9. INNER AMENABILITY 53
(ii) groups that have a finite conjugacy class = {1},
(iii) direct products Γ × ∆, where Γ = {1} is amenable,
(iv) weakly commutative groups (i.e., groups Γ such that for every finite
F Γ there is a γ Γ \ {1} commuting with each element of F ). For
example, direct sums Γ1 Γ2 . . . of countable non-trivial groups,
(v) the Thompson group F = x0,x1,x2,... |xi
−1xnxi
= xn+1,i n .
On the other hand the free groups Fn,n 2, are not inner amenable.
Also ICC groups with property (T) are not inner amenable.
(B) The following result was proved in Jones-Schmidt [JS].
Theorem 9.1 (Jones-Schmidt). Given a free, measure preserving action
of a countable group Γ on (X, µ), if [EΓ
X
] is not closed in N[EΓ
X
], then Γ is
inner amenable.
Proof. Since, letting E =
X
, [E] is not closed in N[E], there is a
sequence Tn [E] with δu(Tn, 1) , some 0, and TnγTn
−1
γ
uniformly, ∀γ Γ.
Put
Tn(x) = α(n, x) · x,
where α(n, x) Γ. Let U be a non-principal ultrafilter on N and define a
positive linear functional on
∞(Γ)
by
ϕ(f) = lim
n→U
f(α(n, x))dµ(x).
Note that if χ1 is the characteristic function of {1}, then ϕ(χ1) = 1 as
ϕ(χ1) = limn→U χ1(α(n, x))dµ(x)
= limn→U
{x:Tn(x)=x}
dµ(x)
= limn→U µ({x : Tn(x) = x})
= limn→U (1 δu(Tn, 1)) 1 1.
We will next see that ϕ is conjugacy invariant. Then clearly
ψ(f) =
ϕ(f
∗)
1 ϕ(χ1)
,
where f
∗(γ)
= f(γ) if γ = 1,f
∗(1)
= 0, is a conjugacy invariant mean on
Γ \ {1}, so we are done.
For γ Γ,f
∞(Γ),
put · f)(δ) =
f(γ−1δγ).
We need to verify that
ϕ(γ · f) = ϕ(f). We have
ϕ(γ · f) = lim
n→U
· f)(α(n, x))dµ(x)
= lim
n→U
f(γ−1α(n,
x)γ)dµ(x).
Let An = {x :
Tn(γ−1
· x)) =
γ−1
· Tn(x)}, so that µ(An) 0. If x An,
then
Tn(γ−1
· x) =
γ−1
· Tn(x),
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