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 = EΓ

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 (γ · f)(δ) =

f(γ−1δγ).

We need to verify that

ϕ(γ · f) = ϕ(f). We have

ϕ(γ · f) = lim

n→U

(γ · 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),