297
Internat. J. h. & Math. Si.
Vol. 6 No. 2 (1983) 297-306
AN AMENABILITY PROPERTY OF ALGEBRAS OF
FUNCTIONS ON SEMIDIRECT PRODUCTS OF SEMIGROUPS
BAO-TING LERNER
Department of Mathematics
U. S. Naval Academy
Annapolis, MD 21402
(Received March 2, 1982 and in revised form June 28, 1982)
ABSTRACT.
Let S
I
and S
2
be semitopological semigroups,
SI S 2 a semldlrect product.
An amenability property is established for algebras of functions on
S
S
I
2.
This
result is used to decompose the kernel of the weakly almost periodic compactlflcatlon
of
SIS 2
into a semidirect product.
KEY WORDS AND PHRASES. Semitopological s emigroup, s emidect product,
compatification, amenay, strongly almost periodic, weary ost
piodic, krnel.
AMS (MOS) subject clasificio (1970).
Secondary 460
i.
INTRODUCTION.
Primary 22A15, 22A20, 43A01;
Let S I, S be semitopological semigroups (in the terminology of
2
Berglund and Hofmann (i)) with identities, each denoted by i.
That is S
have (Hausdorff) topologies relative to which multiplication in S
I
and S
I and S 2
2
is
separately continuous.
s I, t
Let
T
S
I
S I,
s2,t 2
2
x S
S
I
S 2,
T(s2,(t2,tl)), T(s2,1)
We shall assume the map
t
I
S
I.
The
I be a separately
T(s2,slt I)
continuous map satisfying for each
(s2,sl)(s2,tl), (s2t2,t I)
i, and T(I,’) is the identity map.
(Sl,S2)
Sect pOdct
SlT(S2,tl)
S
1
SIS 2 of S I and
x S
S
2
2
S
I
is continuous for each
is the topological space
29 8
S
B.-T. LERNER
S
I x 2 equipped with multiplication
The above conditions on
(Sl,S2)(tl,t2)
SIS 2
imply that
(s
I
(s2,tl) s2t2).
is a semitopological semigroup with
identity (I,i).
Let F be a closed translation invariant sub-C*-algebra of C(S
2 below) containing the constant functions.
I
S2)
(see
In previous papers (2) and (3),
the author has formulated the necessary and sufficient conditions for the decomposition of the F-compactification of (S
I $2)
into a semidirect product.
The
decomposition may be written symbolically as
(S I$2
where G
{f(.,l)
f
F} and H
)F SI G $2 H
{f(l,’): f
0 being another semidirect
isomorphism
(i.i)
F} and equality denotes canonical
product).
Applications of this decomposition were then made to the almost periodic (AP),
strongly almost periodic (SAP) and left-uniformly continuous (LUC) cases.
situation is less well-behaved in the weakly almost periodic (WAP) case.
example, if S
2
For
is any commutative topological semigroup with identity for
AP(SI),
WAP(SI)
which
S
I
The
SIS 2
then (i.i) fails even if
is taken to be the special
case of a direct product (Junghenn (4)).
However, in the present paper we shall prove an amenability property of
algebras of functions on S
I S 2 which generalizes
a result of Junghenn
(5) and
provides conditions under which the kernel of the WAP-compactification of S
S
I
2
can be decomposed into a semidirect product.
PRELIMINARIES.
2.
Throughout this section S denotes a semitopological semigroup
and C(S) the C*-algebra of bounded continuous complex-valued functions on S.
define operators R
R
f(s)
t
t
and
Ls
f(st)
We
on C(S) by
L
s
f(t)
(s,t
C(S))
f
S
Let F be a conjugate closed, norm closed linear subspace of C(S) containing
the constant function i.
R
S
F
F (resp.
L=
F=F)
Then F is
g
(resp.
et) Ya0n inva
fO iv if
if
it is both left and right
translation invariant.
A mn on F is a positive
that (i)
I
II I-
linear functional
in F
the dual of F, such
We denote by M(F) the set of all means on F.
A mean
299
FUNCTIONS ON SEMIDIRECT PRODUCTS OF SEMIGROUPS
mtiplicaive if (fg)
on F is
(f)(g), f, g
F.
We denote the set of
all multiplicative means on F by MM(F).
If F is left (resp. right) translation invariant, then a mean
right) invariat
F
for each f
if
left
(resp.
(f) (resp. (R s f)
we have (L f)
s
S
s
is
The set of all lft (resp. right) invariant means on F shall be denoted by
(f)).
ElM(F) (resp. RIM(F)).
RIM(F) # ).
F is
left
(resp. ,ight)enable if ElM(F) #
(resp.
If F is translation invariant and both left and right amenable,
F is called amenable.
Now suppose F is left translation invariant.
Tv
T
F
/
C(S) by
(Tvf)
F CF for each v
if T
(s)
M(F).
F -F for each
e
(Lsf),
S.
Then F is
If F is an algebra, then F is
MM(F).
defined in an analogous
F, s
f
F
For each v
define
left iroved
if
left-m-ioved
Right introversion and right-m-introversion are
manner.
F
algebra of C(S) then S denotes the spectrum (=space of
If F is a sub-C
nonzero continuous complex homomorphisms) of F equipped with the relativized weak
topology, and e: S
S
/
F
the evaluation mapping.
If F is admissible (i.e. F is translation invariant, left-m- introverted,
containing the constant functions) then a binary operation (x,y)
on S
(i)
F relative to which the
pair
S
y
(li)
e
F
(sF,e)
xy may be defined
has the following properties:
is a compact Hausdorff topological space and a semigroup such that for each
S
F
the mapping x
S
S
each s
(iii) e C(S
E
F
F
/
S
F
is continuous;
is a continuous homomorphism with range dense in S
S
F)
S
xy
the mapping x
e(s)x
S
F
+ S
F
F
such that for
is continuous; and
F.
The pair (S
F,
e) is the ea0/ F-eompaion
of S.
Let K(S), called the ke of S, denote the mnimal ideal of S.
We shall use
the amenability property in the next section to decompose the kernel of the WAP-
compactification of
3.
SIS 2 into a semidirect product.
THE AMENABILITY THEOREM.
identities and
SIS
Let S
I
and S
2
denote semltopologlcal semlgroups with
as defined in
2_ a semldirect product
I.
We shall denote by
B.-T. LERNER
300
SI SI$2
/
ql
(l,s2),
q2(s2)
dual mapping of
and
for s
S
s
I e I, 2 e
(ql(Sl)
the injection mappings
,
Let
$2).
C(SI@S2)
qi
C(SI)
(Sl,l),
denote the
1,2.
i
qi’
SI$2
$2
q2
THEOREM 3. i
Suppose F is a left translation invariant, left introverted closed subspace of
(a)
C(SIS 2)
D
{s
containing the constant functions, and the semigroup
e S
2
"r(s2,S 1)
2
S
1}
is dense in S
,
amenable if ql F and
2. Then F is left
q2 F are left amenable.
Suppose F is a right translation invariant, right introverted closed subspace
(b)
of C(S
and
,I
S 2)
Then F is right amenable if
containing the constant functions.
ql F
q2 F are right amenable.
To prove (a) choose any
PROOF.
,
l(ql
(Uf)(s 2)
F
Then U
TF=F
f) )’
(L(l,s2)
,
/
q2
s2
e S
e LIM(q
i
M(F) where
and for each f e F define
2.
Since F is left introverted,
For let f e F.
F
I*F ),
(L
(Tf)(s I s 2)
(Sl,S2)
f)
(s I
F
f
s2)e(S I@$2).
Observe that
(Uf) (s 2)
Ul(ql
(L(l,s2) f))
(T(I oql* f) (s2
,
q2
Then UF e
,
q2*F
since
T(U 1
and (i)
2
e
f e F.
T(lOql,)
i.
LIM(q2)
Thus
We must show
*
f (i
for any s
linear operator of norm I since
Let
ql
and put
l
’s2)
e S
2
2.
,
F
Furthermore, U
ql F.
is a mean on
2U.
,
e F
Then
+
(f)
q2 F
_>
is a positive
0 for each f
_>
0 in F,
is a mean on F.
e LIM(F).
,
ql (L (s I,I) L (l,s2) f)
Observe that for s
,
,
ql (L (l,s2)(s
ql
I
S
I e S I, s 2 e 2,
(3.1)
,l)f)
(L(T(s2,sl),
s
2)
f).
Furthermore, for any g e F, s I, t I, e S I,
,
ql
(L(sl,l)g)(tl) L(sl, l(tl’l)
(ql* g)(sltl)
Thus,
,
ql (L (Sl,l) L (l,s 2) f)
g(Sltl’l)
Ls l(ql
,
g)(tl) )"
Lsl(ql*L (l,s 2) f).
(3.2)
FUNCTIONS ON SEMIDIRECT PRODUCTS OF SEMIGROUPS
By (3.1) and (3.2) we obtain for d e D, s I e S I, f
l(ql*(e((d,s I),
d)
301
F,
l(ql *(e(s I,I) L(l,d)f))
f))
(3.3)
*
i (ql (n(l,d) f)
(Lslql*(L(l,d) f))
(Uf) (d).
By the definition of D and the continuity in the variable s
I
of the extreme left
side of (3.3), we obtain,
l(ql *(L (Sl,d) f))
S
Since
2
(Uf)(d) (d e D
s
SI)
I
we therefore have
Ul (ql* (L (Sl,S 2) f))
(Uf)(s 2) (s I e S I, s 2 e
S2).
That is,
UL
(s I ,i)
Uf, Ys
f
S
I e I.
(3.4)
Observe that for s 2, t e S 2,
2
U(L(l,s2) f)(t2)
I (q i
*(L (l,t L (l,s f))
2)
2)
i (ql
(L
s
2
(L(l,s2t2)f))
(Uf)(s2t2)
Uf)(t2).
Thus,
U(L(I,s2) f)
L
s
Uf, s
2
(3.5)
S
2 e 2
By (3.4) and (3.5) we obtain for any s I e S I, s 2
(e (s
l,s 2)
f)
(e
U2
e
(l,s 2) (s I ,i)
S
f)
[U(L(I,s2) e (s I ,l)f)]
u2[es 2 U(L (Sl ,l)f)]
2(Uf)
2(e s 2 Uf)
(f).
e LIM(F) and we are done.
Thus
The proof of (b) is done in an analogous manner and is, in fact, much easier.
RIM(qI*F)
and for each f e F, define
Choose any
Ul
(Uf) (s 2)
i (ql* (R(l, s2) f))
Then U
F
e
*
q2 F
s
2
S
2
since F is right introverted.
positive linear operator of norm 1 since
I
Furthermore,
is a mean on
ql *F
q2* F
U: F
Let
2
e
is a
RIM(q2*F)
B.-T. LERNER
302
and put
is a mean on F and we must show
Then
2-U.
Observe that for any s I, t
I
f(t
ql*R(l,t2)R(sl,s2
f[(t
i
S I,
I’ I)(i
t
s2,
t
2
$2,
2
f
RIM(F).
F,
(Sl’ s2)]
f[(tl(t2,Sl), t2s 2)]
f[(tl(t2,s l),l)(l,t2s2)]
ql*R(l, t2s2) f (tl (t2,Sl))
R
(t2,s I) ql*R (l,t2s 2) f(tl).
ql*R(l,t2) R (Sl,S2) f R (t2,Sl) ql*R(l,t 2 s2)f,
Thus,
and therefore since
U(R
(Sl,S 2)
f)(t 2)
RIM(qI*F),
f)
i (ql*R (l,t2) R( Sl,S2)
e
i
l(R(t2,sl)ql*R(l,t2s2) f)
l(ql *R (l,t2s 2) f)
Rs2 Uf(t2)"
Then,
(R(sl,s2)f) 2[U(R(sl,s2)f)] 2(Rs2Uf)
2(Uf)
Hence
4.
(f).
RIM(F) and we are done.
Application to K(S
S
I
2)WAP
Let S be a semitopological semigroup and let SAP(S) denote the closed linear
span in C(S) of the coefficients of all finite-dimensional continuous unitary
representation of S.
functions on S.
SAP(S) is called the space of
{f e C(S)
Let WAP(S)
WAP(S) is called the space of
Rsf
SYong moS p0c
is relatively weakly
compact}.
w amo p0c functions on S.
(See Berglund,
Junghenn and Milnes (6) for properties of SAP(S), WAP(S)).
In (3) it was shown that if
SI,S 2
are semitopological semigroups with
identities then
(SI$2)SAP XS2 SAP
where (S
and S
I s2)SAP
and
S2
sAP
are the canonical SAp-compactifications of S
(4.1)
I
S
2
X is a compact topological group which is a homomorphic image
2 respectively,
FUNCTIONS ON SEMIDIRECT PRODUCTS OF SEMIGROUPS
303
of the canonical SAP-compactification of S I, and equality denotes canonical
isomorphism.
We now prove the following lemma which shall be used in the decomposition
of the kernel.
LEMMA.
4.1
(sWAP,Y)
sWAP/K(SWAP)
Let
Let S be a semitopological semigroup such that WAP(S) is amenable.
be a WAP-compactification of S,
be right multiplication.
K(sWAP), 05:
the identity of
(K(sWAP), p)
Then
is an SAP-compactification
of S.
PROOF.
Since
group (deLeeuw and Glicksberg (7)).
0:
K(S
S
sWAPs.
WAP)
S
Observe that
Let (S
SAP
WAP) sWAP
wAP
Then p maps S
WAP(S) is amenable, K(S
WAP
and is a compact topological
onto K(S
WAP)
and
is a continuous homomorphism with range dense in
0IK(sWAP)
is the identity mapping on K(S
WAP).
y) be the canonical SAP-compactification of S.
By the universal
mapping property of SAP and WAP compactifications, there exist continuous
and
homomorphisms
,
such that
Observe further that since
and the fact that P(S)
jt
:sSAP/KsWAP
sWAP.s SAP and
0,=.
0, then =0 by the continuity of
S
wAP
-
All of the above relations are illustrated in the following commutative
diagram:
SAP
It suffices to show that
compactification of S.
that (x
Since S
I)
SAP
Yl
and
Let
(x2)
WAP)
is one-to-one so that K(S
SAP
.
Then there
Yl Y2 S
(y2
Y2" Suppose (yl
WAP)
exist
Then
will be an SAP-
x
I
x
2
s
sWAP
((x25)).
SAP
Thus O(x
i)
On the other hand,
@(xi)@()
@(xi$)(i=l,2),
such
((x2)).
((Xl))
is a compact topological group and @() is an idempotent in S
@() is the identity of S
((Xl))
K(S
SAP,
so
B.-T. LERNER
304
(8(xi))
so,
Xl x25,
and hence
O(xi)
(x25)
(Xl$)
Yl
xi
(i
i, 2),
//
Y2"
We shall use the relation (4.1), the above lemma and the results in the
following discussion to establish conditions under which we may express
$2 )WAP]
K[(S I
as a semidirect product
$2 )WAP]
K[(S I
X
@ K(S2 WAP)
where equality denotes canonical isomorphism and X is a compact topological group.
WAP(S2) are amenable. By (deLeeuw and
Glicksberg (7), Lemma 5.2) since qi
Si SIS2 is a continuous homomorphism for
1,2, then F
i
ql WAP (S I $2) CWAP (SI), and F 2 q2*WAP(SI S2)WAP(S2).
I
(In fact, equality holds in the latter.) Thus, ql WAP(SIS2) and q2*WAP(SIS 2)
are amenable and if we assume D
Is 2 e S2:
(s2,S I) S 2} is dense in S I, then
We shall assume that
WAP(S I
$2)
WAP(Sl)
and
is amenable by Theorem 3.1.
$2 )WAP]
K[(S I
By ((7), Theorem 4.11)
groups.
Furthermore, by Lemma 4.1,
SI$2,
and
K[(S
K(S2WAP)
I s2)WAP
$2 )SAP
(S I
D
(
{s
S
and K(S
S
(s2,SI)
2
Suppose
S
I}
(S 1
@
()
$2 )WAP]
s2)WAP
X@ K(S2WAP)
and S
2 (symbollically
S SAP respectively
2
WAP
2
WAP
2
WA.P(Sl),
is dense in S
Furthermore, we may represent K[ (S
K[ (S I
is a SAP-compactification of
denoted by
where equality
Thus, we have proved the following
2 a semidirect product.
e
2
K[(SIS2 )wAP]
are compact topological
Let S I, S be semitopological semigroups with identities and
2
PROPOSITION 4.2
1
K(S2 WAP)
is a SAP-compactification of S
denotes canonical isomorphism).
S
and
I
2.
s2)WAP]
WAP(S 2) are amenable, and
Then WAP(S
I@S 2)
is amenable.
as a semidirect product
where equality denotes canonical isomorphism,
are canonical WAP-compactifications of S
1
@
S
2
and S
2
respectively, and X is a compact topological group which is a continuous homomorphic
image of the canonical SAP-compactification of
ACKNOWLEDGEMENT.
Sl.//
This work was supported by a U. S. Naval Academy Research Council
Grant.
The author wishes to thank the referee for suggesting several improvements to the
exposition of the paper.
FUNCTIONS ON SEMIDIRECT PRODUCTS OF SEMIGROUPS
305
REFERENCES
i.
BERGLUND, J. F. and HOFMANN, K. H., Compact Semitopological Semlgroups and
Weakly Almost Periodic Functions, Lecture Notes in Math., oI. 42, SpringerVerlag, Berlin, 1967.
2.
LERNER, B.T., C*
3.
JUNGHENN, H. D. and LERNER, B.T., Semigroup Compactifications of Semidirect
Products, Trans. Amer. Math. Soc., 265 (1981), 393-404.
4.
JUNGHENN, H.D., C*
Algebras of Functions on Direct Products of Semigroups,
Rocky Mountain J. of Math., 10(3) (1980), 589-597.
5.
JUNGHENN, H.D., Topological Left Amenability of Semidirect Products, Canad. Math.
Bull., 24(1) (1981), 79-85.
6.
BERGLUND, J.F., JUNGHENN, H.D., and MILNES, P., Compact Right Topological Semigroups and Generalization of Almost Periodicity, Lecture Notes in Math., vo!.
663, Springer-Verlag, Berlin, 1978.
7.
deLEEUW, K., and GLICKSBERG, I., Applications of Almost Periodic Compactifications
AcLa. Math., 105(1961), 63-97.
Algebra Compactifications of Semidirect Products of Semitopological Semigroups, Dissertation, Geo. Wash. Univ., Wash., D.C., 1979.