De nition 7.7 A groupG is inner amenable if it supports a meanm ∈ L∞(G)∗
satisfying
m(f) =m(g−1fg) for allg∈G, f∈L∞(G).
Such a meanmisinner invariant. (Paterson, 1988)
For eachg ∈ Gconsider the map Ig : f 7→ g−1fgfor all f ∈ ℓ∞(G). e map
g7→Igis theinner automorphism, which is the mapping under which inner invari- ant means are invariant. is justi es the nomenclature. Inner amenability clearly follows very easily from amenability, and is trivial for Abelian groups.
However,all discrete groups are inner amenable. is is easily seen by the fol- lowing. Ife ∈Gis the identity element, andδe ∈ ℓ∞(G)∗is thetrivial meangiven byδe(f) =f(e)for allf∈ℓ∞(G), then it is easy to see that
δe
(
g−1fg)=f(g−1eg)=f(e) =δe(f),
for allg ∈G. An obvious variation on inner amenability is to removeδefrom con- sideration: a groupGis said to betrivially inner amenableifδeis the only such inner invariant mean, and interest lies in nding inner, but not trivially inner, amenable groups. e investigation into inner invariant means was initiated by Effros (1975),
who showed that all PropertyΓgroups are inner amenable but not merely trivially so. ere are, however, non-discrete locally-compact groups which fail to be trivially in- ner amenable, and so sometimes the original de nition is preferable. Another class of groups that are inner amenable are the [IN]-groups (Lau and Paterson, 1991).
An analogue of the above de nition for semigroups is given below.
De nition 7.8 A semigroupSisinner amenableif it supports a meanm ∈L∞(S)∗
satisfying
m(sf) =m(fs) for alls∈S, f ∈L∞(S).
Such a meanmisinner invariant. (Ling, 1997)
Semigroup inner amenability is certainly a weaker condition than amenability, which is itself quite weak already. IfSis a monoid, thenδ1is again a inner invariant mean:
δ1(sf) =f(s1) = f(s) =f(1s) = δ1(fs). IfShas a zero element, thenδ0 :f7→f(0)is also inner invariant:
δ0(sf) =f(s0) =f(0) =f(0s) =δ0(fs).
Conceivably, many such “trivially” inner invariant means would need to be stamped out to create a suitably interesting condition. By analogy with fair amenability, it may be worthwhile starting with a condition that is too strong, and then weaken it. ere may be inspiration in the following: ifSis a semigroup with inner invariant mean
m, then the nitely-additive measureµde ned byµ(A) =m(χA)satis es
µ(s−1A)=µ(As−1) for alls∈S, A⊆S.
7.2.1 Inner
⊛- and∗-invariance
De nition 7.9 LetSbe a semigroup,µ∈[0, 1]P(S)a nitely-additive measure. (i) µisinner⊛-invariantif
µ(sA) =µ(As) for allA⊆S, s∈S.
(ii) µisinner∗-invariantifµ(sA) = µ(As)merely for thoses ∈ SandA ⊆ S
De nition 7.10 LetSbe a semigroup,m ∈ℓ∞(S)∗a mean. (i) misinner∗-invariantif
m(s∗f) =m(f∗s)
for alls ∈Sandf∈ℓ∞(S)such that boths∗fandf∗sare inℓ∞(S). (ii) misinner⊛-invariantif
m(s⊛f) =m(f⊛s)
for alls ∈Sandf∈ℓ∞(S)(both sides always exist inℓ∞(S)).
e main result will be the equivalence of inner⊛-invariant means with inner⊛- invariant measures, and similarly, inner∗-invariant means and measures. But rst, some observations.
Proposition 7.11 Inner⊛-amenability is trivial for Abelian semigroups and groups (assA = Asfor allsandA), and for semigroups and groups supporting a totally- invariant nitely-additive probabiliy measure, in particular, the amenable groups. Additionally, adjoining an identity or a zero does not affect whether a semigroup is inner∗-amenable, since1A=A=A1and0A={0}=A0for all setsA⊆S.
Inner∗-amenability follows trivially from inner⊛-amenablility, but also from fair amenability.
In light of Proposition 7.11, the de nition of inner⊛-amenable seems to attain a good compromise, with no further weakening (e.g. to inner∗-amenability) required. eorem 7.12 A semigroupSsupports an inner⊛-invariant nitely-additive mea- sure if, and only if, it supports an inner⊛-invariant mean.
Proof Letµbe an inner⊛-invariant nitely-additive measure forS. en de ne
m ∈ℓ∞(S)by setting
m(f) :=
∫
fdµ for allf∈ℓ∞(S).
µ(sA) = µ(As)on the basis sets of the simple functions, and therefore m(s⊛f) = ∫ (s⊛f)dµ= ∫ (f⊛s)dµ=m(f⊛s)
for allf∈ℓ∞(S)ands∈S, as required.
Conversely, letmbe an inner⊛-invariant mean. Sinces⊛χA =χsAfor allA, and similarly on the right, we can setµto be a measure given by
µ(A) := m(χA) for allA⊆S, and therefore,
µ(sA) =m(s⊛χA) = m(χA⊛s) = µ(As), for alls∈SandA⊆S, as required.
e next few results show the equivalence of the inner∗-invariant measures and means.
Lemma 7.13 LetSbe a semigroup with an inner∗-invariant measureµ. en
∫
(s∗χA)dµ=
∫
(χA∗s)dµ
for alls∈SandA⊆Ssuch thats∗χAandχA∗sare both inℓ∞(S).
Proof By Lemma 6.2, there exists two nite partitions of A,{Ai}i∈I and {Bj}j∈J, such thatsacts injectively on the le of eachAiand injectively on the right of each
Bj, and in particular,s∗χAi =χsAiandχBj∗s=χBjs. LetCij :=Ai∩Bjfor eachi ∈
tively on the le and the right of eachCij. en, in a similar vein to Lemma 6.4, ∫ (s∗χA)dµ = ∫ s∗ ∑ (i,j)∈I×J χCij dµ = ∑ (i,j)∈I×J ∫ χsCijdµ = ∑ (i,j)∈I×J µ(sCij) = ∑ (i,j)∈I×J µ(Cijs) = ∑ (i,j)∈I×J ∫ χCijsdµ = ∫ (χA∗s)dµ, as required. □
Remark 7.14 Lemma 7.13 extends, via similar working to Lemmas 6.5, 6.6, and 6.7, to show that if boths∗fandf∗s∈ℓ∞(S), then∫(s∗f)dµ=∫(f∗s)dµ.
us the integral with respect to µ again suffices as an invariant mean. is nal result shows the converse.
Lemma 7.15 LetSbe a semigroup supporting an inner∗-invariant meanm∈ℓ∞(S)∗. en there is an inner∗-invariant nitely-additive measureµ.
Proof Letµ ∈ [0, 1]P(S)be given by µ(A) := m(χA)for allA ⊆ S. Ifs acts injectively on the le and right ofA, thens∗χA =χsAandχA∗s=χAs, and so
µ(sA) =m(s∗χA) = m(χA∗s) = µ(As),
as required. □
It would be interesting to see how much further this idea could be stretched.