# The Characterizations of Extreme Amenability of Locally Compact Semigroups

Authors: Hashem Masiha
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We demonstrate that the characterizations of topological
extreme amenability. In particular, we prove that for every locally compact
semigroupS with a right identity, the condition μ⊙(F×G)=(μ⊙F)×(μ⊙G), for F, G in M(S)∗, and 0<μ∈M(S), implies that μ=εa, for some a∈S (εa is a Dirac measure). We also obtain the
conditions for which M(S)∗ is topologically extremely left amenable.


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## Body

## 1. Introduction

LetS be  a locally compact (Hausdorff)  semigroup
such that its multiplication is separately continuous. We denote by m(S) the Banach algebra  under the supremum norm of
all bounded real-valued functions on S.
For a topological semigroup S,
let BM(S) and CB(S) be the closed subalgebras of m(S) consisting of all Borel measurable functions
and all continuous functions on S,
respectively. Let C0(S) be the subalgebra of CB(S) consisting of the functions which vanish at
infinity. Let M(S) be the Banach space of all bounded regular
Borel (signed) measures on S with a total variation norm. Let M0(S)={μ∈M(S):μ≥0and∥μ∥=1} be the set of all probability measures in M(S).It is known thatM(S)≃C0(S)∗ via the correspondence μ→μ¯,
where μ¯(f)=∫fdμ for any f in C0(S) [1, Section 14]. Consider the continuous dual M(S)∗ of M(S).
An element M in M(S)∗∗ is called a mean on M(S),
if M(1)=1 and M(F)≥0,
whenever F≥0.
An equivalent definition for a mean is that(1.1)inf{F(μ):μ∈M0(S)}≤M(F)≤sup{F(μ):μ∈M0(S)},for any F in M(S)∗.
We also note that M∈M(S)∗∗ is a mean if and only if ∥M∥=M(1)=1.
Each probability measure μ in M0(S) is a mean on M(S)∗∗ if we put μ(F)=F(μ),
for any F in M(S)∗.
An application of Hahn-Banach separation theorem shows that M0(S) is weak* dense in the set of all means on M(S)∗.Under point-wise operations and the supremum norm,C0(S) becomes a Banach algebra. Arens product can
thus be defined in C0(S)∗∗.
In particular, we have the following defining formulas for any f,g in C0(S), m in C0(S)∗,
and θ,ϕ in C0(S)∗∗:(1.2)(m⊙f)(g)=m(fg),(ϕ⊙m)(f)=ϕ(m⊙f),(θ⊙ϕ)(m)=θ(ϕ⊙m).This product induces a multiplication inM(S)∗ via the identification M(S)≅C0(S)∗.
Since M(S) is a set of measures beside being the
continuous dual of C0(S),
this multiplication in M(S)∗ is richer in content than just a generic Arens
product in the second dual of a Banach algebra, and it is different from the
point-wise multiplication in C0(S).ForF,G in M(S)∗,
we denote the multiplication of F and G by F×G.
In [2], it is shown that F×G is defined via the following three steps.(a)
For anyμ∈M(S) and f∈C0(S),
the measure μf∈M(S) is defined by(1.3)∫gdμf=∫gfdμ,∀g∈C0(S).(b)
For anyμ∈M(S) and F∈M(S)∗,
the measure F×μ∈M(S) is defined by(1.4)∫fd(F×μ)=F(μf),∀f∈C0(S).(c)
For anyF,G∈M(S)∗,
the functional F×G∈M(S)∗ is defined by(1.5)(F×G)(μ)=F(G×μ),∀μ∈M(S). We can conclude that M(S)∗ becomes a commutative Banach algebra with an
identity [2].For the topological semigroupS,
we define(1.6)(rsf)(t)=f(ts),(lsf)(t)=f(st),where s∈S,t∈S,
and f∈m(S).
Hence, rs and ls are the operators defined on m(S) onto m(S).
A subset X of m(S) is called a left (right) translation
invariant, if lsX⊆X(rsX⊆X) for all s∈S.
It is well known that both BM(S) and CB(S) are left and right translation invariants [1].
Let S be a topological semigroup, which for each
compact subset E of S and a∈S,(1.7)a−1E={s∈S:as∈E},(Ea−1={s∈S:sa∈E})is compact. Then, C0(S) is a left (right) translation invariant.LetX be a left (right) translation invariant
subspace of m(S) containing the constant function 1.
A mean m on X is called a left (right) invariant, if(1.8)m(lsf)=m(f)(m(rsf)=m(f)),for every s∈S and f∈X [1]. If m(S) has a left invariant mean, then S is said to be left amenable [3]. If m(S) has a multiplicative left-invariant mean, then S is said to be extremely left amenable (see
[4, 5] for more details).Suppose thatS is a locally compact semigroup, then for μ,ν∈M(S),
the convolution μ∗ν is defined by(1.9)∫fdμ∗ν=∬f(st)dμ(s)dν(t)=∬f(st)dν(t)dμ(s),where f∈C0(S).
Hence, M(S) with a convolution μ∗ν as multiplication is a Banach algebra.Now, forF∈M(S)∗ and μ∈M(S),
the linear functional lμ:M(S)∗→M(S)∗ is defined by(1.10)(lμF)(ν)=F(μ∗ν),∀ν∈M(S).We denote lμF by μ⊙F.
Similarly, the rμF=F⊙μ is defined by(1.11)(rμF)(ν)=(F⊙μ)(ν)=F(ν∗μ),∀ν∈M(S).A meanM on M(S)∗ is called left invariant (LIM) if M(lεaF)=M(εa⊙F)=M(F) for all F∈M(S)∗ and for all a∈S.
Also, a mean M on M(S)∗ is called topological left invariant (TLIM) if M(lμF)=M(μ⊙F)=M(F) for all F∈M(S)∗ and for all μ∈M0(S).
A topological left invariant mean M on M(S)∗ is called a multiplicative topological
left-invariant mean (MTLIM) if M(F×G)=M(F)M(G) for all F,G∈M(S)∗.
If there is an MTLIM on M(S)∗,
we say that S is extremely topological left amenable (ETLA).
For results concerning ETLA and ELA semigroups, see [1, 6].In this paper, we demonstrate that the Arens product
and multiplication onM(S)∗ defined by (1.3), (1.4), and (1.5) are associative
(see Lemma 2.4). In [7], it is proved that(1.12)(F×G)⊙μ=(F⊙μ)×(G⊙μ)is valid, for all μ∈M0(S) and F,G∈M(S)∗.
This means that Arens product ⊙ distributes over multiplication × on M(S)∗ from right, for all μ∈M0(S) and F,G∈M(S)∗.
Note that the multiplication in M(S)∗ is different from the point-wise
multiplication in C0(S).
We show that Arens product ⊙ distributes over multiplication × on M(S)∗ from left, when F,G∈M(S)∗,
and μ is a Dirac measure (see Lemma 2.2(ii)). Also,
it is shown that if S is a locally compact semigroup with a right
identity and Arens product ⊙ distributes from left over multiplication ×,
then μ must be a Dirac measure (see Theorem 3.1). In
the rest of this paper, we give some characterizations on ETLA of locally
compact semigroups.


## 2. Preliminaries

In this section, we offer some results which are
useful in the sequel. For more details, refer to [1, 2, 8]. Let |μ| be the total variation of μ,
where μ∈M(S) and L∞(S,|μ|)=BM(S)/N(μ)
are the quotient
Banach algebra with a quotient norm ∥⋅∥μ,∞
;
N(μ) is the closed ideal of BM(S) consisting of all locally |μ|-null functions. Consider the product linear
space(2.1)∏{L∞(S,|μ|):μ∈M(S)}.An elementf={fμ}μ∈M(S) is called a generalized function on S,
if the following conditions are satisfied:(a)
∥f∥=sup{∥fμ∥μ,∞:μ∈M(S)}<∞,(b)
forμ,ν in M(S) with μ≪ν,
we have fμ=fν|μ|-a.e. Let GL(S) be the linear space of all generalized
functions on S.
It is known that GL(S) is a Banach space with the norm defined by the
formula (a) and that GL(S)≅M(S)∗ via the isometric Banach space isomorphism, φ:GL(S)→M(S)∗,
where (φf)(μ)=∫fμdμ for any μ in M(S) and f in GL(S) (see [8, 9]). A function f in BM(S) can be treated as an element in GL(S) with fμ=f for all μ in M(S).
The space BM(S) is thus a subspace of GL(S).Forf∈BM(S),μ∈M(S),
and ν∈M(S),
we define f⊙μ and μ⊙f in L∞(S,|ν|) by(2.2)f⊙μ(s)=∫f(st)dμ(t)=∫lsfdμ|ν|-a.e.,μ⊙f(s)=∫f(ts)dμ(t)=∫rsfdμ|ν|-a.e.It is shown thatf⊙μ,μ⊙f∈GL(S) [8]. If f∈CB(S),
then the above equalities hold everywhere, and f⊙μ and μ⊙f are in CB(S).
Also, if f∈BM(S) and a∈S,
then(2.3)f⊙εa(s)=∫f(st)dεa(t)=f(sa)=(raf)(s),(2.4)εa⊙f(s)=∫f(ts)dεa(t)=f(as)=(laf)(s).Hence, f⊙εa and εa⊙f belong to BM(S).Lemma 2.1.
The mapφ:GL(S)→M(S)∗ defined by (φf)(μ)=∫fμdμ for any μ in M(S) and f in GL(S) satisfies the following statements.(i)
For anyf∈BM(S) and μ∈M(S),(2.5)φ(μ⊙f)=μ⊙φf,φ(f⊙μ)=φf⊙μ.(ii)
For anya∈S and f∈BM(S),(2.6)φ(laf)=εa⊙φf,φ(raf)=φf⊙εa.Proof.
We have(μ⊙f)ν=μ⊙fμ∗ν for any ν∈M(S) [8]. Hence,(2.7)φ(μ⊙f)(ν)=∫(μ⊙f)νdν=∫(μ⊙fμ∗ν)(s)dν(s)=∬fμ∗ν(ts)dμ(t)dν(s)=∫fμ∗νdμ∗ν=φf(μ∗ν)=(μ⊙φf)(ν).Thus, φ(μ⊙f)=μ⊙φf.
Similarly, φ(f⊙μ)=φf⊙μ.
This proves (i).
From (i) and (2.4), part of (ii) is trivial.Lemma 2.2.
For eacha∈S and F,G∈M(S)∗,
we have(i)
(F×G)⊙εa=(F⊙εa)×(G⊙εa),(ii)
εa⊙(F×G)=(εa⊙F)×(εa⊙G).Proof.
(i) For eachμ∈M(S),
from (2.3), we have(2.8)((F×G)⊙εa)(μ)=rεa(F×G)(μ)=(F×G)(μ∗εa)=F(G×(μ∗εa)).But, for g∈C0(S),
from (1.4) and (1.9), we have(2.9)∫gd(G×(μ∗εa))=G((μ∗εa)g)(by(1.4))=G(μrag∗εa)=(G⊙εa)(μrag)=∫(rag)(y)d((G⊙εa)×μ)(y)(by(1.4))=∫g(ya)d((G⊙εa)×μ)(y)=∬g(yx)d((G⊙εa)×μ)(y)dεa(x)(by(1.9))=∫gd(((G⊙εa)×μ)∗εa).
Hence, by the Riesz
representation theorem, G×(μ∗εa)=((G⊙εa)×μ)∗εa. Thus,(2.10)((F×G)⊙εa)(μ)=F(G×(μ∗εa))=F(((G⊙εa)×μ)∗εa)=(F⊙εa)((G⊙εa)×μ)=((F⊙εa)×(G⊙εa))(μ).Therefore, (F×G)⊙εa=(F⊙εa)×(G⊙εa).
(ii) For eachμ∈M(S),
equality (2.4) implies that(2.11)(εa⊙(F×G))=lεa(F×G)(μ)=(F×G)(εa∗μ)=F(G×(εa∗μ)).Now, for g∈C0(S),
from (1.4) and (1.9), we obtain(2.12)∫gd(G×(εa∗μ))=G((εa∗μ)g)=G(εa∗μlag)=(εa⊙G)(μlag)=∫(lag)(y)d((εa⊙G)×μ)(y)=∫g(ay)d((εa⊙G)×μ)(y)=∬g(xy)dεa(x)d((εa⊙G)×μ)(y)=∫gd(((εa⊙G)×μ)∗εa).
Hence, by the Riesz
representation theorem, G×(εa∗μ)=((εa⊙G)×μ)∗εa.
Thus(2.13)(εa⊙(F×G))(μ)=F(G×(εa∗μ))=F(((εa⊙G)×μ)∗εa)=(εa⊙F)((εa⊙G)×μ)=((εa⊙F)×(εa⊙G))(μ).Therefore, εa⊙(F×G)=(εa⊙F)×(εa⊙G).Remark 2.3.
(a) In the proof of Lemma2.2, we use the
equalities (μ∗εa)g=μrag∗εa
and (εa∗μ)g=εa∗μlag.
For f∈C0(S),
we have
(2.14)∫fd(μ∗εa)g=∫gfd(μ∗εa)(by(1.3))=∬(gf)(xy)dμ(x)dεa(y)(by(1.9))=∫(gf)(xa)dμ(x)=∫g(xa)f(xa)dμ(x)=∫(rag)(raf)dμ(by(1.6))=∫(raf)(x)dμrag(x)(by(1.3))=∫f(xa)dμrag(x)(by(1.9))=∬f(xy)dμrag(x)dεa(y)(by(1.6))=∫fd(μrag∗εa).Hence, by the Riesz
representation theorem, (μ∗εa)g=μrag∗εa.
Similarly, (εa∗μ)g=εa∗μlag.
(b) The statement (i) of Lemma2.2 has a general form as replacing a
Dirac measure by μ∈M0(S) [7]. It is natural to ask for which μ in M0(S),
the equality
(2.15)μ⊙(F×G)=(μ⊙F)×(μ⊙G),∀F,G∈M(S)∗is valid?
Now, we demonstrate that the multiplication onM(S)∗ defined by (1.3), (1.4), and (1.5) is associative.Lemma 2.4.
The multiplication× defined by (1.3), (1.4), and (1.5) on M(S)∗ is associative.Proof.
We know that the Arens product⊙ is associative [3, Lemma 1, page 527]. Let π:C0(S)∗→M(S) be isometric order-preserving linear space
isomorphism in [1, Theorem 14.10, page 170], namely, for any m∈C0(S)∗ and f∈C0(S),(2.16)∫fdπ(m)=m(f).Now, let f,g∈C0(S)∗, m∈C0(S)∗,
and F,G∈M(S)∗,
then (1.3) implies that(2.17)∫gdπ(m)f=∫fgdπ(m)=m(fg)=(m⊙f)(g)=∫gdπ(m⊙f).Thus, π(m)f=π(m⊙f).
Also, from (1.4), we have(2.18)∫fd(F×π(m))=F(π(m)f)=F(π(m⊙f))=π∗(F)(m⊙f)=(π∗(F)⊙m)(f)=∫fdπ(π∗(F)⊙m).Hence, F×π(m)=π(π∗(F)⊙m).
Also, from (1.5), we have(2.19)(π∗(F×G))(m)=(F×G)(π(m))=F(G×π(m))=F(π(π∗(G)⊙m))=(π∗(F))(π∗(G)⊙m)=(π∗(F)⊙π∗(G))(m).Therefore, π∗(F×G)=π∗(F)⊙π∗(G).
Now, for any F,G,H∈M(S)∗,(2.20)π∗((F×G)×H)=π∗((F×G)⊙π∗(H))=(π∗(F)⊙π∗(G))⊙π∗(H)=π∗(F)⊙(π∗(G)⊙π∗(H))=π∗(F)⊙(π∗(G×H))=π∗(F×(G×H)).So, (F×G)×H=F×(G×H),
and thus the multiplication of × is associative.Remark 2.5.
We note that one can go through a process analogous
to Day's proof [3] and establish the associativity of × via the demonstration of the following
identities one by one.(i)
For anyμ∈M(S) and f,g∈C0(S),(2.21)(μf)g=μfg.(ii)
For anyF∈M(S)∗, μ∈M(S) and f∈C0(S),(2.22)F×(μf)=(F×μ)f.(iii)
For anyF,G∈M(S)∗ and μ∈M(S),(2.23)(F×G)×μ=F×(G×μ).(iv)
For anyF,G,H∈M(S)∗,(2.24)F×(G×H)=(F×G)×H.The proofs of (i), (ii), and (iii) use the Riesz
representation theorem and the relations (1.3), (1.4), and (1.5). The proof of   (iv)
follows from (iii) using definition.


## 3. Main Results

Each probability measureμ in M0(S) is a mean on M(S)∗,
if we put μ(F)=F(μ) for any F in X.
We give a partial answer to the question: For which μ∈M0(S),
is the equality
(3.1)μ⊙(F×G)=(μ⊙F)×(μ⊙G),∀F,G∈M(S)∗valid?Letf∈GL(S),
from the isometric Banach space isomorphism φ:GL(S)→M(S)∗,
we have φf=F which is in M(S)∗,
where F(μ)=(φf)(μ)=∫fμdμ,
for any μ in M(S).
For μ∈M(S) and g∈C0(S), we have(3.2)∫gd(F×μ)=F(μg)=(φf)(μg)=∫fμgdμg=∫fμdμg=∫fμgdμ=∫gdμfμ.Therefore, F×μ=φf×μ=μfμ.
In particular, φf×μ≪μ and so 1×μ1=μ1=μ.
In view of (1.4), if F∈M(S)∗ and μ∈M(S),
then(3.3)(F×μ)(S)=∫d(F×μ)=F(μ1)=F(μ).Hence, if F≥0 and μ≥0,
then(3.4)∥F×μ∥=(F×μ)(S)=F(μ).Also, since ∥G∥1−G≥0,
we have(3.5)(∥G∥1−G)×μ≥0,and hence, G×μ≤∥G∥1×μ=∥G∥μ whenever μ≥0.Theorem 3.1.
LetS be a locally compact semigroup with a right
identity and that 0<μ∈M(S).
If μ⊙(F×G)=(μ⊙F)×(μ⊙G) for any F,G∈φ(C0(S)),
then μ is a Dirac measure.Proof.
Forf,g∈C0(S),
we have(3.6)φ(μ⊙(fg))=μ⊙φ(fg)(byLemma2.1(i))=μ⊙(φf×φg)=(μ⊙φf)×(μ⊙φg)=φ(μ⊙f)×φ(μ⊙g)(byLemma2.1(i))=φ((μ⊙f)(μ⊙g)).Thus, for any f,g∈C0(S),
from (3.6), we have(3.7)μ⊙(fg)=(μ⊙f)(μ⊙g).Now, let er be a right identity of S,
that is, ser=s for any s∈S,
then for any f∈C0(S),
we have(3.8)(μ⊙f)(er)=∫f(ter)dμ(t)=∫f(t)dμ(t).Hence, for each f,g∈C0(S),(3.9)∫fgdμ=(μ⊙(fg))(er)=(μ⊙f)(er)(μ⊙g)(er)=(∫fdμ)(∫gdμ).In (3.9), we put f=g,
then for any f∈C0(S),(3.10)∫f2dμ=(∫fdμ)2,so for each f,g∈C0(S),(3.11)(∫fgdμ)2=[(∫fdμ)(∫gdμ)]2=(∫fdμ)2(∫gdμ)2=(∫f2dμ)(∫g2dμ),and by Holder's inequality,
there exist real numbers α and β,
not being zero, such that(3.12)αf2=βg2a.e.(μ).
Now, ifA and B are the disjoint compact subsets of S with μ(A)>0 and μ(B)>0,
by the Urysohn's lemma, there exist f and g in C00(S) such that(3.13)f(A)=0=g(B),f(B)=1=g(A).But from (3.12) and μ(A)>0,
there is x0∈A such that(3.14)αf(x0)2=βg(x0)2.So, 0=βg(x0)2=β.
Also, μ(B)>0 follows that there is y0∈B,
such that(3.15)αf(y0)2=βg(y0)2,and therefore 0=αf(y0)2=α.
This contradicts the fact that α and β are not both zero. Hence, if A is a compact subset of S with μ(A)>0 and B is another compact subset of S disjointed from A,
then we must have μ(B)=0.
Therefore, μ(Ac)=0,
that is, μ(A)=μ(S).
This proves that if A is a compact subset S,
then either μ=0 or μ(A)=μ(S).
Now, the regularity ofμ follows that for each Borel subset B of S, μ(B)=0 or μ(B)=μ(S).
Hence, either μ=0 or the measure μ/μ(S) is a Dirac measure, say μ/μ(S)=εa [2].
Now, putμ=μ(S)εa,
we have(3.16)μ(S)=∫dμ=∫12dμ=(∫1dμ)2=(∫dμ)2=μ(S)2,and so μ(S)=1.
Thus, μ=εa.Remark 3.2.
IfS is a discrete semigroup, then C0(S)∗=ℓ1(S),
and so M0(S)∗=C0(S)∗∗=ℓ1(S)∗=m(S) [1]. In this case, the multiplication on M(S)∗ is just the point-wise multiplication as in m(S).
Let e be the right identity of S,
then(3.17)(μ⊙(F×G))(εe)=(F×G)(μ∗εe)=(F×G)(μ)=μ(F×G),((μ⊙F)×(μ⊙G))(εe)=(μ⊙F)(εe)(μ⊙G)(εe)=μ(F)μ(G),since both μ⊙F and μ⊙G are in M(S)∗=m(S).
Hence, μ∈M(S) is multiplicative, if the condition of Theorem 3.1 is satisfied. Therefore, μ must be either 0 or a Dirac measure. But, when S is a topological semigroup, the multiplication
in M(S)∗ defined by (1.3), (1.4), and (1.5) is just a generic
Arens product in the second dual of a Banach algebra, which is different from
the point-wise multiplication.It is known thatM0(S) is weak* dense in the set of all means on M(S)∗.
We give some characterizations theorems for the extreme amenability of locally
compact semigroup.Lemma 3.3.
LetM(S)∗ be TLA. The following statements are
equivalent:(i)
M(S)∗ is ETLA,(ii)
for everyF∈M(S)∗ and μ∈M0(S),
there exists a mean M on M(S)∗ such that M(F×F)=M(μ⊙F)2,(iii)
for everyF∈M(S)∗ and μ∈M0(S),
there exists a mean M on M(S)∗ such that M(F×F)=M(F)M(μ⊙F),(iv)
for everyF∈M(S)∗ and μ∈M0(S),
there exists a mean M on M(S)∗ such that M(μ⊙(F×F))=M(F)2.Proof.
(i)⇒(ii)⇒(iii)⇒(iv) are obvious.
(iv)⇒(i). Suppose that F,G∈M(S)∗ and μ∈M0(S).
For F+G by (iv),
there exists a mean M on M(S)∗ such that(3.18)M(μ⊙((F+G)×(F+G))=M(F+G)2)=M(F)2+2M(F)M(G)+M(G)2,and to  expand the right-hand
side, we get(3.19)M(μ⊙(F×G))=M(F)M(G).Since M(S)∗ is topological left invariant, hence, M(F×G)=M(F)M(G).
Therefore, M(S)∗ is ETLA.Theorem 3.4.
LetM be a topological left invariant mean on M(S)∗.
The following statements are equivalent:(i)
M is a multiplicative,(ii)
there exists a net{μα} in M0(S) such that for any μ in M0(S) and G in M(S)∗,(3.20)w∗−limα(G×(μ∗μα)−M(G)μα)=0.Proof.
(i)⇒(ii). Let M be a multiplicative topological left invariant
mean on M(S)∗.
By Lemma 3.3,
for any F∈M(S)∗ and μ∈M0(S),(3.21)M(μ⊙(F×F))=M(F)2.
Let F,G∈M(S)∗,
then(3.22)M(μ⊙((F+G)×(F+G)))=M(F+G)2.We have(3.23)M(μ⊙(F×F+2F×G+G×G))=(M(F)+M(G))2.So,(3.24)M(μ⊙(F×F))+2M(μ⊙(F×G))+M(μ(G×G))=M(F)2+M(G)2+2M(F)M(G).Thus,(3.25)M(μ⊙(F×G))=M(F)M(G).
Note that we apply the commutativity of× in M(S)∗.
Since M is a mean on M(S)∗ and M0(S) is weak* dense in the set of all means on M(S)∗,
hence, there exists a net {μα} in M0(S) such that M=w∗−limαμα in M(S)∗∗.
Now for F∈M(S)∗,(3.26)w∗−limαF(G×(μ∗μα)−M(G)μα)=w∗−limα(μα(μ⊙(F×G))−M(G)μα(F))=M(μ⊙(F×G))−M(F)M(G)=0.Thus,(3.27)F(w∗−limα(G×(μ∗μα)−M(G)μα))=0,that is,(3.28)w∗−limα(G×(μ∗μα)−M(G)μα)=0.
(ii)⇒(i). Since M is a topological left invariant mean on M(S)∗,
there exists a net {μα} in M0(S) such that M=w∗−limαμα in M(S)∗∗.
If μ∈M0(S),(3.29)M(F×G)−M(F)M(G)=M(μ⊙(F×G))−M(F)M(G)=w∗−limαμα(μ⊙(F×G))−M(G)(w∗−limαμα(F))=w∗−limα((F×G)(μ∗μα)−M(G)F(μα))=w∗−limα(F(G×(μ∗μα))−F(M(G)μα))=w∗−limαF(G×(μ∗μα)−M(G)μα)=F(w∗−limα(G×(μ∗μα)−M(G)μα))=F(0)=0.Therefore, M(F×G)=M(F)M(G),
that is, M(S)∗ is extremely topological left amenable (ETLA).Lemma 3.5.
IfM is a multiplicative topological left invariant
mean on M(S)∗,
then there is a net {μβ} in M0(S) such that for any μ in M0(S) and F in M(S)∗,(3.30)limβ∥F×(μ∗μβ)−M(F)μβ∥=0.Proof.
We considerM(S) with the norm topology. Let 𝒫=M(S)M(S)∗×M0(S) with the product of the norm topologies, where M(S)∗×M0(S)
is the set
theoretic cartesian product. Then, 𝒫 is a locally convex topological vector space [10]. Now, by Theorem 3.4
corresponding to M,
there exists a net {μα} in M0(S) such that for any μ in M0(S) and F in M(S)∗,(3.31)w∗−limα(F×(μ∗μα)−M(F)μα)=0.We define a linear map P:M(S)→𝒫 by(3.32)P(ν)(F,μ)=F×(μ∗ν)−M(F)ν,for all (F,μ)∈M(S)∗×M0(S).
Hence,(3.33)w∗−limαP(μα)(F,μ)=0,that is, P(μα)→0 in the product of weak topologies [10].
Therefore, 0 lies in the weak closure of the convex set P(M0(S)),
and so is in the closure of P(M0(S)) in the original topology of 𝒫. So, there is a net {μβ} in M0(S) such that for all (F,μ)∈M(S)∗×M0(S),(3.34)limβ∥P(μβ)(F,μ)∥=0,that is,(3.35)limβ∥F×(μ∗μβ)−M(F)μβ∥=0,and the proof is complete.Theorem 3.6.
LetS be a locally compact semigroup. The following
statements are equivalent:(i)
M(S)∗ is extremely topological left amenable,(ii)
there exists a net{μβ} in M0(S) such that for any μ in M0(S) and F in M(S)∗,(3.36)limβ∥μ∗μβ−μβ∥=0,limβ∥F×μβ−F(μβ)μβ∥=0,(iii)
there exists a net{μγ} in M0(S) such that for any μ in M0(S) and F in M(S)∗,(3.37)w∗−limγ(μ∗μγ−μγ)=0,(3.38)w∗−limγ(F×μγ−F(μγ)μγ)=0.Proof.
(i)⇒(ii). Let M be a multiplicative left invariant mean on M(S)∗.
Theorem 3.4 implies that there exists a net {μα} in M0(S) such that for any μ in M0(S) and F in M(S)∗,(3.39)w∗−limα(F×(μ∗μα)−M(F)μα)=0.By Lemma 3.5, there exists a net {μβ} in M0(S) such that for any μ in M0(S) and F in M(S)∗,(3.40)limβ∥F×(μ∗μβ)−M(F)μβ∥=0.Without the loss of generality,
we may assume that μβ→M1σ(M(S)∗∗,M(S)∗) for some mean M1 in M(S)∗∗.
Therefore, for any F,G in M(S)∗ and μ in M0(S),
we have(3.41)M1(μ⊙(G×F))−M(F)M1(G)=limβ{(μ⊙(G×F))(μβ)−M(F)M1(G)}=limβ[G(F×(μ∗μβ))−G(M(F)μβ)]=limβG(F×(μ∗μβ)−M(F)μβ)=0.In (3.41), we put F=1=G,
then(3.42)M1(μ⊙(1×1))=M(1)M1(1),M1(μ⊙1)=M(1)M1(1),M(1)=1.
Also, for F=1 and G in M(S)∗,(3.43)M1(μ⊙G)=M1(G),and for G=1 and F in M(S)∗,(3.44)M1(μ⊙F)=M1(F).Therefore, for any F in M(S)∗,
we have(3.45)M(F)=M1(μ⊙F)=M1(F).Now, from (3.30) for F=1,
we get(3.46)limβ∥μ∗μβ−μβ∥=0.Also, let F in M(S)∗ and ε>0 be given. Since(3.47)M(F)=M1(F),F(μβ)→M(F),∥μβ∥≤1,it follows from (3.30) that for any μ in M0(S),(3.48)limβ∥F×(μ∗μβ)−M1(F)μβ∥=0.Now fix an arbitrary μ∈M0(S).
This together with (3.46) implies that there exists a β0 such that(3.49)∥μ∗μβ−μβ∥<ε3(∥F∥+1),∀β⪰β0,∥F×(μ∗μβ)−M1(F)μβ∥<ε3.
Also, we may assume that(3.50)|F(μβ)−M1(F)|<ε3.Consequently,(3.51)∥F×μβ−F(μβ)μβ∥≤∥F×μβ−F×(μ∗μβ)∥+∥F×(μ∗μβ)−M1(F)μβ∥+∥M1(F)μβ−F(μβ)μβ∥≤∥F∥∥μ∗μβ−μβ∥+∥F×(μ∗μβ)−M1(F)μβ∥∥μβ∥|M1(F)−F(μβ)|<ε3+ε3+ε3=ε,∀β⪰β0.
Obviously, (ii)⇒(iii).
(iii)⇒(i). Since M0(S) is  weak*  dense in the set of all means on M(S)∗,
by passing to a subnet if necessary, we may assume that μγ→M weakly* in M(S)∗∗ for some mean M.
Thus, the assertion of (3.37) implies that M is a topological left invariant mean. Also, (3.38) implies that M is multiplicative because for any F,G in M(S)∗ and μ in M0(S),(3.52)M(G×F)−M(G)M(F)=w∗−limγ{(G×F)(μγ)−G(μγ)F(μγ)}=w∗−limγ{G(F×μγ)−G(F(μγ)μγ)}=w∗−limγG(F×μγ−F(μγ)μγ)=G(0)=0.Therefore, M(G×F)=M(G)M(F),
that is, M(S)∗ is extremely left amenable.Remark 3.7.
The conclusions of Theorem3.6 are different from the
classical characterizations of extremely left amenable discrete semigroups [4,
Theorem 2]. This difference in the two situations is that any multiplicative
mean on m(S) is the weak* limit of evaluation functionals, while taking
weak* limits of all convergent nets of Dirac
measures in M(S)∗ does not exhaust all multiplicative means on M(S)∗ [2, Theorem 2.7].Theorem 3.8.
LetS be a locally compact semigroup. Define a
function T:M(S)∗→m(S) by(3.53)(T(F))(a)=F⊙εa,∀a∈S.Then,(i)
T is bounded and linear,(ii)
T(1)=1,(iii)
T(F)≥0 whenever F≥0,(iv)
T(F×G)=T(F)T(G) for all F and G in M(S)∗,(v)
lb(T(F))=T(εb⊙F) for all b∈S and F∈M(S)∗.Proof.
(i),(ii),
and (iii) are obvious.
(iv) For any a∈S,
we have(3.54)(T(F×G))(a)=(F×G)⊙εa=(F⊙εa)×(G⊙εa)(byLemma2.2(i))=T(F)(a)×T(G)(a)=(T(F)×T(G))(a)=(T(F)T(G))(a).In the final equality, we use
the fact that multiplication in m(S) is a point-wise multiplication, see Remark 3.2
of Theorem 3.1.
(v) Leta∈S and F∈M(S)∗,
then(3.55)lb(T(F))(a)=(T(F))(ba)=F⊙εba=F(εb∗εa)=(εb⊙F)(εa)=(T(εb⊙F))(a).So, lb(T(F))=T(εb⊙F).Remark 3.9.
From Theorem3.8, it follows that the map T∗:m(S)∗→M(S)∗∗ carries means to means, multiplicative means
to multiplicative means, left invariant means to left invariant means, and
multiplicative left invariant means to multiplicative left invariant means. But T∗ does not carry a type of means in m(S)∗ onto the same type of means in M(S)∗∗.
Indeed, if M is a multiplicative topological left invariant
mean which is not weak* limit of all convergent nets of Dirac measures
in M(S)∗,
then M does not belong to T∗(ℳ),
where ℳ is the set of all means on m(S).



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