405
Internat. J. Math. & Math. Sci.
VOL. 15 NO. 2 (1992) 405-408
A NOTE ON THE SUPPORT OF RIGHT INVARIANT MEASURES
N.A. TSERPES
Department of Mathematics
University of Patra
Patra, Greece
(Received June 6, 1990)
ABSTRACT. A regular measure p on a locally compact topological semigroup is called
(K) for every compact K and x in its support. It is
right invariant if (Kx)
shown that this condition implies a property reminiscent of the right cancellation
law. This is used to generalize a theorem of A.
MukherJea
and the author (with a new
’proof) to the effect that the support of an r*-lnvariant measure is a left group
the measure is right invariant on its support.
KEY WORDS AND PHRASES. Topological semigroup, left group, rlght tnvarlant (Borel)
measure, r*-invartant measure, support of a Borel measure, locally compact semtgroup.
1980 AMS SUBJECT CLASSIFICATION CODE.
22A15, 22A20,
43AOfi,
28C10.
1. INTRODUCTION.
S
In what follows
will denote a T
2
.locally
a positive regular (Radon) measure on the
(Jointly continuous multiplication) and
Borel o-algebra of S with support
in
[1]
and
[2
F
{s
We shall use the notation
denoting the right continuous translation
ant on S
if
(Bx
-1)
compact topological semigroup
S; for every open V s, v(V) > 0 }, as
-1
{s
B}, t x
S; sx
Bx
s
tl(B)
sx
The measure is called r*-invari-
for all Borel B and x in S. Such measures received
(B)
considerable attention in the past in connection with the (still unsolved) conjec-
ture of L.N. Argabright (Proc. Amer. Math. Soc. 17 (1966), 377-382)
port is a left group
i.e.,
(Kx)
In
[lJ
is called
p(K)
(Fx =F
that their sup-
for all x in F)
and right
if it is left simple and contains an ldempotent element).
cancellative (equivalently
The measure
F is left simple
Fight invariant on its suppor t if
F and every x
for every compact K
(1.1)
F
A. MukherJea and the author proved the "rather tight" result
The support of an r-invariant measure on S is a left group
THEOREM 1.
iff
the
measure is right invariant on its support.
Professor Mukherjea in a meeting at University of South Florida asked the questions
(i)
some
whether the "intriguing" condition (1.1) (introduced by himself) implies
sort of right cancellation on F
(1.1) plus that the
tx’S
x
in view of the fact proven by Rigelhof
F, are open maps, imply
[3]
that
right cancellation on F.
hether Theorem 1 (above) can be substantially generalized. In this note we show:
As
for question (i) indeed there is a generalized" right cancellation on S (See
_
406
N.A. TSERPES
cannot substantially be general-
Lemma 1, below) but as for question (ii), Theorem 1
-1
_>
izcd except that we may only assume that (Bx
for every Bore1 B
(B)
and
every x c F. (Unlike condition (1.1), no extra generality is obtained whether we
F ). Moreover, our proof (although patented on that of
S or B
assume B
[1])
[I]
does not use the functlonal analytlc apparatus of
of
since it uses a version
cancellation from the intrinsic properties of the measure.
2.
S
We begin by showing in what sense
LEMMA 1.
right cancellable mod
F,
fl’ f2’ f3
for
If
F
for every f
ff3
ffl
then
flf2 f3f2
F.
satisfies (1.1)). Then
be right invariant on its support (i.e.,
Let
(i)
pre-
is
that is, we can cancel on the right by premultiplying by any element of
the support.
If
(ii)
F
is also a right ideal of S, then for
s3f
Slf
tion
3
2
I
2
and n particular for any Idempotent element
PROOF. We shall argue by contradiction as in
U
that we can find disjoint compact neighborhoods
W
neighborhood of the form
C
K
v
u,lf
slf 2
s3f 2
V
175
p.
but
fs
We prove
I
fs
so
3
respectlvely of these two
1
must contain a compact
FF ), i.e.,
F (f
for some
1
,oh
(Ks I) + (Ks 3)
(K) + (Z)
[3,
uslf Vs
). Now
closure(FF)
F.
e
and
F, the equa-
f2
FF
f
which in turn must contain a right translate of some compact
f
of
F-
f some point in
distinct points (with
neighborhood
S,
3
Rigelhof
Assume
(ii): (The proof of (i) is dre slmilarly).
s
Sl
for all
fs
fs
implies
(Zs
1U
(K L/
Ks3)f 2)
K)Slf 2)
(K),
which is a contradiction.
satisfy (1.1). Then
Let
COROLLARY 1.
For any pair of idempotents
(ii)
For any Idempotent
(iil)
If
zyx
yzyx
e
for
F,
el,
eF
e
is right cancellable.
F,
x,y,z
by the above Lemma.
e
I
ele 2 ele2e2
and use the fact that e I
(ii): First cancel
by premultlplylng by
z
and obtain
x
zy
then
(i): It l’ollows since
ROOF.
by premultiplying by
zy
is idempotent.
and by Lemma I
THEOREM 2.
Suppose
(Bf
F
zyzy
and
f
2
and then cancel
zy
as follows:
satisfies
-1
> (B)
is a left group
B
e
(ii):Similarly
y
by premultiplying by
for every Borel B and every
p
iff
F
so that we have
Also (2.1) implles that
Ff
F
f
F
(2.1)
satisfies (1.1).
PROOF. Clearly (1.1) plus inner regularlty of
Borel
we ,r.ay c,cel
is idempotent.
Nc, we are ready to give the generalization of Theorem 1
Thea
so that
e
F, we have ele 2
I
2
the idempotex ts in F form a left-zero subsemlgroup of F.
(i)
for all
(Bf
f
-I)
imply
p(B)
F.
p(Bf
for every
-1
f
< p(B)
F
for all
and Borel B.
In the proof of Theorem I
in
[1],
SUPPORT OF RIGHT INVARIANT MEASURES
e
we produced an idempotent
so
F
Fa
all
for
F
a
Fa, for
in
[l,p.
(cf.
F,
a
407
so that
Fe
Fe
Fa
F
and
974). Now, the same proof goes through with-
out any difficulty except that instead of the right cacellatton on
F, we use
Fa, a
Corollary 1 (iii) above.
We give next a result summarizing certain important conditions on
that are equivalent to
COROLLARY 2.
F
F
and
being a left group.
For a locally compact second countable semtgroup S admitting an r*-in-
variant measure
,
F
(1)
these are equivalent:
is right cancellable
is right invariant on its support, i.e., satisfies (1.1))
(ii)
S is pre- right cancellative with respect to F, i.e.,
(iii)
F,
Sl,S2,S 3
fs I
implies
fs
3
(iv)
F
is a left group.
(v)
F
has the right translations
(vi)
F
has the right translations open and
REMARK. It is not
tf
closed for all
p
SlS 2
with
s3s 2
F.
for all f
F.
f
satisfies (1.1).
known to our knowledge if (v) and (iv) are equivalent in the
absence of second countability.
Most of these follow from Theorem 1 or
PROOF.
cellation implies that
tf
are
one-to-one
Theorem 2.
K,
and for compact
Note that right can-
Kxx
that right invariance on its support follows from r*-invariance, so
(tii) => (iv) (cf. Theorem 2).
[4]
Kx ,so
F
(i) => (ii) =>
Since F is metrizable being regular the technique in
for producing an idempotent in
By the result of Rigelhof,
-1
Fx
F
applies and thus
(i).([3]p.
(vt) implies
becomes a left group.
175). For
(iii), see
our Lemma
2, below.
REARK. The following will show the "tightness" of the conditions of Theorem 2.
It is well kvown that a property that
"melds" naturally (at first sight) with condi-1
tion (1 1) is that of lower r-.-invariance, i.e.,
that V(Bx
(B) for all Borel
B C S
[5,
x
and
S,
for
and
(1.1) are equivalent to the condition
(cf.[2J
and
92])
p.
(K)
(Kx)
for all compact K
S
g
x
and
K and x
this inequality becoming equality whenever
(2.2)
with
are in F.
This condition (2.2) implies that F is a right ideal and
tent
[0
e
)
Fe
F for very idempoS, but these are not enough to make Theorem 2 valid, for the example of
with addition and Lebesgue measure
does not satisfy (2.1)
is not r*-invariant (it
shows that
of Theorem 2). However this S is pre- right cancellative as
the following Lemma generally indicates.
LEMMA 2.
S.
all
Suppose
fs
Then
f
1
satisfies (2.2)
fs
3
for all
f
FF
and suppose that
If moreover
SlS 2
SlS 2
F,
then
for
fs
Sl,S2,S 3
1
fs
3
for
F.
PROOF.
Suppose first that
Ltmma 1 (ii) with
SlS 2
F.
Then the second equality in the proof of
becomes less or equal
replaced by s
2
equality and thus a contradiction obtains. Next assume that
F.
s3s 2
f2
and the last remains
SlS 2 s3s 2 and SlS 2
Again, as before(See proof of Lemma 1) there are disjoint compact neighborhoods
N.A. TSERPES
408
-1
-1
and Vs
respectively, such that the intersection of Us
3
I
3
contain a compact neighborhood W of f (we use W f F instead of W). Then we have
U, V
of
fs
fs
I
again the inequality
(W) + (W) < (Ws 1) + (Ws 3) < (Ws I (,}
again a contradiction.
RARK.
F
Ws3)s 2)
(W
V W)SlS 2)
(W),
The most difficult part in problems involving the nature of the support
Fx
is producing an idempotent element in
or in F itself. For this, it would be
inteSting to have a "survey paper" giving all known methods for producing an idempo-
tent in the presence of measure and/or
haviffg some compact subsemigroup
or
topological invarlance conditions.
a compact fiber
xx
-I
version of (2.2) and a subsemigroup of positive finite inner measure,
the technique in
d Kalisch
[1]
which is in some
(Canad. J. of Math. 4
is onto
t
x
(1952), 396-4J),
x
and the technique of
?
[4]
which
are closed mappings, can the "onto-
)
f(xs) on
f(x)
s
the
that
to
prove
suffice
(that will
L2(S,
be used to prove that the operator
in the non-second countable case
we know only
:nse an adoptatlon of a method of Gelbaum
needs metrlzabillty !. (For example, when the t
hess" of the
Apart for
or a two-slded
support of an r*-Invariant measure is a left group
when the
tx’S
are closed).
REFERENCES
1.
compact semigroups,
2.
TSERPES, N.A. and
on semigroups,
3.
A problem on r-invariant measures on locally
Indiana Univ. Math. J 21 (1972), 973-977.
MUKHERJEA, A. and TSERPES, N.A.
RIGELHOF, R.
Soc.
MUKHERJEA, A. On
Semigroup Forum
certain conjectures on invariant measures
(1970), 260-266.
Invariant measures on locally compact semigroups, Proc. Amer. Math.
28 (1971), 173-176.
and MUKHERJEA, A. Mesures de probabilite r*-invariantes sur
semigroup metrique, C.R. Acad. Sc. Paris Ser. A. 268 (1969), 318-9.
un
4.
TSERPES, N.A.
5.
BERGLUND, J.F. and HOFMANN, K.H. Compact semltopoloIcal semis.ups aid weakly Flmo.st periodic functions, Springer 1967, Lecture Notes n Math. no 42.