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ABSTRACT

EXTREMELY AMENABLE SEMIGROUPS

BY

Eitoku Goya

Various characterizations for left amenability
(for groups or semigroups) have been attempted by several
authors. They are the monotone extension property, the
fixed point property and others. Here we characterize
extremely left amenable semigroups by the multiplicative
monotone extension property, or equivalently, it can be
characterized by some type of ideal extension property.
These characterizations are closely related to the structure
of the extremal points of LIM(S), the set of all left invar-
iant means on LUC(S). Therefore we shall investigate the
structure of such an extremal point for a particular topolog-

ical semigroup.

Next characterizations for a compact topological
semigroup S, for which LUC(S) is n-extremely left amenable
(n-ELA), will be investigated, and we shall show how this

works to determine the structure of the kernel.

A necessary condition on a topological group G, for
which LUC(G) is n-ELA, is given by A.T. Lau and E. Granirer.
We shall consider sufficient conditions and generalize a

few theorems concerning left amenability.

Eitoku Goya

Finally, we consider the fixed point property on
locally compact topological semigroups G with respect
to separately continuous linear actions of M(G) on a

(£,c) space.

EXTREMELY AMENABLE SEMIGROUPS

BY

Eitoku Goya

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1972

TO YUSUKO, MASATO
AND SANEYUKI

ii

ACKNOWLEDGEMENT

My thanks are due to Professor J.C. Kurtz for
suggesting this fresh area of research, giving much infor—
mation and discussing freely. I wish especially to

acknowledge his sustaining mental support throughout my

work.

I am also grateful to Mrs. Haubert and Miss Stewart

for doing an excellent job of the typing.

iii

Introduction

CHAPTER

I.

II.

III.

IV. .
is

V.

BIBLIOGRAPHY

Preliminaries

TABLE OF CONTENTS

iv

Compact topological semigroups

Invariant extensions of multiplicative
linear operators

Locally compact topological semigroups

Topological groups G for which LUC(G)
n-ELA

10

22

31
4O

46

INTRODUCTION

Invariant means on spaces of functions have been
studied by Von Neumann [27], Banach [1], Day [2], [3] and
others. Day's paper on amenable semigroups [3] presents
a comprehensive summary of the earlier work, as well as

many of the more recent results.

Let S be a topological semigroup, that is, a
semigroup with a Hausdorff topology such that the multi-
plication map (x,y) 4 xy is separately continuous. Let
m(S)[C(S)] be the space of all bounded [bounded continuous]
real functions on S. If f E m(S), a 6 S, we set
Hf” = suplf(s)| (sup norm) and define (Laf)(s) = af(s) = f(as)
and (::§)(s) = fa(s) = f(saL Let X be a translation
invariant (i.e. af, f3 6 X whenever f 6 X, a E S),
normed closed subalgebra of m(S) containing constants.

X* will denote the conjugate space, and La:X* 4 X* is

the adjoint of la. An element m E X* is a mean if (1)
m(ls) = l and (2) m(f) 2 O for all f 2 0, where 15 is

the characteristic function of S. m is a multiplicative
mean if an addition, m(fog) = m(f)m(g) for all f,g e X.

m is a left invariant mean (LIM) on X if m is a mean and

Law = m for all a E S.

Definition 0.1: A function f G C(S) is left uni-
formly continuous if for any net 8a 4 5(Sa’5 6 S),
H Sf - sf” a O. Denote by LUC(S) the space of all left

uniformly continuous functions (Nanioka ) .

Let A(S) be the set of all multiplicative means

on LUC(S) and COA(S) be the convex hull of A(S).

Definition 0.2: A semigroup S is left amenable
(LA) if m(S) has a LIM and extremely left amenable (ELA)

if m(S) has a multiplicative LIM.

The first to consider LA semigroups was M.M. Day
[2], [3]. ELA semigroups were considered by T. Mitchell
[16] and later by E. Granirer [8]. Various characterizations
for LA(ELA) semigroups have been considered by several

authors.

R.J. Silverman and Ti-Yen [25] characterized LA
semigroups by the monotone extension peoperty (MEP) and
Hahn-Banach extension peoperty (HBEP) for an order complete
vector lattice whose positive cone is sharp. Day obtained
similar results with a weak HBEP for order compact spaces
[6]. In Chapter II, we characterize ELA semigroups by the

multiplicative MEP for order compact algebras.

Now let S be a compact topological semigroup.
NUmakura [19] showed that the kernel K of S can be written

as a disjoint union of closed minimal left ideals in S.

W.G. Rosen [23] proved that K is a minimal left ideal if
LUC(S) has a LIM. In Chapter III, we characterize those 8
for Which LUC(S) has a LIM in COA(S), in terms of minimal
left ideals and determine the structure of the kernel.
Furthermore, wegivea partial answer to the following question

posed by A.T. Lau [l4].

(*) If LUC(S) has a LIM in COA(S), then is every
extremal point of the set of all LIMEson LUC(S) in
COA(S)? If S is discrete, the answer is positive

(Sorenson). However the general case is still open.

A.T. Ian and E. Granirer [11] also showed that if
LUC(G) has a LIM in COA(G), then there is an open normal
subgroup N of G such that IG/N|<+m and LUC(N) has a
multiplicative LIM; here G is a topological group. In
Chapter IV, we consider the converse of this statement and

generalize theorems concerning left amenability.

Several authors have recently studied topological
semigroups S for which LUC(S) has a LIM. Mitchell [18]
recently considers a fixed point property on a topological
semigroup S for which LUC(S) has a multiplicative LIM.
J.C.S. Wong considers a locally compact topological semigroup
S for which LUC(S) has a LIM. In Chapter V, we consider
a fixed point property with respect to separately continuous

linear action of a convolution semigroup M(S).

CHAPTER I

PRELIMINARIES

In this chapter, we list some basic concepts and
known results. For a more complete exposition, the reader
is referred to Day [3], Granirer [8], [9], [10], A.T. Lau

[l4] and Dunford-Schwartz [5].

For any set A we shall denote by [A] the car-
dinality of A. If A is a subset of a semigroup S, then
-1
a A = {s E SIa-s E A] and 1A is the characteristic

function of A. Let S be a semigroup, X a translation

invariant norm-closed subalgebra of m(S) containing constants.

Definition 1.1: m E X* is a point measure if there

is some 30 in S such that m(f) = f(so) for all f E X.

It is well known that the set of all point measures
on X is w*-dense in the set of all multiplicative means
on X. X is left amenable if X has a LIM and X is
extremely left amenable if X has a multiplicative LIM.
Similarly right amenability is defined using right translation
ra. X .is amenable if X is left and right amenable and
X is extremely amenable if X is left and right extremely

amenable. We now list some characterizations for ELA semi-

groupsand M.M. Day's fixed point theorem.

Theorem 1.1: The following statements are equivalent.

(a) S is ELA.

(b) For any a,b 6 S, there is some c E S such that
ac = bc = c. (E. Granirer [9]).

(c) S has the common fixed point property on compacta,
that is , for any compact Hausdorff space X and any

homomorphic representation 8' of S as a semigroup of

continuous maps from X to X (under functional composition),
there is some xo 6 X such that s'(xo) = x0 for all
s' 6 S' (Mitchell [16]).
(d) S is a LA semigroup and m(Sf .g) = m(f -g) for
all f,g 6 m(S), s E S and LIM m on m(S) (E. Granirer [9]).
(e) S is a LA semigroup and each extremal point

of the set of all LIMHson m(S) is multiplicative (E. Granirer

[9])-

Theorem 1.2: Let A be a ring (not necessarily
commutative) and {Ts:s E 8] an anti-representation of the

ELA semigroup S as ring homomorphisms of A_ into A. Then

KA = [2: (ai - Tsiai): ai 6 A, si 6 S] satisfies
KA = [a - T a : a 6 A, s E S] = U T -1(o). Furthermore,
S 368 S

K is a two sided ideal of the ring A which coincides with
H = [23bi(ai — Tsi) ci : si 6 S, ai’bi’ci E A (some of
b or C1 need not appear) n = l,2,--']. Consequently, if T

is an additive map of A to the abelian group B which

satisfies TTSa = Ta for all s E S, a E A, then

T[b(TSa)c] = T(bac) for all s 6 S and a,b,c E A (where

 

b or c need not appear) (E.Granirer [10]).

Let K be a convex subset of a linear space V which
is compact in some topology for which A(K), the space of real
affine continuous functions on K, separates the points of K.
Let /A(K) be the space of all affine continuous maps of
K into itself. Then /h(K) is a semigroup under composition
of mappings and if the product topology in KK is used, it
can be checked that the multiplication is separately con-
tinuous . The next theorem is M.M. Day's famous fixed point

theorem.

Theorem 1.3: A semigroup S is left amenable if
and only if for each convex,compact subset K, Where A(K)
separates points, and each homomorphism h of S into
/A(K), there is in K a common fixed point of all mappings

hs: s c.- s [6:p. 18].

We remark that if K is a convex, compact subset
of a locally convex T2 space V, then by the Hahn-Banach
theorem, the space of all real affine continuous functions

on K separates the points of K.

Let S be a topological semigroup. Then LUC(S)
is a translation invariant, norm-closed subalgebra of m(S)
which contains constants and is left introverted, that is,

*
m(sf) E LUC(S) for all f E LUC(S) and m E LUC(S)

(Namioka [20]). Hence for ¢,w e LUC(s)*, we can define

m 0 W E LUC(S)* (the Arens product of m and W) by

(m 0 Y)(f) = m(h) where h(s) = T(Sf)(Day [3: p. 540]). The
Arens product renders the set of means on LUC(S) or A(S)

into a semigroup.

Definition 1.2: For any topological semigroup S,
LUC(S) is n—extremely left amenable (n-ELA) if there is
a subset Ho of A(S), [Ho] = n, which is minimal with

respect to the property: LaHo = HD for all a E S (A.T. Lau

[14: p- 71]).

Remark 1.1:

(a) It is important to note that if H1 is another
finite subset of A(S) Which is minimal with respect to the
property L3H1 = H1 for all a E S, then IHOI = [H1].

(b) If Ho = [ml,---,mn], it is easy to see that
(l/nbélil cpi €COA(s) is a LIM on LUC(S). Conversely,
if LHC(S) has a LIM of the type (l/n) E) mi in COA(S),
then LUC(S) is m-ELA, where m divide:=1n, and there is a
subset H of {m1,-¢-,mnl which is minimal with respect to

LaH = H for all a E S.

Definition 1.3 (Ljapin): An equivalence relation E
on a semigroup S is two sided stable if a E b implies

ca E cb and ac E be for all c E S.

Let S be a topological semigroup and HCAKS) such

that L.H = H for all a E S. Define on S the two sided
stable equivalence relation E: a E b if and only if

Lam = me for all m 6 H(Sorenson). Denote by S/H the
factor semigroup defined by this equivalence relation (see

[14; p. 72]).

Theorem 1.4 (A.T. Lau [14]): If S is a topological
semigroup such that LUC(S) is n-ELA, then there exists
a collection 3 of n disjoint open and closed subsets of
S with union 8, such that 1A 6 LUC(S) for all A E 3,

and 3 is the decomposition of S by cosets of S/H for

any finite subset HCA(S) satisfying LaH = H for all a E S.

Let S be a topological semigroup such that LUC(S)
is n-ELA with 3 a decomposition of S as obtained in
Theorem 1.4. The proof of Theorem 1.4 shows that there exists
an open and closed sub-semignaq> T E 3 such that lT E LUC(S),
and if H is any finite subset of A(S) with LaH = H for
all a E S, then T = [s E SILS w = m for all m 6 H].
Furthermore, if {a1,"',an] is a coset representative of
S/Ho, then 3 = {ade i = l,2,-°°n]. If S = G a topological
group, by A.T. Lau and E. Granirer [ll], LUC(T) has a multi-
plicative LIM. In the case of a topological semigroup we
set pLUC(S) = [pf]f e LUC(S)] where (pf)(t) = f(t), t C T.
Then pLUC(S) is a translation invariant, norm—closed sub—
algebra of LUC(T) containing IT, and pLUC(S) has a multi—

plicative LIM (A.T. Lau [14]).

We close this chapter by stating the relation between

the set of LIM's on LUC(S) and the set of LIM's on

pLUC(S).

Theorem 1.5 (A.T. Lau [14]): For any topological
semigroup S, if LUC(S) is n-ELA, then there exists a
linear transformation mapping the set of LIM's on pLUC(S)
one to one onto the set of LIM's on LUC(S). Indeed, let
H be a finite subset of A(S) such that LaH = H for all
a 6 S, T = [s E 51LS m = m for all m E H], and
P: LUC(S) 4 LUC(T) Where (Pf)(t) = f(t) if t 6 T. For
any coset representative p of S/H, the mapping

F (4)) =(1/n) Z (L P*) (Cb) is such a linear transformation.
P aEp a

CHAPTER II
INVARIANT EXTENSIONS OF MULTIPLICATIVE

LINEAR OPERATORS

We begin this chapter with a few definitions to
explain the monotone extension property. Throughout this
chapter we assume that the ordered linear space A under
consideration has an order unit, i.e., an element e 6 A
such that for each f e A, there is a positive number d(f)

with f s o(f)e

Definition 2.1: Let A be a linear space or an alge—
bra over the real field R, and S be a semigroup. A repre—
sentation (anti—representation) h of a semigroup S on A
is defined in this chapter to be a homomorphism (anti- ) of
S into the space of all algebraic homomorphisms of A into

itself.

Definition 2.2: An algebra A over R is called an
ordered algebra if, regarded as a linear Space,

(1) A is an ordered linear space and x ° y ? 0 for
any X 2 O, y g 0 ,

(2) The order unit e in A is simultaneously a

multiplicative identity [13].

Definition 2.3: An ordered algebra A is called an
ordered locally convex algebra, simply (0,2,c)-algebra, if

(1) A is an (0,1,c)-space,regarded as a linearspace.

10

ll

(2) Multiplication (x,y) 4 xy is separately contin-
uous.
An (o,L,c)-algebra (linear space) is called an order compact
algebra (space), simply (0*,L,c)-algebra (space), if A is
order complete and each order interval is compact with respect

to the (L,c) topology on A
‘We now give examples ofHan (o*,L,c)-algebra (space).

Example 2.1: If u is any measure, and if V is
L1(u) or any abstract L-space, Kakutani showed that V is
order complete and that every order interval is weakly compact.
That is, V is an (0*,L,c)-space relative to the weak topology

(see Day [4:p. 107-108]).

Example 2.2: Let X be an arbitrary set and m(X)
be the set of all bounded real functions on X . m(x) can

be considered as a subalgebra of the (o,z,c)-algebra URX ,

xex
where RX = R. m(x) is an (o*,L,c)—algebra with respect to

the product topology. If X = [x], a single point, then m(X) = R.

Definition 2.4: The pair [S,A], where S is a semi-
group and A is a commutative (0*,L,c)—algebra (space), has
the monotone extension property, simply MEP, if and only if
for any collection [B,BO,C,h(s),mo] where

(a) B is an ordered algebra (linear Space) with posi-

tive cone C

12

(b) B0 is an ordered subalgebra (subspace) of B,
ordered by cone BO 0 C such that (f + BO) n C # 0 for all
f E B and Bo contains the order unit e of B.

(c) h is an anti-representation of S on B and
hs(Bo) g Bo’ hS(C) ; C, -e E hs(e) e e for all s E S.

(d) mo: Bo 4 A is a monotone linear operator such
that mo(e) = e', the order unit in A, and ¢O(hsf) = mo(f)

for all f 6 Bo and s E S. Then there exists a monotone

linear operator m: B 4 A such that wIBO = m , m(hsf) = m(f)

o
for all f E B and s E S. In particular, if we consider
multiplicative monotone linear operators ”0 and m, the

pair [S,A] has multiplicative MEP, simply mult MEP. In What
follows, we simply say that S has MEP or mult MEP if

[S,A] has MEP or mult MEP for arbitrary A.

The next theorem is a characterization of ELA

semigroups by mult MEP .

Theorem 2.1: A semigroup S is ELA if and only
if s has the mult MEP. To prove the theorem we need the

following lemma, which is interesting in itself.

Lemma 2.1: A semigroup S is LA if and only if

S has the MEP.

Proof: We shall make use of the idea in the proof
of Theorem 15.2 ([6:p. 49]). Consider the pair [S,V] and

[L,LO,C,h(s),mo], where L,LO and V are linear spaces.

13

For each f E L, there is d(f) p 0 such that -a(f)e s f s

m(f)e. Let If = [-d(f)eg d(f)e'] and K = HIf. Consider the
fEL A
product topology on V = nvf where Vf = V. Then V is
fEL
a locally convex space. By assumption, each If is com-

pact. Consequently by Tychonov's theorem, K is compact in
V . By [4:p. 105], there exists a monotone linear extension
m of mo . Define Hs : L(L,V) 4 L(L,V) by (HSW)(f) =
m(hsf), where L(L,V) is the space of all linear operators

from L to V . Then ¢:HST E K for all s E S . Indeed,

—d(f)e g f g 0(f)e, -e g hs(e) g e and hS(C) g C implies

HA

hS(—n(f)e) Sh (f) §h8(o(f)e) n(f)e i.e., —o(f)e §h5(f) g

s
o(f)e . Since m is a monotone linear operator, —u(f)e' S
(Hsm)(f) é m(f)e' . Hence K', the closed convex hull of
[Hsm : s E S], is a compact convex set in K carried into
itself by all the continuous affine mappings, HS, 5 E 8
Since H is a homomorphism of S into [A(K') and S is

LA, by Theorem 1.3 (Day), there exists a point $ E K' such

L 4 V

that Hsé = $ for all s E S . In other words, $
is an invariant linear operator. We now show that $ is a
monotone extension of mo. Let KO = C0[Hsm| s E 8]. Clearly
every element in K0 is a monotone extension of (po . For Q,

. A .
we have a net [mo] in KO such that ma 4 m WIth respect

A A
to the product topology in V . Since V is a TZ-space,
$ is a linear extension of mo . Finally, let f e V and

O S f g 0(f)e. Since each ma is a monotone linear

14

extension of ”0’ we have wa(f) E [o,a(f)e'| . By hypothesis

[o,o(f)e'] is compact, thus $(f) E [o,a(f)e'] , since

ma(f) 4 m(f) . Conversely, assume that S has the MEP, and

consider V = R, B = m(s), B0 = the set of all constant
functions on S . Define ”0 by ¢O(c ' l) = c, where

h = L . Then m is a LIM on B . By the MEP, there
5 s o o

is an invariant mean on m(S), which was to be shown.

The Krein-Milman theorem is required for the proof of

Theorem 2.1.

Definition 2.5: Let K be a non-empty subset of a
real or complex linear space V . A point p E K is said to
be an extremal point of K if whenever p = okl + (l-o)k2

With 0 < a < l and kl’k2 E K, then k1 = k2

Theorem 2.2 (Krein—Milman): If K is a compact,con-
vex subset of a (L,c)—space, and E is the set of extremal

points of K, then CENE):; K . Consequently 66(E) = (MK)

and CD(E) = K if K is convex.

Proof of Theorem 2.1: Let KO be the set of all mono—
tone invariant linear operators m : B 4 A such that
cp|BO = mo . Then as was shown above, K0 is a non—empty,
convex and compact set. By the Krein—Milman theorem, there is an
extremal point m of K0 . To show that m is multiplica-

tive, we use the idea in the proof of Theorem 1.1 (e) and

15

Theorem 1.2 repeatedly.

Case 1: Suppose o s f

:M

e and f 6 BO . We define
a linear operator Uf : B 4 A by Uf(g) = m(f ° g) -

m(f)w(g) . By Theorem 1.2, Uf(sg) = Uf(g), Where Sg = hs(g)'

Furthermore, Uf(g) = o for g E 30’ since m is multipli-
cative on B0 . Let g E B and g g o . Then (T + Uf)(g)==
m(g)(e' - m(f)) + m(f ' g) s o and (m - Uf)(g) = m(g) -<p(f -g)+
w(f)-m(@ e 0 since m is monotone and f ° g e 9 implies
m(f - g) e m(g). Thus w i Uf e K0 and w = 1/2(cp + Uf) +
1/2(m - Uf) implies uf E 0 on B, i.e., m(f ° 9) = m(f)m(g)
for all g E B .

Next let f be an arbitrary element in B0 . Then

there is 0(f) > 0 such that f

HA

a(f)e, so f = de —

(0e - f) = afl - bf2 where a,b 2 o and o g f. g e,fi E B

m((afl - biz) - g) = a.,(f1- g) - bcp(f2 - g) =

0'
Thus T(f ° 9)

m(f)m(g) for all f 6 B0 and g 6 EL

Case 2: Suppose o E f E e and f G B . We define
of = B 4 A by uf(g) = m(f ° 9) - m(f)m(9) . By case 1,
Uf(g) = o for all g E BO . Hence analogously, we have

m(f ° 9) = m(f)w(g) for all f,g e B .

In Theorem 2.1, if we replace A,B,hS by R,m(S),pS
and 30’ $0 by any invariant subalgebra of m(S) containing
constants and any multiplicative LIM on 80’ we get the fol-

lowing corollary.

16

Corollary 2.1: S is ELA if and only if any multi-
plicative LIM on B0 can be extended to a multiplicative

LIM on m(S)

In the corollary, if we replace the multiplicative
LIM mo by an ideal, we obtain another characterization of an
ELA semigroup. To show this, the Gelfand-Mazur theorem is

required.

Theorem 2.3 (Gelfand-Mazur): If A is a complex
Banach algebra with unit in which each non-zero element is
invertible, then A is isometrically isomorphic to the com-

plex field.

Theorem 2.4: S is ELA if and only if any ideal I

A
in A containing K can be extended to a maximal ideal I

A
in m(S) containing Km(S)’ where A is a left invariant
subalgebra of m(S) containing constants and KA = {f — Sf|

f E A, s E S] .

Proof: We may assume that A is closed and I is
maximal in A, since each is is continuous and any ideal in
A can be extended to a maximal ideal in A because 1 E A
We use the idea in the proof of lemma [9:p.99] . Let
AC = A + iA and I = I + iI. Then AC is a commutative

C

Banach algebra with identity with respect to the sup norm

topology, and IC is a maximal ideal in AC . Thus by the

Gelfand—Mazur theorem, Ac/IC a C . Consequently there is

17

a non-zero multiplicative linear functional u on AC such
that Ker(u) = IC . Let “0 = ulAc . Then ”0 is a multi-
plicative invariant linear functional on A such that

“0(1) = 1, Since Ker(uo) = I and KA c I . Furthermore

H0 is real. To show this, let 9 E A and u

iB,B# 0 . Then f = (g-orlm-l is in A. The series

O .
ZZl/n!(—if)n is norm convergent to e-lf in AC . Thus
n=0 6
-'f °’ . . —' f
Me 1 > = Z 1/n3u((-1f)“) = Z 1/n: (-1u0(f))n = e 1910‘ H
n=o n=o

-i2 _ -if
e — e > 1 . However He H E l, consequently "n” > 1,
which is contradiction since ”u” = 1. This shows that ”o is

a multiplicative LIM on A . By Corollary 2.1, “0 can be
extended to a multiplicative LIM a on m(S) . Clearly

A

I = Ker(fl) is a maximal ideal in m(S) containing I and

Km(S) . Conversely, let mo be any multiplicative LIM on
A . Then Ker(¢o) = I is a maximal ideal in A containing
RA . By the assumption, there is a maximal ideal I in m(s)
containing Km(S) and I . By the same argument as above,

there is a multiplicative LIM m on m(S) such that

A .
Ker(¢) = Io . Clearly Ker(¢[A) = I

there is a number k such that w/A

(oIA)(1)

Ker(¢o) . Consequently

kwo . However

”0(1) = 1 implies k = 1, which was to be shown.

Next we consider a similar type of problem for
topological semigroups. An action of S on a topological

space X is a map SxX 4 X that satisfies (3132)x==sl(szx)

18

for all 51,52 6 S and x E X . In what follows, we will
consider only those actions for which the map SxX 4 X is
jointly continuous. The next theorem is a generalization of

Theorem 1.1(c)

Theorem 2.5 (T. Mitchell [18:p. 633]): Let S be
a topological semigroup. The following properties are
equivalent.

(P1) LUC(S) has a multiplicative LIM .

(Fl) Whenever 8 acts on a compact Hausdorff space Y,

Y contains a common fixed point of S .

Now we are ready to state the multiplicative MEP

style problem.

Theorem 2.6: LUC(S) has a multiplicative LIM if
and only if any multiplicative LIM m on a left invariant
subalgebra A of LUC(S) containing constants can be extended

to a multiplicative LIM on LUC(S)

Proof: We need to show only that if LUC(S) has a
multiplicative LIM, then ”0 can be extended to a multi—
plicative LIM on LUC(S) . By the classical Hahn-Banach
extension theorem, mo can be extended to a mean on LUC(S)
by taking p(f) = “f” as a positive sublinear functional.
Let K be the set of all extensions of $0 to means on

LUC(S) . Then K is non-empty, convex and w*—-closed .

19

By Alaoglu's theorem, K is w*-compact and by the Krein—

A
Milman theorem, K has an extremal point qp . As in the

proof of Theorem 2.1, $ is a multiplicative mean. Next
let Y be the set of all multiplicative extensions of $0
to means on LUC(S) . Then Y is non—empty and w*—compact
again by .Alaoglu's theorem. For each s e S and n E Y,
we define an action of S on Y by sn = 2*u . As

T. Mitchell showed in the proof of Theorem 2.5, this multi—
plication is jointly continuous. Thus by Theorem 2.5 (F1),
Y has a common fixed point of S, that is, there is a n

*
in Y such that su = u or ‘5” = p for all s E S

This n is a required multiplicative LIM .

We note that the technique used in the proof of
Theorem 2.6 will give a second proof of Theorem 2.1. Since
S is ELA, we can use T. Mitchell's common fixed point pro—
perty on compacta (Theorem 1.1(c)) instead of (F1) in the
Theorem 2.5, and (fisw)(f) = m(hsf), where m E K = the
set of all multiplicative monotone linear extensions of mo,
as a homomorphic representation of S as a semigroup of

continuous maps of K into itself.

We close this chapter by giving examples of ELA and

n-ELA semigroups.
Example 2.3: A finite group G is [G] — ELA .

The next theorem will give a sufficient condition

20

for LUC(S) to be n-ELA

Theorem 2.7: Let S be a topological semigroup such
that

(l) LUC(S) separates the points of S .

(2) S has a finite ideal I Which is a minimal
right ideal.
Then LUC(S) is n-ELA for some n . Consequently, a finite

left amenable semigroup is n-ELA for some n

Proof: For any x E S, x1 = I since I is a mini-
mal right ideal. Let J c I be a minimal left ideal. Then
xJ = J for all x E S . Let J = {al,...,ak} and
H = {m1,...,wk} Where ¢i(f) = f(ai) . Then H is minimal
with respect to LaH = H for all a E S since LUC(S)
separates points . Thus LUC(S) is |H| - ELA . If S
is a finite left amenable semigroup, by R.G. Rosen [23], S

has an ideal which is a minimal right ideal.

If LUC(S) is n—ELA and LUC(T) is m—ELA, then
LUC(SxT) is nm-ELA (see proposition 6.4[14]). Hence one

can construct many examples.

Example 2.4 (A.T. Lau): E. Hewitt has constructed a
Hausdorff regular topological space SO such that the only
continuous real functions on SO are the constant functions.

Define on SO the binary operation a - b = a for a,b E 50'

21

Clearly S0 is a topological semigroup and LUC(SO) = C(SO)
is ELA . Let To be any n—ELA (discrete) semigroup and
A A A

T = S )(T . Then LUC(T ) is n-ELA even though T

o o o o o

is not even left amenable as a discrete semigroup. This
A
says that multiplicative LIME; on LUC(TO) can not be ex-

A
tended to multiplicative LIM's on m(TO) for n = l . ,

Example 2.5: Let G be a discrete group. Denote I
by GC the semigroup of all finite or countable subsets with i
the usual multiplication A ° B = [abla e A,b E B] . For any
A,B E Gc’ let C be 'the group generated by A u.B. Then
C E GC and AC = BC = CA = CB = C . Thus GC is extremely
amenable (E. Granirer). If G is not countable, GC does
not contain zero. Next let S be any discrete semigroup
with the same multiplication as above. Sc is ELA, ERA or

EA if s is ELA, ERA or EA .

Example 2.6: If S is a semigroup generated by one
element, then S is ELA if and only if S is finite and
has a zero. Since if S = [an]: is ELA and p > o is
such that aap = azap = ap, then an = ap(n 2 p). Thus S
is an ELA semigroup with zero ap . More generally, we can

show that any finitely generated ELA semigroup contains a

T
I
l
i
l
I
right zero (E. Granirer).

CHAPTER III

COMPACT TOPOLOGICAL SEMIGROUPS

To begin with, we shall start with a problem posed by

A.T. Lau , that is

(*) If LUC(S) is n-ELA, then is every extremal
point of LIM(S), the set of all LIM's on LUC(S), in COA(S)?
Let S be ELA. Then every extremal point of LIM(S) is
multiplicative by Theorem 1.1(e). However generally this is

not true. Here we attack this problem under strong conditions.

Throughout this chapter, we consider only semigroups
with compact Hausdorfftopologies, unless otherwise stated
explicitly. Let A(S) be the set of all multiplicative
means on C(S) and LIM(S), the set of all LIM's on C(S).
This is reasonable since LUC(S): C(S) if S is a compact
topological semigroup with jointly continuous multiplication

(Namioka [20]) .

The following proposition is the key to treat the

compact topological semigroup.

Theorem 3.1: Let X be a compact Hausdorff space.
If m E C(X)* is non-zero and multiplicative, then m is a

point measure (5, p. 278).

22

23

Lemma 3.1: If C(S) has a multiplicative LIM,

then every extremal point of LIM(S) is multiplicative.

Proof: Let Q be a multiplicative LIM on C(S).
By Theorem 3.1 m is a point measure. Hence there is some
a in S such that m(f) = f(a) for all fe§c(S). Since
m is a LIM and C(S) separates the points of S, sa = a
for all s E S. By Theorem 1.1(b), S is ELA. By Lemma 2.1
every LIM m on C(S) can be extended to a LIM $ on m(S),
and by Theorem 1.1(d), {5(f ~89) = $(f - g) for all f,g e m(S)
and s E S. Hence m(f - Sg) = m(f - g) for all f,g E C(S),
m E LIM(S) and s E S. Thus as in the proof of Theorem 2.1,

every extremal point of LIM(S) is multiplicative.

Theorem 3.2: Let S be a compact topological semi—
group with jointly continuous multiplication. If LUC(S) is

n—ELA, then every extremal point of LIM(S) is in COA(S).

Proof: The proof depends mainly on Theorem 1.5.
Notice that pLUC(S) = LUC(T) since LUC(S) = C(S), where
T is a compact topological sub-semigroup of 8. Hence
pLUC(S) = LUC(T) has a multiplicative LIM. Thus every
extremal point of LIM(T), the set of all LIM's on pLUC(S),
is multiplicative by Lemma 3.1. Let u be an extremal
point of LIM(S). Since u is a LIM, by Theorem 1.5, there
is a Y in LIM(T) such that FO(V) = p. Clearly Y is

an extremal point of LIM(T), thus multiplicative.

24

Hence u = FO(W) = l/n. ZiLa(p*V) E COA(S).
an

In the proof of Lemma 3.1, we got a left ideal I
such that 51 = I for all s E S, where I1 = [a]. By a

slight modification of this idea, we get the following theorem.

Theorem 3.3: The following conditions on S are
equivalent.

(a) There is a subset H of A(S) which is minimal
with respect to the property: LaH = H for all a E S, and
[HI = n.

(b) S has a minimal left ideal consisting of n-
elements and x1 = I for all x E S.

n n

Proof: (a) a»(b) Let H = {miil and m = l/n Kiwi.

i=1

Then m is a LIM on C(S). Since S is compact, by
Theorem 3.1, mi is a point measure for i: 1,2,°-',n. Let
wi(f) = f(ai) for all f E C(S) and I = {a1,---,an].
Since m(xf) = m(f) for all x E S and f E C(S), we have

n n

(3.1) l/n Zf(xa.) = l/n 23f(a.).

. i . 1

i=1 i=1

Claim: (l) x1 = I for all x E S. (2) I is
minimal. Suppose xaj E I for some x E S and j. By

Urysohn's lemma, there is a f E C(S) such that o s f S l,

f(ai) = 0 (i = l,2,-'-,n) and f(xaj) 1. This yields a
contradiction since 0 = m(f) = m(xf) e 1/n because of (3.1).

Thus xI c I for all x E S. Similarly we obtain x1 = I

25

for all x E S. Next let J = [b1,°°',bk] be a left ideal
in I and H0 = {Y1,---Wk], where Yi(f) = f(bi)' Clearly
L H = HD for all a E S. The minimality of H implies

H = H, which means minimality of I. (b) h (a) is trivial.

We can define n—extremely right amenable (n~ERA)

as well as n-ELA. ‘We also can show that IICA4S) is minimal

with respect to RaH = H for all a E S, and [H] = n, where
Ra = ra*, if and only if there is a minimal right ideal I
such that [I] = n and Ix = I for all ix E S. Such ideals

are groups.

Corollary 3.1: If H and H are subsets of A(S)

1 2
Which are minimal with respect to L H = H and R H = H
a 1 1 a 2 2

for all a E S, and [HI] = n, [H2] = m respectively, then
n = m and HI = H2.

Proof: Let I1 and 12 be a minimal left ideal and
a minimal right ideal obtained from H1 and H2 as in
Theorem 3.3. Then x11 = II and 12x = 12 for all x E S
implies 12 = Ile = 11' Thus n = m and H1 = H2.

From Corollary 3.1 we get the following immediately.

Corollary 3.2: The following conditions on S are
equivalent.
(a) A(S) has a subset H which is minimal with

respect to LaH = RaH = H for all a E S and [HI = n.

26

(b) S has a minimal two sided ideal I such that

[II = n and xI = Ix = I for all x E S.

In what follows, we assume that S is a compact
topological semigroup with jointly continuous multiplication.
Then LUC(S) = C(S). Furthermore, by Numakura [19], S has
kernel K, that is, the unique minimal two sided ideal.

From Theorem 3.3 and Corollaries 3.1, 3.2 we get the following.

Theorem 3.4: Let S be a compact topological semi-
group with jointly continuous multiplication.

(a) LUC(S) is Il-ELA if and if only S has a minimal
left ideal I such that x1 = I for all x E S and [I] = n.

(b) If LUC(S) is n-ELA and m-ERA, then n = m.

(c) LUC(S) is n-extremely amenable, that is,
n—ELA and n-ERA, if and only if,the kernel K is a group
consisting of n elements and xK = Kx = K for all x E S.
Furthermore, LUCTS) has a unique invariant mean of the form
l/n igami where H = [mi]?=1<=A(S), and RaH = LaH = H for
all a E S.

(d) If LUC(S) is n-ELA and A(S) has a unique
subset H which is minimal with respect to LaH = H for
a E S, then H is minimal with respect to RaH = H for all

a E S. Consequently, LUC(S) is n-extremely amenable.

Proof of (c): Let I be a minimal two sided ideal

as in Corollary 3.2. Since K is the unique minimal two

27

sided ideal, K = I. The converse also follows from
Corollary 3.2. m = 1/n(ml+---+mn) is a LIM. The conclusion
follows from Corollary 1. (W.G. Rosen [23]) since K is a

group.

Proof of (d): Let I be the minimal left ideal
obtained from H. For any x,y E S, y(Ix) = Ix. Ix is a
minimal left ideal, for suppose not. Then there is a minimal
left ideal J in Ix such that yJ = J for all y E S. By (a)
LUC(S) is [J[(<[Ix[s [I[) -ELA which is a contradiction
since LUC(S) is [II -ELA. By the uniqueness of H, Ix = I.

The conclusion follows from Corollary 3.2.

Next we shall discuss the relation between LUC(S)

and LUC(J), Where J is a right or left ideal in S.

Theorem 3.5: If LUC(S) is n-ELA, then

(a) LUC(J) is n-ELA for an arbitrary right
or left ideal J.

(b) If LUC(J) is n-ELA for some closed left

ideal J, then LUC(S) is n-ELA.

Proof: Let I be a minimal left ideal such that
x1 = I for all x E S, and [I[ = n. Let J be a right ideal
in S. Then J 2.JI = I. Since I is a group, I is a
minimal left ideal in J. Let Ho = [m1,"°,mn}, where
3
I = [a1,°°',an] and mi(f) = f(ai) for all f E LUC(J).

Since I is minimal and LUC(J) separates the points of J,

28

HO(CA(J)) is minimal with respect to LaHo = HO for all
a E J. Thus LUC(J) is n-ELA. Next let J be a left
ideal in S and x be any fixed element in J. Then

Ix ; J and IIxI = III. Furthermore, Ix is minimal in J.

Thus LUC(J) is n-ELA.

Proof of (b): Let LUC(J) be n-ELA, where J is
a closed left ideal, and I be a minimal left ideal in J
such that x1 = I for all x E J, and III = n. Clearly I
is a minimal left ideal in S and x1 = I for all x E S.

The conclusion follows from Theorem 3.4(a).

Let LUC(S) be n-ELA, and J be a left ideal.
We have shown in the proof of Theorem 3.5(a) that for each
x E J, Ix 9 J, moreover, Ix is a group and IIxI = n.
Thus if J is a minimal left ideal in S, then J is a group
and [JI = n. Since the kernel K is a disjoint union of
minimal closed left ideals, we can summarize in the following
manner.

Corollary 3.3: If LUC(S) is n-ELA, the kernel
K is a disjoint union of groups consisting of n-elements.
In particular, if LUC(S) is ELA, then K consists of all

right zeros of S.

Finally, we give a characterization of n-ELA LUC(S)

by L*—invariant measures.

II 2.

'9 Definition 3.1: A Borel measure

HIE) = “(S-1E), m(S) = l for all

 

H on S is an

 

z*-invariant measure if

u
I Borel sets E E B and s E S.

 

Theorem 3.6: The following conditions are equivalent.

I (a) LUC(S) is n-ELA.

(b) There is an L*-invariant measure H of the

I'1

are regular

‘— ——--_..-‘ ‘

multiplicative Borel measures, and {pi} is minimal in the

{Ili:'°'fl}'<i C {Hi} and 1/k(ui+°--+u]'() is an

 

sense that if

I £*-invariant measure, then k = n.

a _ 3'. tun—II:_‘
“.1

 

1 Proof: (a) s (b) Let H = (oi) <:A(S), Where H

is minimal with respect to LaH = H for all a E S, [HI = n,

for all f E LUC(S). Then

. ‘ 1.: c“ 41...”:

and let mi(f) = f(ai)

I = [a1,a2,--°,an] is a minimal left ideal such that xI = I

for all x E 8. Define “i by

 

 

 

.‘.—. _'_
'yau -. -

pi(E) = for E E B.

Clearly the ].'s are multiplicative, regular Borel measures
I1

and
oi(f) = [fdni for f e LUC(S) (i = 1,2,---,n).

Claim: n = l/n(u1+--°+un) is the required

z*-invariant measure. We only need to show that u(s-1E) = u(E)

50

for all E E B and s E S. Clearly u(s-1E) = H(s-l

s-lEflI=[tEI[stEE]= {t EIIst E1530 1}, since I is

EDI).

a left ideal. Let e be an identity in the group I. Then
st = s(et) = (se)t and se E I. Thus s-lE 0 I = (se)-1(E 0 I),
where (se)-1 is an inverse of (se) in I, and

-1 . . . —1
[(se) (E f] I)I = [E n II, which implies u(se) (E 0 I)) =
11(EIW I) or H(s-lE) = u(E). The minimality follows from

the definition of Hi.

(b) = (a): Set m = 1/n(cp1 + m2 +---+ mn) Where
m(f) = [Sfdu and mi(f) = ISdei° Clearly m is a LIM and
each mi is a multiplicative mean, which implies that LUC(S)
is k—ELA for some k(sn). Let H = {@1’...’$k} c [mi] be
minimal with respect to LaH = H for all a E S. If
¢i(f) = f(ai) = Ifdui, then regularity and uniqueness of

ui implies

ui(E) = <

 

for all E E B. As in the proof of (a) = (b), 1/k(ui+---+ui)
is an L*-invariant measure. The minimality of {ui} implies

k = n.

CHAPTER IV
TOPOLOGICAL GROUPS G FOR WHICH

LUC(G) IS n—ELA

Throughout this chapter, G will be a topological
group. A.T. Lau and E. Granirer obtained the following lemma

in [11:p. 255].

Lemma 4.1: Let G be any topological group and

n
assume that LUC(G) has a LIM m of the form m = Z Giwi’
i=1

n
where mi E A(G), Oi > o for all i, Eéffi_=:l, and 91 # wj
if i #’j . Then there is some open normal subgroup N such

that IG/NI < + m and LUC(N) has a multiplicative LIM .

If we check the proof of Lemma 4.1 and Theorem 1.4

carefully, we see that the following result holds.

Proposition 4.1: If LUC(G) is n-ELA, then there
is an open normal subgroup N such that IG/NI = n and

LUC(N) has a multiplicative LIM .

We shall consider the converse of this proposition, or
more generally, if LUC(G/N) is n-ELA and if LUC(N) is
m-ELA, then is LUC(G) k-ELA for some k ? What is the

relation between k and m - n ?

Remark 4.1: It is well known that if G is a local-

ly compact group and LUC(G) is n—ELA, then [GI = n IllI.

31

32

Theorem 4.1: Let G be a topological group and N
an open normal subgroup such that LUC(G/N) is n-ELA and
LUC(N) is m—ELA. If LUC(G) is k-ELA, then m - n is a

divisor of k.

Proof: Since G/N is discrete, by Remark 4.1,

IG/NI = n. By Proposition 4.1 there is an open normal sub-
group NO in N such that IN/NOI = m. Let
[ai: i = l,2,"°,n ° m] be a coset representative of G/No
n-m
and G = U aiNo Claim: IN E LUC(G). Since G is a group,
i=1 0
we only need to show that Us I - I H 4 o for any net
' a No No
sq 4 e, where e is an identity in G. Since NO is
open in G, we may assume sa E No for all d. Then
_ '1 _ _
INo(sat) _ 1 s sat e No e t e sa No - No. Thus IsaIno (t)
|N (t)I = o for all a and t e G i.e. IN E LUC(G).
o 0 _
Since LUc«3) is translation invariant, Ia N = ailIN E LUC(G).
i o 0

Let H = [m : i = 1,2,°°-k] CA(G) and LaH = H
for all a E G. Then m = 1/k(m1 + °°° + wk) is a LIM.
n

-m
= m(IN ) and l = m(IG) = m ( Z31
0 i=1
n - me (I ). Hence m(I ) = l/n - m = z/k for some
No No
L, which was to be shown.

Thus ”(Ia.N ) a.N )
i o i 0

Remark 4.2: To my best knowledge, under the assump-
tions of the theorem, it is an open question as to whether or
not LUC(G) is k-ELA for some k if G is a non-locally

compact or nonabelian group and m > 1. If m = l, we get

33

the exact converse of Proposition 4.1.

Theorem 4.2: Let N be open normal subgroup of G
such that LUC(G/N) is n-ELA and LUC(N) has a multipli-

cative LIM . Then LUC(G) is n-ELA .

Proof: Several techniques for discrete amenable
groups are useful here (see [7I,[22I). Let ml be a mul-

tiplicative LIM on LUC(N) and m2 be a LIM on
LUC(G/N) where m2 = 1/n(cpl + ... + wn) and mi 6 A(G/N).
Let f E LUC(G). Then the restriction of xf to N is in
LUC(N). Let Nx = Ny. Then x = yn for some n e N and
ml(xf) = ml((yn)f) = m1(n(yf)) = m1(yf). Thus F(NX) =
m1(xf) is well defined and F E LUC(G/N) since G/N is

discrete. We define m, éi by m(f) = m2(F) and

A .—

¢i(f) = mi(F). Then m is a LIM on LUC(G) and

.... I)

m = 1/n(c?3l + ... + én)’ where mi E A(G). By Remark 1.1(b),
LUC(G) is k-ELA for some k s n. By Theorem 4.1 we get

k = n.

As an immediate corollary of Proposition 4.1 and

Theorem 4.2, we have:

Corollary 4.1: LUC(G) is n-ELA if and only if
there is an open normal subgroup N such that IG/NI = n

and LUC(N) is ELA.

Theorem 4.3: Let H be a dense subgroup of G.

34
Then LUC(G) is n—ELA if and only if LUC(H) is n—ELA.

Proof: Let LUC(G) be n-ELA. By Corollary 4.1,
there is an open normal subgroup N such that IG/NI = n
and LUC(N) is n-ELA. Since H is dense, IH/HONI = n.
We only need to show that LUC(HnN) is ELA. By Wiley
[15], each f E LUC(HnN) has a unique extension f e LUC(N).
Let m be a multiplicative LIM on LUC(N), and define

_ A _
m(f) = m(f). Then m is a multiplicative LIM on LUC(NnH).

Conversely, let m be a LIM of the form
l/'n(cp1 +...+ wn) Where mi E A(H). Define m on LUC(G)
by m(f) = m(f/H). Then 8' is a LIM on LUC(G) and
_ a n a
m = 1/n(cp1 + ...+ mm), where ¢i(f) = ¢i(f/H). By
Remark 1.1(b), LUC(G) is k-ELA for some k s n. From the

first part of this theorem, LUC(H) is k-ELA. Thus k = n.

Theorem 4.4: If LUC(G) is n-ELA, then LUC(H)

is k-ELA for some k s n, Where H is an open subgroup.

Proof: Let N be an open normal subgroup such
that IG/NI = n and LUC(N) is ELA. Let N0 = NnH, then
N0 is an open normal subgroup of H and IH/NOI = k s n.
We only need to show that LUC(NO) is ELA. Let N =

uN?xO where {xa:o E A] are coset representatives of N/NO.
O. E '

If x E N, there is unique d e A such that x E Nox so

a)

let x = nxo. Define f(x) = f(n), where f; LUC(NO). We

want to show that f e LUC(N). Let YB 4 e. Since no is

35

open, we may assume yB 6 NO for all B. Thus
A A A A
Iny(x) - ef(x)I = [f(yBx) — f(x)I
= [f(an) — f(n)I g “ny - fHNo 4 0 Where x = nxa, thus
f E LUC(N). Define m(f) = m(f), Where m is a multipli-

cative LIM on LUC(N).

Next we consider the n—ELA relation between G and

its dense semigroup S.

Definition 4.1: Let S be a topological semigroup.
Then f E C(S) is uniformly continuous if f is left and
right uniformly continuous, that is, if sa 4 s, then
HsOf - sf” 4 o and Hfsa -fs” 4 o. Denote by UC(S), the
set of all uniformly continuous functions on S. Then UC(S)
is a translation invariant norm-closed subalgebra of C(S)

containing constants.

Lemma 4.2: (a) If UC(G) has a LIM u of the type
n n
u a 2049,, 2a. = 1, a. > o, and cp. are all different multiplicative
i=1 i 1 i=1 i i 1
means on UC(G), then there is an open normal subgroup N
such that IG/NI < + m and UC(N) has a multiplicative LIM.
Furthermore, IG/NI is a divisor of n.

(b) IG/NI = n if and only if H = [ml,...,mn] is minimal

with respect to LaH = H for all a E G.

Proof: The proof for LUC(G) in Lemma 4.1 still
works for UC(G). Thus there is an open normal subgroup N

in G such that IG/NI < + m and UC(N) has a multiplicative

36

LIM. We only need to show that IG/NI is a divisor of n.
Let H = {ml,...,nn] and IG/NI = k, then LaH = H for
all a e G will be shown. Hence u = l/'n(cp1 +...+ on) is

a LIM. Since 6 UC(G) and H is a LIM, as in the

IN
proof of Theorem 4.1, l/k = L/h for some I.

(b) (c) Let Iaiui = 1,2,...k) be a coset representative of

G/N, where a = e, and H0 = [La ..La Since

1 1‘01“ km}-
N = [a E G [Lawi = mi;i = 1,2,...n], LaHO = HO for all
a E G. The minimality of H implies k = n . (m) This

follows from the proof of Lemma 4.2 (a).

The following corollary will be obtained from 4.2

directly.
Corollary 4.2: If UC(G) has aIM‘s ul,u2 of the
i . . . i .
form u. = 7 ail)mil) where oil) a 0, Z oil) = l and
1 -= J J J -= J
_ J 1 J 1
H(l) = [¢{})...,mé[)J is minimal with respect to LaH(1) = H(fl
i
for all a E G, then k = k

1 2'

Now we define n—ELA for UC(G).

Definition 4.2: UC(G) is n—ELA if and only if
UC(G) has multiplicative means m1,m2,...,mn such that
H = [ml,...,¢n] is minimal with respect to LaH EIH for all
a E G.

As in the case of Corollary 4.1, UC(G) is n-ELA

if and if only there is an open normal subgroup N such that

37

IG/NI = n and UC(N) has a multiplicative LIM

Theorem 4.5: Let S be a dense sub—semigroup of G

(a) If S has the finite intersection property for
right ideals and LUC(G) is n—ELA, then UC(S) has a
LIM of the form n = l/'n(cpl +...+ mn)’ where the mi's are
all different multiplicative means on UC(S) and H = [mi]
is minimal with respect to LSH = H for all s E S

(b) If S has the finite intersection property for
right ideals and left ideals and UC(G) is n—ELA, then
UC(S) also has the property of part (a)

Proof: (a) Let f be in UC(S) . By A.T. Lau [15],
f has a unique extension f in LUC(G) . Let u be a
LIM on LUC(G) of the form u = l/n(cp1 +...+ wn)’ where
H = [mi] is minimal. Define a on UC(S) by &(f) = u(f).
Then a is a LIM on UC(S) and a = 1/n(c:)l +...+ Q),

where $i(f) = mi(f) . Claim: All $1 are different and

A
H = [gi] is minimal. Let N be an open normal subgroup of
G such that IG/NI = n and G = SilNLJSEIN...US;1N, where
51 = e and si E S (i = 2,3,...,n). Since IN and
[Si—lN = ‘siIN are in UC(G), Ian = [NIS and

- are in UC(S). Furthermore

Is;1(NflS) ‘ ‘siIInns)
T _

fl
INnS — Since u 18 a LIM,

-l = I -l and I .
si (N05) si N N

and the mi are multiplicative means, for each i there is

38

a unique ki E [l,2,...,n} such that

[o if 3's: ki
mi(Isle) = I
J

 

wi(ls;l(NflS)) = I

 

This implies that all Ti are different and [Qi] is mini—
mal.

(b) By Corollary 2.9 [15], each f e UC(S) has a unique
extension E E UC(G). The conclusion follows from Lemma 4.2

and proof of (a) above.

Theorem 4.6: Let G be a metric group with compact
subgroup K . If LUC(G/K) is left amenable, the LUC(G)

is left amenable.

be a LIM on LUC(K) and m be

Proof: Let m 2

l
a LIM on LUC(G/K) . We define F(Kx) = m1(xf) for
f E LUC(G). We need to show F E LUC(G/K). Since G/K is
metric, it suffices to prove left uniform continuity on

sequences. Let Kxn 4 Kx in G/K. It is shown in [26]

that the coset representatives xn and x may be so chosen

39

that xn 4 X . Then

[(KXn)F(KY) - (KX)F(KY) I = IF(KXnY)- F(KXY) I = Iml(Xan) —m1(xyf) I
g H(xny)f--(xy)fIIK é H(xny)f--(xy)fIIG==IIxnf--xf”G 4 0

thus F E LUC(G/K). Clearly' m(f) = m2(F) is a LIM on
LUC(G)

CHAPTER V

LOCALLY COMPACT TOPOLOGICAL SEMIGROUPS

T. Mitchell obtained fixed point theorems concerning
jointly continuous actions of S on a compact space X for

which LUC(S) has a multiplicative LIM.

Here we attack the same type of problem concerning
separately continuous linear actions of M(G) on a (z.c)-
space E. Mitchell's results neither include, nor are included
in the theorem obtained below. Technically we rely heavily

on J.C.S. Wong [28].

Throughout this chapter we assume that S is a locally
compact topological semigroup with jointly continuous multi-
plication. M(G) will denote the set of all bounded regular
Borel measures on G. (See Hewitt and Ross. [12]). M(G) is
B-space with respect to the total variation norm. It is
known that M(G) is a convolution algebra if we define the
convolution u * v of two measures u,v in M(G) by the
formula

(5.1) .Gf(z) d(u*v) = IG IGf(XY)dudv
for f E CO(G), the continuous functions vanishing at
infinity. It follows immediately that the same formula

is valid for any f E Ll(G,(hA*IVI) (since in the proof of

40

41

Theorem 19.10 [12], only the property that (x,y) 4 xy is

jointly continuous, is used).

Let MO(G) be the set of all probability measures,
i.e., u E MO(G) if and only if u e 0 and H(G) = l.
MOO(G) will denote the set of all multiplicative probability

measures, i.e., u E MD (G) if and only if u E MO(G) and

o
u(Er] F) = H(E) - u(F) for all E,F E B. MO(G) and

MOO(G) (to be shown later) are convolution semigroups.

For each f E LUC(G), define the semi—norm pf on
the linear space M(G) by pf(u) = [IGfduI. The locally
convex topology on M(G) determine by these semi—norms is

denoted by w. (See [24])

Note that each p E M(G) can be regarded as a linear
functional in LUC*(G) (But this embedding might not be

one-to-one, in other words, T is not separated).

Lemma 5.1: M00(G) is a convolution semigroup.

Proof: Let u, v E MOO(G). For any Borel set E,

by (5.1)

(u*v)(E) = IxEd(u*V) = [éxE(xy)dudv = [Gu(Ey_1)dv(y)

where E = {xIxy E E].

y—l

Since u is multiplicative, for any bounded f,g E L1(G,u).

42

p
I f - gdu = ] fdu I gdh
G G G

Thus

(u*\D(E n F) = [G IGXEnF‘XY’ dudv = IGu(Ey_1) s(Fy_1hmu

= [Gu(EY4)d\2’ IGu(Fy_1)dv = (u*v)(E)'(u*v)(F) i.e.,

w*v E Moo(G)°

Lemma 5.2 (J.C.S. Wong [28]):

(a) For each u in M(G), the map m 4 E 0 m is
w* - w* continuous on any norm-bounded subset of LUC*(G).

(b) For each m E LUC*(G), the map u 4 a o m is
continuous with respect to the T—topology on M(G) and
w*-topology on LUC*(G).

(c) If u. venue), then E 05:53? on LUC(G).

(d) A multiplicative mean m on LUC(G) is a

LIM if and only if E C>m = m for any n E MD (G).

o
(e) LUC(G) has a multiplicative LIM on LUC(G)
if and only if there is a net {um} in MOO(G) such that

T
* _
u “a na4 0 for each u E MOO(G).

Proof: (d),(e) are modified statements of (d)
(e) in Lemma 31 [28]. We prove only (e). Let LUC(G)
have multiplicative LEM m. By (d) E C>m = m for all
u E Moo(G). Since the set of all point measures in LUC*(G)
is w*-dense in the set of all multiplicative means, there

is a net [pg] of point measures in MOO(G) such that

43

 

m by (a),(c). We have u ”a ”a u ”a ”a
— — — - T
u o ”a “a 4 u C>m m 0. Thus u ua ”a 4 o for

each u E MOO(G).

Conversely, let “a be a net of measures in
T .
MOO(G) such that p*ua - ”a 4 0. Since the set of all

multiplicative means in LUC*(G) is w*-compact, we have

_ * .
uaw4 m (if necessary, take subnets of ua).

- — = - * — ' - — * — . - =
u 0'm m u 0 (w lim ua) (w 1&m UH)

w* — im - - - w* - im — = w* - i - - - - =
( 1a u 0 L10) ( 1a ua) 1am(u G um um)

* _ ' ‘?"I"' = '
w l&m(u ua ua) 0 by (a),(c). Thus m is a

multiplicative LIM by (d).

Definition 5.1: Let G be a locally compact tOPO-
logical semigroup and E be a separated (L,c) space. An
action T of M(G) on E is a homomorphism of M(G) into
the algebra of linear operators in E. Thus we have a bilinear

map T: H(G) x E 4 E (where (u,x) 4 Tu(x), u E M(G) and

x E B) such that Tu*v = Tu . TV for any u, v E M(G).

If S is a compact subset of E, we say that S is

MOO(G) invariant under T if TU(S) g S for any u E MOO(G).
In this case, T induces an action T: MOO(G) x S 4 S of the

convolution semigroup MOO(G) on 8.

Now we are ready to state the fixed point theorem.

44

Theorem 5.1: Let G be a locally compact topological
semigroup. The following conditions are equivalent.
(a) LUC(G) has a multiplicative LIM.
(b) If T: M(G) x E 4 E is any action of M(G)
on a separated (z,c) space E and S any compact MOO(G)-
invariant subset of E such that
(i) for each u E MOO(G), Tu: S 4 S is continuous
and
(ii) for each s E S, the map u 4 Tu(s) from
M(G) into E is continuous When M(G) has the topology r,

then the induced action T: MOO(G) x S 4 S has a fixed point.

Proof: Assume that LUC(G) has a multiplicative
LIM. By Lemma 5.2 (e) there is a net of measures Ina] in

w
s -
MOO(G) such that u Ua ”a 4 0 for all u E MOO(G).

Let T: M(G) x E 4 E be any action of M(G) on E and S
a compact Moo(G)-invariant subset of E satisfying con-

ditions (i) and (ii) of (b). Consider the net {TH (s)] in S
a
where s is arbitrary,fixed. By the compactness of S,

there is some so in S such that Tu (s) 4 50 (if nec-

a
essary, take a subset of um).

Claim: 50 is a required fixed point<mfthe action T
T:M (G)XS4S.
oo

TU.(SO) = Tu(l&m Tuq(S)) = 182m TU (TU(.Y'(S))= 157m (TUOTUO’) (S)

= 1&m Tu*u (s) = lém (T

a Hutu "L1 (S) + Tu (5)) = lcjim TU (s) =

(1 C1 C1. C1

45

for all u (EMOO(G), since for fixed u E MOO(G), Tu is
continuous, the map u 4 Tu(s) is linear for a fixed 5

and u*ua- ”a 1 0 in M(G). Conversely, let (b) be true and
E = LUC*(G) with the w*—topology and let S be the set of
all multiplicative means in LUc*(G). Define an action T

of M(G) on E by Tu(m) = a C>m. By Lemma 5.2 (a),(b),
(c), T satisfies the continuity conditions (i) and (ii)

of (b). Thus an induced action T: MOO(G) x S 4 S has a
fixed point in S, i.e., E C>m = m for all u E Mo (G).

0
by Lemma 5.2 (d), m is a LIM.

 

BIBLIOGRAPHY

[1]

[2]

[3]

I4]

[5]

I6]

[7]

[8]

I9]

I10]

I11]

I12]

[13]

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