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Amenable Semigroups and Invariant Means, Study notes of Algebra

The concept of amenable semigroups and the existence and uniqueness of invariant means in the context of torsion groups and Burnside's conjecture. It also covers the isometry of m(2) with 1(2)*, the w*-topology, and the semigroup of left invariant means.

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bg1
AMENABLE
SEMIGROUPS
BY
IAHLON
IVY.
DAY
1.
Introduction
We
begin
with
the
definitions
needed
to
formulate
the
results
of
this
paper
and
then
survey
the
known
results
on
existence
and
behaviour
of
invariant
means
on
semigroups.
Then
follow
new
results,
among
which
are
some
cri-
teria
for
existence
of
invariant
means;
these
are
found
in
4.
In
5
it
is
proved
that
an
amenable
semigroup
is
strongly
amenable;
this
settles
a
ques-
tion
that
first
arose
in
an
earlier
paper
[10].
In
8
this
result
from
5
is
plied
to
improve
other
results
of
the
paper
[10]
on
the
relationships
between
means
and
ergodicity.
In
5
the
semigroup
algebra
11(2)
is
discussed;
it
is
used
as
the
principal
tool
in
the
proof
of
the
result
on
strong
amenability.
In
6
is
discussed
the
specialization
to
the
semigroup
algebra
of
a
semigroup
of
an
idea
of
Arens
[1];
Arens
has
given
a
construction
which
makes
an
gebra
out
of
the
second
coniugate
space
of
a
Banach
algebra,
and
has
con-
structed
an
example
of
a
commutative
algebra
whose
second-conjugate
gebra
is
not
commutative.
We
Show
in
6
that
the
semigroup
algebra
of
the
additive
semigroup
of
positive
integers
has
this
pathological
property;
the
proof
depends
on
showing
that
if
an
abelian
semigroup
has
at
least
two
in-
variant
means,
hen
they
cannot
commute
in
the
second-coniugate
algebra.
7
discusses
this
necessary
condition
for
commutativity
in
more
detail.
The
best
result
there
is
that
an
abelian
group
G
has
a
unique
invariant
mean
if
and
only
if
G
is
a
finite
group.
For
general
torsion
groups
the
question
of
uniqueness
and
existence
of
invariant
means
is
dependent
on
whether
Burn-
side’s
conjecture,
that
every
finitely
generated
torsion
group
is
finite,
is
true
or
not.
9
contains
the
proof
that
a
theorem
of
G.
G.
Lorentz
[17],
about
the
set
where
all
invariant
means
are
uniquely
determined,
carries
over
to
amenable
semigroups.
10
introduces
the
concepts
of
amenable
and
introverted
subspaces
of
m(2)
and
shows
how
many
of
the
preceding
results
have
depended
only
on
these
properties
of
m(2).
In
11
these
results
are
applied
to
the
space
C(2)
of
bounded
continuous
functions
on
a
topological
semigroup.
2.
Preliminary
definitions
All
of
the
present
study
will
start
from
the
relationships
between
a
set
which
shall
usually
be
a
semigroup
or
group,
and
certain
function
spaces
de-
Received
November
5,
1956.
The
major
part
of
this
work
was
done
at
the
University
of
Illinois
under
the
sponsor-
ship
of
the
Office
of
Ordnance
Research,
U.
S.
Army.
5O9
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
pf1f
pf20
pf21
pf22
pf23
pf24

Partial preview of the text

Download Amenable Semigroups and Invariant Means and more Study notes Algebra in PDF only on Docsity!

AMENABLE SEMIGROUPS

BY IAHLON IVY. DAY

1. Introduction

We begin with the definitions needed to formulate the results of this paper

and then survey the known results on existence and behaviour of invariant

means on semigroups. Then follow new results, among which are some cri-

teria (^) for existence of invariant means; these are found (^) in (^) 4. In 5 it is

proved that an amenable semigroup is strongly amenable; this settles a^ ques-

tion that first arose in an earlier paper [10]. In 8 this result from 5 is

plied to improve other results of the^ paper [10] on the^ relationships between

means and ergodicity. In 5 the semigroup algebra 11(2) is discussed; it is

used as the (^) principal tool in the proof of the result on strong amenability.

In 6 is discussed the specialization to the semigroup algebra of a semigroup

of an idea of Arens [1]; Arens has given a construction which makes an

gebra out of the second coniugate space of a Banach algebra, and has con-

structed an example of a commutative algebra whose second-conjugate

gebra is not commutative. We Show in 6 that the semigroup algebra of the

additive semigroup of positive integers has this pathological property; the

proof depends on^ showing that^ if^ an^ abelian^ semigroup^ has^ at^ least^ two^ in-

variant means, hen^ they^ cannot^ commute^ in^ the^ second-coniugate^ algebra.

7 discusses^ this^ necessary^ condition^ for^ commutativity^ in^ more^ detail.

The best result there is that an abelian group G has a unique invariant mean

if and only if G is a finite group. For general torsion groups the question of

uniqueness and existence of invariant means is dependent on whether Burn-

side’s conjecture, that every finitely generated torsion group is^ finite, is^ true

or not.

9 contains^ the^ proof^ that^ a^ theorem^ of^ G.^ G.^ Lorentz^ [17],^ about^ the^ set

where all invariant means are uniquely determined, carries over^ to amenable

semigroups.

10 introduces^ the^ concepts^ of^ amenable^ and^ introverted^ subspaces^ of

m(2) and^ shows^ how^ many^ of^ the^ preceding^ results^ have^ depended^ only^ on

these properties of m(2). In 11 these results are^ applied to^ the^ space

C(2) of^ bounded^ continuous^ functions^ on^ a^ topological^ semigroup.

2. Preliminary definitions

All of the present study will^ start^ from^ the^ relationships^ between^ a^ set

which shall usually be a^ semigroup or^ group,^ and^ certain^ function^ spaces^ de-

Received November (^) 5, 1956. The (^) major part of this work was done at the University of Illinois under the sponsor- ship of^ the^ Office^ of^ Ordnance^ Research, U.^ S.^ Army. 5O

(^510) MAHLON Io DAY

termined by 2. These spces re defined s follows; see Bnch [2], pp. 11-

for the (^) cse where 2; (^) is countable.

1(2) is^ the set of ll those rel-wlued functions defined on 2 for which

is finite.

m(2) is^ the set of ll bounded, rel-vlued functions x on 2 with norm

lub Ix()[.

1(2) nd^ m(2) re Bneh sprees.

As in Bnaeh [2], p. 188, eeh Bneh spree B hs conjugate spree B*

eonsi.sting of^11 the^ liner, rel-vlued functions on B; B* is Bneh spree

under the norm

lubllll_, [B(b)1.

We shall be interested in certain elements of m(21)*, but first we remark

that the proof of isometry of m(2) with 1(2)*, given in Banaeh [2], p. 67,

for countable Z, is valid in general; specifically:

For each x in m(2) there is a Tx defined for 11 0 in 1(2;) by

such that

() for^ ech^ x^ in^ m(2), Tx^ is^ in/(2;)*,

(b) T^ is^ liner; that^ is, dditive, homogeneous,^ nd^ continuous,

(c) for^ each^ x, Tx^ x^ [I; that^ is,

lublloll_ Tx(0)^ lub Ix(z)I,

(c) T^ crries^11 of^ m(2) onto^ ll^ of^ 1(2)*.

As in Banch [2], p. 100, ech linear operator U from one Bnch space B

to nother such spce B’ determines coniugte or djoint operator U* from

B’* to B* by means of the formula:

For each y^ in B (^) , (^) Ufl’ is that element of B* for which

(U*y)(b) y(Ub) for 11 b in B.

Bnch shows that U* is lso linear operator nd that U* U [I.

In the special cse of the isometric operator T from m(2) onto 1(2)*, the d-

ioint operator^ T^ is^ lso^ n^ isometry^ from/(2)^ onto^ ll^ of^ re(Z).

The wek topology of^ Bnch spce B is defined, for example, in Hille

[15], p. 23, by means^ of^ neighborhoods; for^ our purposes it will often be con-

venient to^ think^ of^ it^ in^ terms^ of^ convergence; for^ discussion of^ general, or

Moore-Smith, convergence see^ G.^ Birkhoff^ [3], Tukey [22], nd^ Kelley [16];

we shll use the terminology of Kelley [16].

DEFINITION 1. If {b.}, where n runs over directed system 9, is net of

elements in Bnch spce B, then Ibm} converges weakly to b (in symbols,

512 MAHLON M. DAY

{U} has the strong limit U in 2 (in symbols, s-lim U U) means that

lim. Us^ x^ Ux^0 for^ every^ choice^ of^ x^ in^ B.

{U.} has the^ weak limit U in 2 (in symbols, w-lim U U) means^ that

lim [’(U.^ b)^ ’(Ub)]^0 for^ every^ choice^ of^ /’^ in^ B’*^ and^ b in^ B.

In the special case in which B B, the set 2(B) of linear operators from

B into B has still more structure; it becomes an algebra if we define multipli-

cation in 2(B) as follows: For each S and T in 2(B), ST is that element of

2(B) for^ which

ST(b) S(Tb) for^ all^ b^ in^ B.

It is easily seen (Hille [15], page 33) that this multiplication is continuous

in the norm topology;in fact,

so 2(B) is a Banach,algebra.

Three other elementary processes will^ be useful in^ several^ later^ sections.

(1) Let 2 and Z^ be sets, and^ let^ f be^ a function^ carrying 2 onto^ all^ of

This determines a^ linear^ operator, which^ we^ call^ F, from^ m(2’) into^ m(2):

For each x’ in m(Z’), Fx’ is that element of m(2) such that

(Fx’) (o’) x’(fa) for^ every^ a^ in

It can be checked that F is a linear operator carrying m(2 ) isometrically into

m(2). Hence^ F^ is^ a^ linear operator of^ norm^ one^ carrying m(2) onto^ re(Y,’)*.

(2) Let^ 2V^ be^ a^ subset^ of^ Z; then^ there^ is^ a^ natural^ mapping^ II^ of^ m(2)

onto re(Y,’) in^ which^ for^ each^ x in^ re(Z), IIx^ is^ that^ function^ on^ 2V^ which^ agrees

with x on 2;’;

(IIx)(a’) x(z’) for^ all^ a’^ in^ Z’.

Then it can be verified that II is a linear operator of norm one and that^ II*

is an isometry of m(Z’)* into m(Z)*.

(3) If^ is^ an^ element^ of^ 2, it^ determines^ an^ element^ Iz^ of^ ll(Z)^ by^ the

formula 1 if ’

,

(i) (,)^

0 if a’ a.

We shall often inject 2 in this way into ll(Z) and^ identify the^ image

with a and use the same label for both. This^ simplifies the^ notation^ much

more than it adds to the confusion.

  1. Means on m(:)

In the common usage of sophomore calculus, a^ mean^ value, or^ average

value, of^ a^ function^ is^ a^ number^ chosen^ in^ some^ reasonable^ fashion^ between

the least upper bound and greatest lower bound of^ the^ function.^ Here^ we

ask that the choice be made simultaneously for all^ functions^ in^ m(Z) and

made in a linear way.

AMENABLE SEMIGROUPS 513

DEFINITION 1.

each x in m(2;)

A mean on m(2;) is an element of m(2;)* such that for

glb z x() -_<^ t(x) -<^ lub,^ z x().

(A) Each^ mean^ t on^ m(2;)^ has^ the^ following^ properties:

(a) t is in the unit sphere in^ m(Z)*.

(b) If e is the function whose^ value^ is^1 at every point of 2;, then

(c) If x() >-^0 for all^ in Z, then^ (x) (^) => 0. lt ll=i.

(B) If^ an^ element^ t of^ m(2;)*^ satisfies^ (a)^ and^ (b),^ or^ if^ t satisfies^ any

two of the conditions (a’), (b), and^ (c) of^ (A), then^ t is a^ mean on m(2;).

A useful corollary of^ this^ is

(C) The set of^ means^ on^ m(2;) is^ nonempty,^ convex, and^ w*-compact.

DEFINITION 2. An element 0 of ll (2;) is called a countable mean on 2;^ if

0() >-^0 for^ all^ in^ 2;^ and^ if^ 0(’)^ 1.^ A^ countable^ mean^ is^ called

a (^) finite mean on 2;^ if, in (^) addition, the set { (^) #() > 0} is a finite set.

Clearly the^ set^ of^ finite^ means^ is^ norm-dense^ in^ the^ set^ of^ countable^ means.

This nomenclature is^ a^ slight^ abuse^ of^ language,^ since^ the image,^ QO^ or^ Q#,

should, perhaps, more^ properly^ be^ called^ the^ countable^ or^ finite^ mean.^ See

Day [10] for^ the^ next^ result.

(D) If (I)^ is^ the^ set^ of^ finite^ means^ on^ 2;,^ then^ Q(I)^ is^ w*-dense^ in^ the^ set

of means^ on^ m(2;).

Consider next^ the^ operations^ between^ sets^ which^ were^ introduced^ in^2 and their effect on means.

IbEMMA 1. If f maps onto Z’, then F* maps M, the set^ of means^ on

onto M’, the set of means on m(2;’).

If t is in M and

Ft,thenll#’ll -<^ IIFII I]1] 1.^ AlsoF(e’) e, so (^) ’(e’) (F*t)(e’) (Fe’) t(e) 1.^ By^ (B), (^) t’ is^ a^ mean, so

F*M

M’.

If (^) t’ is a mean on m(2;’), let^ m0 {Fx’lx’ e^ m(2;’)}, and^ let^ t0 be^ defined on (^) m0 by o(Xo ’(F-Ixo for^ each^ x0 in^ m0. Then^0 is^ a^ linear^ func-

tional on m0 of norm one; by^ the^ Hahn-Banach^ theorem^ (Banach [2], page

27) 0 has at least one extension^ of^ norm^ one.^ Also

#(e) o(e) t’(e’) 1;

by (B), t is^ a^ mean^ on^ m(2;). But

(F*t)(x’) (Fx’)^ o (Fx’)^ ’(F-1Fx^ ’)^ t’(x’)

for all x’ in^ m(2;’); hence^ Ft t’,^ and^ F^ maps^ M^ onto^ M’.

:LEMMA 2. (^) If 2;’^ is a subset (^) of ,^ then II*M’ M.

AMENABLE SEMIGROUPS 515

DEFINITION 2. A semigroup 2; is called amenable if there is a mean t on

m(2;) which^ is^ both^ left^ and^ right invariant.^ In^ case^ only a^ left^ [right] in-

variant mean exists, 2;^ is called l-[r-] amenable.

We give in this section the many properties of invariant means which had

been announced with or without proofs before this paper and give references

to at least one source for each. These are listed in order with capital letters

to label them; the results called lemmas and theorems later in the section

are new. The first two properties simplify many calculations.

(A) If 2;^ is^ both^ l-^ and^ r-amenable, then^ it^ is^ amenable.

This was proved for groups by Day [10]. To prove it for semigroups is

easiest after 6 of this paper; if and p are, respectively, left and right in-

variant, it^ will^ be^ shown^ in^ 6,^ Corollary^ 2,^ that^ k^ (R)^ p^ is^ both^ left^ and^ right

invariant.

(B) An 1-[r-] amenable^ group is^ also^ r-[1-] amenable; and^ therefore is^ ame-

nable.

This also was proved in Day [10]; basically it depends on the fact that the

operation g

g-1 transposes the order of products, and therefore inter-

changes left^ and^ right.

One of the earliest studies of invariant means is that of yon^ Neumann

[18]. The^ groups which^ he^ calls^ measurable^ can^ be^ seen^ to^ be^ those^ which

are called/-amenable here; (A) and (B) show that this class coincides with

the class of amenable groups, which shows that many of the results in Day

[10] are^ consequences^ of^ results^ in^ vsn^ Neumann^ [18].

An example, (4) at the beginning of this section, shows that nothing like

(B) is^ true^ for^ semigroups^ in^ general.^ In^ that^ semigroup,

( x)(’) x(’) x(’)

so every l is the identity and every mean is left invariant. (Means always

exist.) But^ (r^ x)(’)^ x(’z)^ x(z)^ for^ all^ z’,^ so^ r x^ x(z)e,^ and^ if

is right invariant, then (r x) (x) x(z)(e) for all z and x. Therefore

x is a constant function for all x in m(2), or else t(e) 0 and p(x) 0 for

all x in m(2;). But if 2;^ has more than one element in it, then m(2) has non-

constant functions in it, so a semigroup of the type in example (4) has no

right invariant linear^ functionals on^ it unless it has^ but one element.

Next come techniques^ for^ creating new^ amenable^ semigroups from^ given

ones.

(C) if is^ an^ (/-)[r-] amenable^ semigroup and^ f a^ homomorphism of

onto Z’, then Z’^ is^ (/-)[r-] amenable.

In (^) (B) two possibilities, left or right, are considered. In (C) and in 5, three choices left or right or (^) both, are possible, the same choice to be used all the way through the sentence.

516 MAttLON M. DAY

One proves that

F’t, is^ left^ invariant^ on^ m(Z’) if^ is^ left^ invariant

over 2, and similarly for right invariant means. See Day, [12] for groups.

(D) If G^ is^ a^ (/-)[r-] amenable^ group,^ so^ is^ every^ subgroup.

See Day [12]. Also this has been published recently by F01ner [14].

The proof will be given in connection with a stronger result in 7, Theo-

rem 2. It has not been published before.

This result too may fail for semigroups. As an example let Z be any

non-amenable semigroup, and let 2]^ contain 2V and one new element 0 such

that 0’^ ’0 00 0, and^ 2’^ is^ a^ subsemigroup of 2:.^ 2:^ has^ an^ invariant

mean: (x) x(0). The subsemigroup 2V has not.

(E) Let^ H^ be^ a^ normal^ subgroup^ of a^ group^ G^ such^ that^ H^ and^ G/H are

amenable; then^ G^ is^ amenable.

See von Neumann [18] for left amenable; (B) and (A) complete the proof

(see Day^ [10]).

(F) Suppose that {Z is a set of amenable^ subsemigroups of a^ semigroup

such that (a) (^) for each m, n there exists p with ,,^ ,^ u ,^ and^ (b) 2 tJ, ,^ Then is amenable.

yon Neumann [18] has this for a well-ordered system of subgroups of a

group. In the present generality it is^ in^ Day^ [10].

To be sure these methods of construction have^ some^ value, we^ need^ ex-

amples. We know already a^ non-amenable^ semigroup^ but^ we^ need^ also

(G) A^ free group on^2 generators^ is^ not^ amenable.

This can be gotten from yon^ Neumann [18]; it^ is also in Day [10].

with (D) it asserts that

Used

(G’) A^ free group^ on^2 or^ more^ generators^ is^ not^ amenable.

group has^ a^ free subgroup^ on^ more^ than^ one^ generator.

No amenable

We have two basic families of amenable semigroups.

(H) Every^ abelian^ semigroup^ is^ amenable.

For groups this is^ in yon^ Neumann^ [18]; for^ semigroups^ in^ Day^ [9].

(I) Every^ finite group^ is^ amenable.

More precisely, for later use note that there is exactly one invariant^ mean

(left or^ right)^ on^ a^ finite^ group;if^ G^ has^ n^ elements,^ then

(x) (^) n-la x(g) for^ all^ x^ in^ m(G) is that mean.

A finite semigroup need not have any invariant mean.^ If^2 is^ a^ finite

semigroup in which ’

’, 2 is not amenable if^ it^ has^ more^ than^ one^ ele-

ment.

518 MAI-ILON M. DAY

(Locally finite means thut every finite subset of G generates a finite sub-

group of G) (Day [10].)

R. Baer calls a group G "supersolvable" if every nontrivial homomorphic

image of G has a nontrivial, abelian, normal subgroup; we shall use the

term Baer group for a group such that every nontrivial homomorphic image

of G hs a nontrivil normal, amenable subgroup.

THEOREM 1. Every Baer group is amenable.

This (^) depends on

LEMMA 1.^ Every^ group G^ contains^ a^ normal, amenable subgroup G which

contains all other normal, amenable subgroups of G.

Let H} be the family of normal, menble subgroups of G. The family

is closed under the (^) process of (^) taking unions of (^) increasing simply ordered sub-

sets, so^ Zorn’s^ lemma^ (see Kelley^ [16], page^ 33) applies to^ give a^ normal,

abelian G not included in any other H in /H}. If, now, H e {H} and H is

not a subset of G1, let G smallest subgroup of G spanned by G and H;

then G is normal in G’ and G’/G is isomorphic to H/G H. Hence G

and G’/G re amenable. By (E), G’ is amenable.

But if g’ is a word in G’ and (^) g e (^) G, ggg-^ is word in G (^) too, since (^) G and

H are both normal. Hence G’ is a normal, amenable subgroup of G which

contains G this^ contradiction^ shows^ that^ H^ G.

To prove Theorem 1 we suppose that G is a Baer group and that G c G.

Then G/G contains a normal, amenable subgroup A {1}; also G’, the in-

verse image of A, is an extension of G by A. Because A nd G are me-

nable, (E) asserts^ that^ G’^ is^ amenable.^ G’^ is^ also^ normal.^ This^ contradicts

Lemm 1.

Note again how the free group furnishes an example to prevent the as-

sumption that a group must have a largest amenable subgroup. If a and b

are the generators of free group G, then the infinite cyclic subgroups on

these generators are both amenable. But G is the only subgroup of G con-

taining both a and b, and G is not amenable.

Since not every subsemigroup of an amenable semigroup is amenable,

the following partial results add some information.

THEOREM 2. Let F be^ a semigroup with^ a^ left invariant^ mean^ t. Suppose

that Z is a subsemigroup of F^ such^ that^ t(x) O, where^ x is^ the^ characteristic

function of^.^ Then^ is^ left^ amenable.

Proof. Let^ h^ denote^ the^ left^ translation^ operator^ in^ m(2),^ and^ for^ each

in F define (^) i from m(F) into m(F) by

ix (lx)x^ for^ all^ x^ in^ m(r).

AMENABLE SEMIGROUPS 519

Define T from m(2) into re(r) by

(Tx)(,)

{:(/)ifif 2,2,

and let v T*/t(x). Then v is a mean on m(2).

that for each in 2 and x in m(2)

T(X x) i. (Tx).

It can easily be checked

Now let us fix a in 2. Let v l x i x, so v(/) x(z/) x(o)x().

This shows that v(,) can either be 0 or i and takes no other value, and,

therefore, that^ v^ is^ the^ characteristic^ function^ of^ a^ set^ E.^ It^ is^ clear^ that

Let us take any (^) /in F and consider the sequence (^) z i (^) => 0}. If possible, suppose that there exists (^0) =< /^ <: j such that ak, and a,^ both are in E. It follows from (^) (b) and e E that a,^ e 2, which again with (b) shows that , (^) e E, and this is a contradiction. Thus, either no belongs to E, or

there is exactly one j such that (^) av e E. Now let n (^) > 0 be an integer, and let us consider

)n EO<_i<_n l v.

Then for each /in 1

w. (,)^ 0_<__<^ v(z)^0 or^1

by our previous considerations; therefore,

thus

(n-t 1)(v)- t(w ___<^ 1.

As this is true for every n, g(v) 0. Now if we take any x in m(2) such

that x (^) --< 1, then we can easily check that

and therefore

or

--v (^) <= l (Tx) i(Tx) (^) <= v,

t(l Tx) t(i Tx) O,

t(i Tx)^ t(l^ Tx)

and by homogeneity it follows that

(i Tx) (l Tx) (Tx)

for all x in m(2). From^ (a) this^ can^ be^ written^ as

t[T(ho x)] [Tx],

and this is the same as

x)

forllxl] -<^ 1,

AMENABLE SEMIGROUPS 521

For every a in 2 and x in m(2;), r* m(x) m(x) is a subnet of (^) r* t(x)

t (x);^ hence^ it^ also^ tends^ to^ zero.^ By^ (A),^ t is^ right^ invariant.

In Day [10] it^ was^ proved that

(C) A semigroup 2;^ is^ (/-)Jr-] amenable^ if^ and^ only if there^ exists a^ net

{.} of^ finite^ means^ such^ that^ the^ net^ {Q} is^ w*-convergent^ to^ (/-)[r-]

invarianceo

This follows from (A) and (B) of the present section and (C) and (D) of

Observe that [Q0} is w-convergent to zero in m(2;) if and only if

is w-convergent to 0 in/1(2;); hence we can convert this to

(C’) A^ semigroup^ is^ (/-)[r-] amenable^ if^ and^ only if^ there^ exists^ a^ net

[} of^ finite^ means^ such^ that^ {} converges^ weakly^ to^ (/-)[r-] invariance.

A condition formally stronger than amenability was used in Day [10] and

was named, for groups, in an abstract of that period, Day [11].

DEFINITION 2. is called (r-)[/-] strongly amenable if there exists a net

{} of^ finite^ means^ convergent^ in^ norm^ to^ (right)^ [left]^ invariance;

that is such that for each

(lim r^ Q^ Q^ o)^ [lim^ I*^ Q^ Q^ 0].

The notation^ was^ so^ unwieldy^ that^ while^ many^ properties^ of^ amenable

groups could be shown to have^ analogues for^ strongly amenable groups, it

was not then possible to decide whether every amenable group is strongly

amenable, nor was it convenient to discuss strong amenability of semi-

groups. This can^ be^ handled^ by changing the^ problem to^ one^ stated in 1(2;), and^ this^ in^ turn^ requires a^ discussion^ of^ a^ multiplicatio (^) n operaion

which makes a Banach algebra out of 11(2;). This definition of multiplica-

tion is a familiar one in the classical case where Z is a finite semigroup; see,

for example, van der Waerden [23], page 49.

DEFINITION 3. For each choice of 01 and 02 in ll(Y,), define 01 02 by the

formula

If ech element in Z is identified with the vector Iz in l(Z) (see defi-

nition in 2, (3)), then it is easy to check that for ll z in

Hence if we drop the I, it will cuse no confusion in the multiplication in

Z, since I is an^ isomorphism of Z into l(Z). Hereafter^ we^ shll^ use the

symbol a both for^ in^ nd Iz in l(Z). This^ gives the^ formul

for every 0 in l().

Then this multiplication in l(Z) lso determines right nd left transla-

tion operations 0z nd z0 in/(Z);

(0)(’) ,^ 0(z2) nd^ (0)(’) ,=,^ 0(z).

522 MAHLON M. DAY

Direct calculations with the definitions prove

(D) r(Qo) Q(o0-) and^ l*(QO)^ Q(0-o) for^ all^ in^ 2:^ and^0 in

Therefore the elements^ r* Q^ Q.^ and^ l* Q^ Q^ which^ were^ dis-

cussed in the^ definitions^ of^ weak^ and^ strong^ amenability are^ images^ under^ Q

of the elements^. and^ 0-. Under^ the^ mapping^ Q, norms^ are

preserved and^ weak^ convergence^ to^ zero^ in^ 11(2) is^ equivalent^ to^ weak*

convergence to^ zero^ of^ the^ images^ in^ m(2:)*. This^ proves^ the^ following^ re-

formulation of the preceding amenability conditions.

LEMMA 1. A semigroup is amenable (strongly amenable) if and only if

there exists a net {q,} of finite means such that for every (r^ in

lim(--) 0 lim(--)

in the weak (norm) topology of

This displays clearly that^ strong^ amenability is^ not^ less^ of^ a^ restriction on a semigroup than^ is^ amenability. The^ purpose of^ this^ section^ is^ to^ prove these two^ conditions^ equivalent, but^ we^ now^ turn^ aside^ from^ the^ main^ stream of that proof to^ give some^ information^ about^ the^ semigroup algebra which will be needed.

LEMMA 2.^ Suppose^ that^01 and^ 0.^ are^ in^ 11(), or^ are^ countable^ means, or

are finite means; then^ the^ same^ property^ is^ possessed^ by^1 0. Hence^0 and

0(r have for each (r in the same of these properties as has O. Also multiplica-

tion in 11() is^ associative, so^ (0-0)# ((0#). Finally,^01 02 01 [1[1 02

and

01 02 ae21 01(O")0"02^ ae21 02(IT)01^ 0".

Hence 01 0. is an element of 1(2:) if the 0 are, and 11(2:) is a Banach algebra,

possibly without^ unit.^ When^ the^ numbers^ 0(0-) are^ all^ nonnegative,^ then

the only possible proper inequality in the above chain is prevented from

occurring, and then 01 0 01 0 [[;in particular, 01 0 is a countable

mean if the 0 are countable or finite means. When the 0 are finite means,

01 0.() 0 except^ in^ the^ finite^ set

and (^) 0(0-) (^) > 01.

COROLLARY 1. The set of countable means^ and^ the^ set^ of finite means^ are

subsemigroups in the multiplicative semigroup of the^ Banach^ algebra 11().

524 MAHLON M, DAY

THEOREM 1. A semigroup is amenable if and only if it is strongly ame-

nable.

The characterizations of Lemma 1 show that a strongly amenable semi-

group is^ amenable.^ If, on^ the^ other^ hand, 2 is^ amenable, Lemma 1 asserts

the existence of net q^ (q of finite means such that in the weak topol-

ogy of^ 11(2) we^ have^ for^ each^ of^2

lim ( ) 0 lim ( ).

Let be (^) any finite subset of ,^ and enumerate the elements of in some order as , , ..., (^). Then tends to zero (^) weakly in (^) /(Z);

by Lemma^4 there^ is^ a^ net^ of^ finite^ averages^ of^ elements^ far^ out^ in^ such

that lim 0. By Lemma 3 the weak limit of

is zero for j 2, k; hence there is u subnet {} of such that

lim 0 for^ j^ 1,^ 2,^ while^ a still^ tends^ weakly

to zero for j (^) 3, ..., k. Continuing by induction there exists a subnet

{q} such that

lim aq^0 for^1 j^ k.

If Z is finite, this net will do to show one side of strong amenability if

Z. If Z is (^) infinite, let g^ be the cartesian product of ,^ the directed

system of integers, with A, where A is the net of all finite subsets of Z or-

deredby so (n, ) >^ (n’, ’) meansn >^ n’and

Then for each

i (n, ) let (i) (n, ) be so chosen that

(1) (n, 8) is finite average of elements ,^ m^ n, and

(2) for each element of

(n, ) (n, )[ < 1/(number of^ elements^ in^ ). Such an element (^) (n, ) can be chosen from the net ,^ associated to by the

construction of the preceding paragraph, for each ,q is a finite average of

finite means , and is therefore a finite mean (^) itself, and, once is chosen

and n given as^ well, q for q large^ enough^ uses^ only^ elements^ with^ m^ n

and can be taken as close to zero in norm as may be desired. This net (^) {} is a net of finite averages of elements far out in ,^ and lim 0 for^ each^ in^ Z.^ By^ Lemma^3 the^ weak^ limit^ of^ a^ , still is zero for each in.^ Hence the argument just used will yield a net ’

which is norm convergent to right invariance as well as to left in-

variance. This proves the theorem by displaying a net with the charac-

teristic property which Lemma 1 says is equivalent to strong amenability.

It is worthy of note that there is truly something that^ needed^ proof^ in

this theorem. It is well-known (Banach, page 137, gives the^ case^ where^ Z

is countable, but the proof does not depend on^ that^ property of^ Z) that^ for

sequences in lx(Z) weak convergence to an element is^ equivalent to^ strong

convergence to the same element. But this is^ a theorem^ for^ sequences; for

nets in general the facts that (a) weak and norm^ topologies are^ distinct^ in

AMENABLE SEMIGROUPS 525

11(2:) if^21 is^ not^ finite, and^ (b) these^ topologies can^ both^ be^ determined^ by

convergence of nets, show that a net {0.} might converge weakly to zero

while at the same time it need not con.verge to zero in norm.

A recent theorem of Flner [14] gives two new characteristic properties of

amenable groups. THEOREM OF (^) FLNER. Amenability (^) of a group G is equivalent to each (^) of the following conditions: (a) For each number tc^ such that 0 <-^ tc^ < 1 and each (^) finite subset (^) "r of G,

there is a finite subset E of G such that for each g in ".

(no. (^) of elements common to E and to gE)/(no, (^) of elements in E) (^) > t. (b) There is a number ko, 0 < ]Co^ < 1, such^ that^ for each^ choice^ of finitely

many, not^ necessarily distinct, elements^ gl g., g,,^ e^ G^ there^ is^ a^ finite set

E <- G such that

n-l_, (no. (^) of elements^ common^ to^ E^ and^ g^ E) >-^ leo(no,^ of elements^ in^ E).

For groups this yields another proof that left amenability is equivalent to

strong amenability. For a given finite subset of G and a given e (^) > 0, take E by (^) FOlner’s condition (^) (a) with lc 1 s; then set ,(g) (^) 1/I (^) EI

if g e E, 0 if g e E. This net converges in norm to left invariance.

It is not now clear whether Flner’s condition can be derived from strong

amenability in general. A related question is" How much tampering can a

net of means strongly convergent to invariance take before it loses its de-

sirable property. In this vein we have two results

LEMMA 5. If {q,} is a net of finite means which is^ weak^ [norm] convergent

to right lleft} invariance, then for each 0 in 11() such^ that^ e(O) 1,

in the weak [norm] topology in l(Z).

For one typical case of the proof assume that w-lim (

for each^ in^ Z. Then^ for^ each^ finite^ mean^ we^ have

e 2 ()^2 ()(^ ); therefore (^) b. tends weakly to zero. But each mean 0 in^ 11(21) can^ be^ ap- proximated arbitrarily closely in norm by a finite mean b, and^ for^ all^. in wehave[l- -01] (^) --< I1- 011. For^ each^ x^ in^ m(2)^ and^ each^ e^ >^ 0, take (^) I1 0ll < e, and then (^) takeso that lx( )1 < e.^ Then

hence {. 0 } tends weakly to^ zero.^ Similar^ proofs^ yield^ the^ corre-

sponding results for {0 .} and^ for^ norm^ convergence.

LEMMA 6. If is a^ semigroup and^ q^ {q,}^ is^ a^ net^ of finite means^ con-

AMENABLE SEMIGROUPS 527

and 0 in 11(2), the first definition gives for all 0’ in 11(2;),

=

o()[( e )(o’)] o()x .

Ts shows that for ech x nd 0,

x (^0) O()z.

But

(x )(o’) x(o’). x()o’()

, (^) x(’)o’(’) ( x) (o’).

:SO

x=l.x.

Hence

( x)() (x ) ( x) : (x).

We take this as our basic definition, now that we have checked that it agrees

with Arens’s definition; that is, we rewrite the definitions for our case as"

xa=lx.

( x)() (z ) ( x) ( )(x).

( )(x) ( ).

We add two new definitions

o() (^) r. O()r

We are now ready to^ describe^ the^ properties of this^ multiplication in^ m()*

and to show^ how^ invariant^ means^ appear.

LE 1. (Arens) is associative and distributive; also, the norm of the

product is^ not^ greater^ than^ the^ product^ of the^ norms.

For each x

Ix (^ )](x)^ x[,(.^ )^ x],

and

(. ) x (^ )(x^ )^ .[^ (x^ )]^ .[^ x]

for all x and a. Also

(x .) (x )( x) x[. ( x)],

and

( ) [( x)]

for all x and a. But^ for^ all^ a’, x, and^ a

( ) ( x)(’)^ (.,^ x)^ (,^ x)^ (

528 MAHLON M. DAY

Hence for all and x

o( ( z) (o ).

Hence the last expressions in the second and fourth equations of this proof

are equal for all x and a; hence the last expressions in the first and third

equations are equal for^ all^ x.^ Hence E)^ is associative.

For distributivity we^ check^ first^ that ( + v) (R)^ x x (^) + v x for all , (^) v in m()* and (^) x in re(Z); this is (^) true because for all a in Z

( (^) + ) x ( + )(x ) (x ) (^) + (x ) ( x)() (^) + ( x)() [ x (^) + zl(). Then for all x

IX (^ + ,)](x)^ x[(^ + )^ x]^ x[^ x^ + l x(, x) (^) + x( x) (x )(x) (^) + (x )(x)

=x,+x.

To prove the boundedness, if g, e m()* and x e m(Z), then

I( )(x)l^ ( x)l II,^ llll x^ II, and for^ each^ a

Henceif.]]x] 1,^ then]]^ x^ ]],so

LEMMA 2. (^) U 0 e/I(Z) and e m(Z)*, then QO l, a Qo re.

For each x in m(Z)

$)(x) ( x) ( 0(), x) 0()( x)

o()(x ) o()[( x)()]

(qo)( x) (qo )(x).

For the other conclusion, start with in m()* and^0 in/(); then

Qo o() Q.

Then for^ each^ a^ in^ Z^ and^ x^ in^ re(Z)

( q)(x)^ (Q^ x),

and for each r in Z

(Q x)() (Q)(x ) (Q)(, ) ( x)()

x() (r, )().