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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.
- 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)
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, )().
