without
(Y,3)
factorizatlon
property
mapping
the
P,
property
that o8=.
universal
respet:t
whe,lever
pcoperty P, a,d
the
to
mapping
Theorem
it.
2.4].
For
a
This
fixed
admissible subalgebra F of CtS), all F-compact[flcations of S are algebraically and
topologically isomorphic, and hence, we speak of the F-compactlflcat[on of S. If F is
a norm closed, conjugate closed subspace of C(S) containing constants, then a
(dual of F)is called a rean on F if
multipli.cation (pointwise), a mean
III.
(I)=
F
If F is further closed under
on F is called multlpllcative if (fg)
We denote the set of all means [multlplicatlve means] on F by
(f)(g), f, g e F.
Wi+h
M(F) [MM(F)].
w-topology, MM(F) is compact and it is the w-closure of e(S),
We note that (MM(F), e) is an Ff(s)}.
where e is the evaluation map {e(s)(f)
We
compactification of S, and we call ths the canonical F-compactlflcatlon of S.
{f e C(S):s/(Lsf) is continuous for
wi[[ need the admissible subalgebra LMC(S)
a[[’
MMC(S)} in the sequel.
We note that the LMC(S)-compactlflcation is maximal
with respect to the property that it is a right topological compactlficatlon of S [2,
III Theorem 4.5].
A flow is a triple (S,X,), where S is a semitopologlcal semlgroup, X is a
X
is a continuous homomorphism such that t(s):X+X
oten write (S,X) for (S, X, ) and sx
4e
for each s e S.
continuous
is
X is a
X
for (s)x.
compact right topological semigroup (with respect to the product
We denote the Ellis
and
function
composition) of all self maps of K.
topology
X
a compact right
then
X).
E(S, X) is
by E(S,
semigroup, the clsoure of (S)in X
If X is a convex subset of a real or a complex vector space,
topological semigroup.
compact topological space, and :SX
(S, X) is ca[led an affine flow. A point
x in X is called a fixed point of the flow (S, X) if s--X for each s in S. If Y is a
closed Invariant subspace of X, then (S, Y) is a flow under the restricted action. A
and ,(s):X+X is afflne for each s in S, then
,
lira slY for
X such that lira six
of
y. Let f e LMC(S) and Z be the closure
some net (s i) in S, then x
RSf in the
Z
Then Z is
topology of polntwise convergence on C(S). Define :SZ by (s)
f is called a
pointwlse compact [6], and (S, Z, )is easily seen to be a flow.
flow (S, X, ) is called distal if, whenever
y
Rs Z’I
distal function if the flow
(S, Z, ) is distal.
H. D. Junghenn has shown that D(S),
is an admissible subalgebra of C(S) and that a
E(X), the
uv(f) for u, v in X and e
LMC(S) is distal iff uev(f)
he has
S.
of
Also,
LMC(S)-compactlfication
idempotents of X, where (X,a) is the
proved that the D(S)-compactification (Y,B)is maximal with respect to the property
that xey=xy for all x, y in Y, e e E(Y) [5, Theorem 3.4].
the
set
of all distal functions,
function
2.
GENERALIZED DISTAL FUNCTIONS.
Let (S, X, ) be a flow and E(S, X), the Ellis semlgroup. Define
n
n
n
E(S, X) are both
Then E(S, X) and
X)}.
E(S, X)
{glg2
n
E(S, X) is nonempty as compact
compact right topological semlgroups. We note that
gnlgiE(S,
right topological semlgroups have Idempotent elements [4].
SEMIGROUP COMPACTIFICATIONS BY GENERALIZED DISTAL FUNCTIONS
,
DEFINITION I.
A flow (S, X, 7) is n-distal (-distal) if for
y
X, whenever
)n
$(y) for some F. e ES, X), then ()
(y) for every r. e E(S x
$(x)
(
l
E(S,
x)n).
A function f
DEFINITION 2.
LMC(S) is said to be n-distal (-distal), if the
flow (S, Z, ), where Z is the closure of
on C(S), and (s)
RslZ, Dn(s}
PROPOSITION
3.
A
x, y e X such
that
llm
sn--
{sis 2
every s e
PROOF.
iDa(S)].
that sx
six
s
n
(Sk)
llm
n.
j
lira
ij
Let
n.
.n
E(S, X)
net
(si)
llmln
(y).
o
i 2
o
S 2, then
Dn(s)
SlnX)
II
(y).
lm
lm (st
SlnY
e (S, x)n,nand (x)
llm
i(x) lira
i
i)
only
if,
then
sx
2
If
whenever
for
sy
sly.
Then,
by
it follows
sy for
On(X)
io2 o
for each x e X.
n
n,
then
lira
E(S, X)
i’
(y). This completes the proof.
DI(s) D2(S)
Din(S).
Trivially all distal
n
llm
where
n
e
i(y)
n,
Then by hypothesis, sx
o
-(SilSi2
n(X)
1o2 o
an identity, then D(S)
S,
in
Since (S n) c_ E(S, X)
Let x, y e X and
By induction one can easily show that
We note that if S
and
if
n.
Sufficlency.
Then,
lim l2m
EXAMPLES.
n-dlstal
some
Let x, y e X and (s i)
S such that llm
six
[llm w(s i)](x)
we have
[llm (s i](y).
e E(S, X) such that (x)
(sij).
(x)
for
sly
(y) for every
e E(S, X)
DI(s) D2(S)
S}.
s
sy for every s e S
s e S
llm
necessary,
if
(x)
hypothesis,
We denote the set of all n-distal
c
c_
c_
c_DS).
D(S)_
Clearly,
flow (S, X, n) is
Necessity.
subnet
Taking
in the topology of potntwise convergence
Rsf
is n-distal (-distal).
(-distal) functions by
Thus
255
Dn(S)
functions
D(S),
are n- and
and that if S has
distal
functions.
(ll)
Let S be the semlgroup of all strictly upper triangular matrices (elements on
the diagonal and below are zero) of order n+2 with entries from reals. With discrete
topology, it is a topological semlgroup and
g(s)
(cl,n+2vO)
(ill)
Let
h 2, s
(N, +)
be
(cl,
the
Dn(s)
LMC(S)
j) e S, one verifies that g e
semigroup
of
C(S).
Defining g:S/R by
Dn(S)
and g e
Dn-I (S).
positive integers with discrete topology.
Define f (t)
I/t if t < n+l and
0 if t > n+l. Again it is easily verified that
n
f e Dn(N) and f e
we give an example f e D(S) but
Later
Dn(s) for any
n
n
Dn-I(N).
n.
Using
the
structure theory of compact right topological semlgroups, one may
readily prove the following result of R. Ellis: (S, X) is distal if and only if E(S,
X) is a group with respect to function composition and with identity, the identity
function
[4, Proposition 5.3].
We have a more general
result
corresponding
to
generalized distal flows.
PROPOSITION
4.
A
flow
X)
is
(S,
n-dlstal (-distal) if
and
only
n
if E(S, X)
(0 E(S, X) n) is left simple.
PROOF. We first prove the n-case. Necessity. Let Z
It suffices to
E(S, X)
prove that pe
p for all p e Z and e e E(Z).
Let x e X and e e E(Z).
Then e is
n.
256
R.D. PANDIAN
E(S, X), and e()= e(e()).
Therefore by delnlt[on of nfor all p
Z, and hence, p
Sufficiency.
Let
pe.
X such that p(x)
pn(y) where
p(y) for some p
Then pn(x)
x, y
E(S, X).
n
As Z is left simple, Z
E E(S, X)
For any q
zp
Z, q
rp where
n
n
(rpn)(x) r(p (x)) r(p (y)) rpn(y) q(y).
r g Z, and q(x)
Hence, the flow
also an dempotent
p(x)
distal,
n.
Pn
is
of
p(e(x))
n.
The proof
n-distal.
of the -case is similar.
supply the sufficiency part.
p(y).
p(x)
Sufficiency.
pn(y)
pn(x)
Then,
We omit the necessity part and
Let x, y
X and p E EiS, X) such that
readily verified that
Zqo.
simple, Z
q0
Let
g
n
fiE(S, X) and
E
Z.
Then
(qn),
q0(x)
lq0
pn
As
for every n.
n
there exists a subsequence of (p), call it
g
such that
q0(y).
for some
ES, X), a compact space,
qn/q0 in E(S, X). It is
N E(S, X) n is left
Since Z
in Z.
I
Now,
(x)
l(qo(X)) l(q0(y)) lq0(y) (y), and this completes the proof.
Let S be a semitopological semigroup, (X, a)the canonical LMC(S)LEMMA 5.
compactlflcatlon of S, and f
LMC(S).
i)
The following statements are equivalent.
a)
f E
Dn(s).
b)
T
f
c)
uev(f
uev
T
uv
f for all. u
n
X
v E
xn+l
uv(f) for all u
X, and
e E
v e X and e
E(X)
E(X).
The following statements are equivalent.
a)
f e
D(S)
b)
T
f
ue v
T
c) uef(f)
PROOF.
{Tx
f: x E
>
b)
i.e., k(ue)
>
u
X
2
n
c)
n
v e
N
X, and e e E(X).
v E X, and e E E(X).
xn),
X n,
n.
k(x)(Tyf)
v e
n,
v e X, and e
E(X).
As E(S, Z) n is left simple (hypothesis), k(u)k(e)
In particular,
k(ue)(Tvf)
k(u)(Tvf)
where
Tv f
is
k(u),
e Z, i.e.,
topology, it follows that
Tuevf
Tuevf Tuvf
X, and e e E(X).
X
uev(f)
.
E(S, Z) satisfying koa
and k(e)
Then, k(u)
E(S, Z)
homomorphism of X onto
is right topological with w
Let u
and
E
E(S, Z)
k(u).
X
ontinuous
Let u
Sinc___e X
for all u
b)
X.
n
X, let Tx be the left-m-introversion operator determlned by x.
the closure of Rsf in the topology of pointwise convergence on C(S)
X} [2,
Lemma
4.19].
Defining k: X/E(S, Z) by
Txyf, one
an idempotent of
Tuvf.
O X
uv(f) for all u
verifies that k is a
i) a)
f for ai[ u
For x
Z
Then,
uv
n+l,
v e
X, and e e E(X).
UlU2ev(f)
u
Then, u
l(Tuvf) l(Tuf)
u
X,
UlU2, where
u
UlU2V(f)
uv(f).
xn+
Thus,
uev(f)
uv(f). It is easily verified that uev(f)
uv(f) for u
n
c) ===> a)
Let p
There
E(S, Z) and let d be an idempotent of E(S, Z)
n
exists u e X
e e E(X)such that k(u)
p, and k(e)
d.
Such a choice of e is
-I
possible as k (d) is a compact subsemigroup of X. Let v e X.
For any w
X,
w(T Beyf)
wuev(f)= wuv(f) (hypothesis)
Therefore,
W(Tuvf).
Tuevf Tuvf. Now
k(ue) (Tvf)
which implies that k(ue)
k(u) (Tv f)’
k(u).
Thus,
k(ue) --k(u)
As E(S, Z) n is right topological, it
k(ue)
pd
k(u)
p pd
p.
Tuevf
follows that pd
n.
Tuvf
p for all
n
E(S, Z) proving that E(S, Z) n is left simple. Consequently, the flow
(S, Z, ) is n-distal, and thus, f g Dn(s). ii) The proofs of a) ==> b) and b)
p
E
c) in i) are easily modified to prove the corresponding results in il).
==>
Let us prove
257
SEMIGROUP COMPACTIFICATIONS BY GENERALIZED DISTAL FUNCTIONS
==>
the case c)
n
It suffices to show that f E(S, Z)
a).
n
Let
is left simple.
E(S, Z)
n.
There
exists
fiE(S, Z)
n
k(u)
that
such
in
fiX
u
an
of
d. We prove the existence
an e e E(X) such that k(e)
n
E(S Z) n
lim Pl for some (pi)
Then, p
Let n be fixed and p e E(S, Z)
p.
n
n
has a
Now (xi) (c_ X
For each Pi’ there exists x E X such that k(x i)
Pi"
k(x
j)_whlch
j)
in
X
converging to an element, call t x n,
convergent subnet (x
pj
xn),
We now have a sequence (Xn)C X, (x n
k(x ).
converges to k(x n) and hence p
p
d
let
and
be
of
[dempotent
an
fl
n.
One readily verifies
u in X.
having a convergent subsequence (x’ n) such that x’ n
n
is smlar to the
whlch
p. We omit the rest of the proof
that u g0 X and that k(u)
===>
proof of c)
a) in i).
a)
THEOREM 6.
Dn(s) -(D(S)
Dn(s)
b) The
and D(S) are admissible subalgebras of C(S).
-) compactificatlon (Y,B)of S is maximal with respect to the property
that
uv for
uev
yn+l
u e
(u e Y.
flyn),
v
Y, and
e e E(Y)
a) Let (X,a) denote the canonical LMC-compactificaion of S. That Dn(s) is
It is easily verified that
a linear subspace of C(S) is immediate from Lemma 5 (i).
Then,
S.
E(X), and s
v
X, e
Dn(s) is norm closed Let f Dn(s), u
PROOF.
xn+1
uev(L f)
(s)uev(f)
s
invariant.
In a
uv(L f).
(s)uv(f)
similar
s
one
manner,
Dn(s)
Hence,
verifies
that
left
translation
right
translation
is
Dn(s)
is
The fact that X is the set of all multiplicatlve means proves that
invariant.
Dn(s)
uv(1), Dn(S) contains all the constant functions Let
There
Let 8:X MM(Dn(S)) be the restriction map.
and f
w
Dn (S).
uevw(f)
T f and uev(Tw,f)
T f
w’
exists a w in X such that 8(w)
uev(Twf)
W
W
As uev(1)
is an algebra
MM(Dn(S))
uv(Twf --uv(Tw,f).
uvw(f)
Thus, Dn(s)
introverted.
Thus,
Tw,f
Dn(s)which
proves that
Dn(s)
is
left m
algebra of C(S). The proof that D (S) is
We give the proof for the -case, and omit the
is an admissible
b)
Let (X,a) denote the canonical LMC-compactiflcatlon of S.
The e is a continuous homomorphism of X
Let 8:X/Y denote the restriction mapping.
onto Y such that eoa
B. First, we prove that Y has the property (I). Let
There exist
v
u e Y.O
Y, and e e E(Y).___ u
UlU 2 where u e Y and u 2
n
O(y)
v,
u
2),
(x
I,
(i
that
such
X
x
e
d
and
y
eft
E(X),
X,
i) i
Xl,
2
and 8(d)= e.
xdy(f)
xy(f)
(xdy)(f)
Therefore, for any f e Dm(S), uev(f)
an admissible algebra is similar,
proof
for
the
n-case.
yn.
yn,
[Lemma 5 ii)] --e(xy)(f) --uv(f).
Hence, uev
uv, and thus, Y has the property
(I).
To prove that (Y, B) is maximal with respect to this property, it remains to
show
that
B0
C(Y0)c_ D(S)
for
any
right
topological
compactificatlon
(Yo’ BO)of
S
C(S) is the adJoint of B
It is shown
(I), where Bo:C(Y O)
O.
c_ LMC(S) [5, page 385].
Therefore, there exists a continuous
homomorphism :X
6oa.
such
that B
f.
Let g e C(Y 0) and Bog
0
a(s) (Bog)
Now, a(s) (f)
By
taking
BOg(s) g(B0(s)) g((a(s))).__
Let u e X. O X
g((x)) for x e X.
limits, x(f)
v
X, and e e E(X).
n
Clearly, d(u)
d(v) e
and d(e) is an idempotent of
Y0" Then ue(f)
g(d(uev))
g(d(u)d(e)d(v))
has
g(d(u)(v)) (since
(I))
property
g(d(uv))
uv(f).
Thus f e D ( S) and this completes the proof
having
that
property
B0 C(Yo)
Y0
n,
Y0" OY0
YO
YO
258
R.D. PANDIAN
3.
INVERSE I.[M[TS AND
Dns)-COMPACTIF[CAT[ONS.
In this section, we prove tlat
(X,)
its compact[f [cation
Xn,an)
DnS)-compactification
the
is
Dn(s)
U
is an admissible subalgebra of CS), and
is the inverse limit space of the spectrum
S and
of
{Xn,nm
(n;m) is the
nm:Xn/Xm
where
restr[ctlon
[3].
Fo definition and terminologies in inverse [imlts, we shall follow Dugundji
Let 1 be a preordered set and
be a family of topologcal spaces.
Fo
;
map.
)
{K}$eI
,
q ,q
n,
)
spectrum over I.
p:
,. :XX: X
assume there [s given
X
X
o
Then
The subspace {x e
the
is
ma,
projection
a continuous map such that whenever
{x :} is called an
==> P[x"
H o P(x)}, where
the
family
<
inverse
called the inverse limit spectrum of the
is
spectrum and is denoted by X
THEOREM 7.
Let X be a compact topological space, and
g I, indexed by a
directed set I, be a family of topological spaces. Assume there are given
X
for every
)
and for
surJective, continuous, and consistent maps
)
(i.e., for
such that for any two distinct points x|, x e X,
[X}
,o
there exists $ e
ll, mit
)
F.(xl) # .,$(x2).
{X: }.
space of the spectrum
.
The
{X:w}
Therefore
hypotheses
{x e
Xas
Define 8:X
8(x)=
homeomorphism.
maps)
If
=op(e(x)).
FtXr:
==>
<
4
p(O(x))
then
Therefore, z
If Xl,
e I such that
-I
g
(p(y)).
Thus
(2)
)
we
see
t
(I
yt) c_
is
continuous
for
each
rl
0
T
x }.
(by
and
of
,
that,
if
t,
tl, t2,
in,
are
n), then
i
(
(Y)
{yt}
consistency
x 2 are two d[stlnct points of X,
(xl) $(x2). This implies that
we prove that
X. For
and
(z) p(y)
o p(y) p(y) (since y
maps)
of
arbitrary members of I such that t
t
where p (x)
w(x)= wow(x)
x.
Hence, e(x) e
(P)) c-l(p(y)).
(z) o (z) (consistency
w
Eop(x)}
8(x 2) and hence, 8 is inJectlve. Let y e
Let z e
(p(y)). Then,
8(i)
F,q
Set
(x
x
> q,
)
We complete the proof by showing that 8 is a
(,i(x))igl.
,
for
==> p(x)}
X:
then by hypothesis, there exists
As
that
imply
is an inverse spectrum over I.
{x e
X
2
Then X is homeomorphlc to the inverse
such that
PROOF
:
, : X/ X
0
-I
{wt (Yt):t
e I} is a class of
closed sets in X, a compact space, with every finite intersection being nonempty (by
(2)).
hence
e
Therefore, t sfl I
is
surjective.
t
l(Yt)
#
Clearly 8 is
#"
is
closed,
For
any x e
continuous
flteIt-l(yt )’
Since
X
is
O(x)
y
compact, 8 is
and
a
homeomo rphism.
THEOREM 8.
U
space of the
((p)
Let
of__C(S)
subalgebras
S. Then F
F
[F}e I
indexed by a directed set
such that
uIF
() and
(X, e)
is the
is admissible and the F-compactification
s%ectrum {X:w}
).
Frl F(
I, be a family of admissible
where
n:X X
(
F-compactificatlon
(X,e)
of
is the inverse limit
) is the restriction map
259
SEMIGROUP COMPACTIFICATIONS BY GENERALIZED DISTAL FUNCTIONS
PROOF.
o
that
n
(x)
any two distinct points
n tx2).
*
an
e F such
that
there
#
(x 2)
x
x[, x2e
,(x2 )"
F
that
and
such that
such that
e X there exists n e
2
an f e F such that
I and g e
g
Xl,
that
fact
x[)
that
eKists
there exist
(x I)
The
Define
Then, in view of theorem 7, it suffices to prove
as the restriction map.
X
:X
it is easily seen that F is admissible.
is directed,
Since
X and x
fact
The
*
x
2
implies
there
that x I, x 2 e X and x
By continuity of
xl(f) x2(f).
](Xl)(g) xl g)
x
x|
exists
2 impl[es
and x2,
n (x2)(g)"
theorem
x2 (g)
Hence,
is an immediate
The following
completes the proof.
that
corollary to theorem 8.
THEOREM 9.
(X,)
is
O
the
Dn(s)
is an admissible subalgebra of C(S) and [ts compactlflcatlon
inverse
the
of
space
limit
spectrum
{Kn:nm}
where
(Xn, en)
is
Dn(s)-compactification of S and nm
Dn(s) c_D=(S). Now, we give an example of a function f e D(S) but
Defining f(t) as f(t)
D N)for any n.
Let fn be defined as in Example ill.
C_D(N).
Dn(N)
Clearly,
U
f
e
(norm). Thus,
I/t, t e N, we see that n
We remark that at this point we do not know whether the containment in
f .D N).
O Dn(s) c_D(S) is proper.
:X
the
(n a m) is the restriction map.
m
X
It is
clear that
4.
FIXED POINT THEOREM.
(right) translation invariant
on F is called left (right)
subspace of C(S) containing constants.
invariant if for each f e F, s e S,
(Lsf) (f) [(Rsf) (f)]. A left (right)
translation invariant subspace F of C(S) is said to be left (right) amenable if there
Let F be a norm closed,
conjugate
closed,
[eft
Then a mean
(right) invariant mean on F, and amenable if F is translation invariant and
both left and right amenable. L. N. Argabright [I] has proved that F is left amenable
if and only if every affine flow (S, X, ) such that {x e X:U x A(X)C_F} # # has a fixed
point, where A(X) denotes the Banach space of all continuous complex valued afflne
is a left
functions on X, and U :C(X)
S, h e C(X), and
h(sx), s
C(S) is defined as U h(s)
We make use of this result to prove the following fixed point theorem.
x e X.
M(F)
us prepare a lemma for proving the theorem. Defining : S
M(F)
as
Let
,
(s)(x)
L s x, where L s denotes the adjolnt of L :F
F, one verifies that, relative
s
,
to the action (s, x)
If in addition, F is an
L x, (S, M(F), w) is afflne flow.
s
algebra, then MM(F) is a closed invariant subspace of M(F), and relative to the
restricted action, (S, MM(F), ) is a flow.
These actions of S are called the natural
actions of S on M(F) (MM(F)).
let (Z
MM(Dn(S), B) denote the canonical Dn(s)
compactificatlon of S. Then relative to the natural action, (S, Z, ) is a flow.
LEMMA 10. The flow (S, Z, 7) is (n+1)-distal.
PROOF.
Let z z e Z and (si) c_ S such that lim siz
lim siz 2.
I.e.,
2
lira
8(si)z
llm 8
(si)z 2.
Taking subnet if necessary,
ZoZ 2
z0z
where
z
e Z.
llm 8 (s
Hence yZoZl yz0z for every y
Z, from which it follows
0
2
that zz
zz for each z e Zz
a
left
ideal in Z, a compact right
Now,
being
Zz0,
2
0.
ez 2. For
topological semigroup, has an idempotent element e. Thus ez
i)
s e
sn+l
(S, Z, )
sz
8(s)z
8(s)ez
8(s) ez
(Theorem 6)
(n+l)-distal (Prop. 3).
THEOREM II.
(Fixed Point Theorem)
2
sz
2.
Therefore,
is
affflne flow
(S, Y, ) containng
(n+1)-dlstal has a fixed point.
Dn(S)
is
left amenable if and only if every
a closed invariant subspace
Z such that (S, Z, ) is
260
R.D. PANDIAN
PROOF.
Dn(S)
Let
left
Suppose that iS, Y, ) [s an affine flow
containing a closed invariant subspace Z such that iS, Z, ) ts (n+l)-distal.. Ue show
that {x
Y:U A(Y) r Dn(S)}
0. Let x Z and h e A(Y). It is easily seen that
x
RsUxh Ush. Let tU
subnet (s]) converging
ssjx
that
sx
0
be
h
x )be a net in U
to some point
every s e S
for
Usjxh Ux
tpology,
h (po[ntwise).
the pointwIse
amenable.
Itence,
0
S
in
The net
h.
o
(six)
As ts):
h(sxo)
h(ssj)
This proves that
and thus, U h e LMCtS).
for
in Z has a convergent
ts continuous,
It
eery s e S.
Ush RsUxh
follows
Is relatively compact
in
Let (X,) denote the canonical lC-
compactiftcatton of S.
As (E(S, Z),)[s a right topological compactiflcation of S,
by the universal apptng property of (X,a), there exists O:X
EtS, Z), a continuous
,
.
o a
Then a(s)(U h)
h(sx)
h((s)(x))
x
Taking limits (a has dense range in X, and h,
are continuous),
we
get u(U h)
h((u)(x)) for each u X.
Let u
v e X, and e
E(X)
n+l
Then (u)
E(S, Z)
(v) E(S Z) and (e) is an Idempotent in E(S
Z)
As E(S, Z) n+l is left simple O(u)O(e)
Hence,
uev(U
h)
(u).
h(O(uev)(x))
x
h((u)(e)(v)(x)) h((u)(v)(x))
As
h((uv)(x)) uv(U x h).
X
is
right
top, ologtcal it follows that uev(U h) uv(U h) for any u e Xn+l
Thus,
U h
Dn(s). This proves the necessary part For Sufficiency, let Y M(Dn(S)) and
homomorphism, such that
h{(O
o
a)(s)(x)}.
xn+l
,
x
define (s):Y
Y as (s)(x)
L x, s
S,
s
Let (Z, B) denote the canonical
flow.
flo (S, Z, )is
point
Yo(f)
Y0
Y.
x
Dn(s)
Then (S, Y, ) is
compactification
(n+l)-distal (Lemma 10).
such that
Y0
LsYo(f)
Yo(Ls
a left invariant mean on
LsY
(=
O)
f) for every s
sY0
So by hypothesis, (S,
for every s e So
Hence
S and for every f
Dn(s).
,
of
S.
an
affine
Then
the
)has a fixed
Therefore,
Dn(s).
YO
is
REFERENCES
I.
ARGABRIGHT, L.
2.
BERGLUND, J., JUNGHENN, H. and MILNES, P.
Invariant Means and Fixed Points, A sequel to Mitchell’s paper,
Trans. Amer. Math. Soc. 130 (1968), 127-130.
Compact Right Topological Semlgroups
and Generalizations of Almost Periodicity, Lecture Notes in Mathematis 663-,
Sprlnger-Verlag, New York,
’|9.
3.
DUGUNDJI, J.
4.
ELLIS, R.
5.
Distal Compactlflcations of Semlgroups, Trans. Amer. Math. Soc.
JUNGHENN, H.
274 (1982), 379-397.
6.
MILNES, P. Compactlflcatlons of Semitopologlcal Semlgroups, Austral. Math. Soc.
15 (1973), 488-503.
_Topology, Prnetice-Hall Internatlonal, Inc., London, 1966.
Lectures on Topological Dynamics, Benjamin, New York, 1969.