Download Amenable Groups: Invariant Means and Characterizations and more Lecture notes Algebra in PDF only on Docsity!
AMENABLE GROUPS
BRENT NELSON
- Amenable Groups
Throughout
1 we let Γ be a discrete group. For f : Γ → C and each s ∈ Γ we define the left translation
action by (s.f )(t) = f (s
− 1 t).
Definition 1.1. A group Γ is amenable is there exists a state μ on l
∞ (Γ) which is invariant under the left
translation action: for all s ∈ Γ and f ∈ l
∞ (Γ), μ(s.f ) = μ(f ).
Example 1.2. Finite groups are amenable: take the state which sends χ{s} to
1 |Γ| for each s ∈ Γ. One can
also see that abelian groups are amenable through the Markov-Kakutani fixed point theorem. Furthermore,
the class of amenable groups is closed under taking subgroups, extensions, quotients, and inductive limits.
Hence we can construct further examples from finite and abelian groups.
Example 1.3. Suppose Γ is finitely generated by S = {s 1 ,... , sd} (and that S
− 1 = S). One can then
consider the Cayley graph of Γ where vertices are group elements and edges connecting two group elements
imply they differ by one of the generators in S. We place a metric on this graph by letting d(s, t) by counting
the “word length” of s
− 1 t (with the generators as our usable letters). A property of interest is how |B(e, r)|,
the ball centered at the identity element of radius r, varies with r; that is, the growth rate of the group.
It turns out that groups with subexponential growth are always amenable.
Example 1.4. Γ = F 2 is non-amenable: let a, b ∈ F 2 be the two generators then let A
is the set of all
words starting with a. A
− is the set of all words starting with a
− 1 , and we define B
± similarly. Lastly, we
set C := { 1 , b, b 2 ,.. .}. We note that we can decompose F 2 in the three following ways:
F 2 = A
t A
− t (B
\ C) t (B
− ∪ C)
= A
t aA
−
= b
− 1 (B
\ C) t (B
∪ C).
If we had a state μ on l
∞ (Γ) which was invariant under left translation then we would obtain:
1 = μ(1) = μ(χA+ + χA− + χB+\C + χB−∪C ) = μ(χA+ ) + μ(χA− ) + μ(χB+\C ) + μ(χB−∪C )
= μ(χA+ ) + μ(a.χA− ) + μ(b
− 1 .χB+\C ) + μ(χB−∪C ) = μ(χA+ + χaA− ) + μ(χb− (^1) B+\C + χB−∪C )
= μ(1) + μ(1) = 2,
a contradiction.
Our goal is to prove the following theorem:
Theorem 1.5. For Γ a discreet group, the following are equivalent:
(1) Γ is amenable;
(2) Γ has an approximate invariant mean;
(3) Γ satisfies the Følner condition;
(4) The trivial representation τ 0 is weakly contained in the the regular representation λ (i.e., there exist
unit vectors ξi ∈ l
2 (Γ) such that ‖λs(ξi) − ξi‖ 2 → 0 for all s ∈ Γ);
(5) there exists a net (ϕ) of finitely supported positive definite functions on Γ, with ϕi(e) = 1 for each i,
such that ϕi → 1 pointwise;
(6) C
∗ (Γ) = C
∗ λ
(7) C
∗ λ (Γ) has a character (i.e., one-dimensional representation);
1 These notes were adapted from [1], to which we direct the reader for more information and less detail.
1
2 BRENT NELSON
(8) for any finite subset E ⊂ Γ, we have
∥ ∥ ∥ ∥ ∥
|E|
s∈E
λs
The main obstacle to proving this theorem is that we don’t understand what most of it is saying. Con-
sequently we’ll parse the theorem as we go (rather than drowning the reader in definitions). The plan is to
prove the cycle (1 2 3 4 5 6 7) and then (4 8). We shall additionally prove (4) ⇒ (6) in case the reader finds
condition (5) distasteful.
Definition 1.6. For a discrete group Γ, let Prob(Γ) be the space of all probability measures on Γ:
Prob(Γ) := {μ ∈ l
1 (Γ) : μ ≥ 0 and
t∈Γ
μ(t) = 1}.
Then we say Γ has an approximate invariant mean if for any finite subset E ⊂ Γ and > 0, there exists
μ ∈ Prob(Γ) such that
max s∈E
‖s.μ − μ‖ 1 < .
Proof of (1) ⇒ (2). Let μ be an invariant mean on l ∞ (Γ). We claim there is a net (μi) ⊂ Prob(Γ) which
converges to μ weak
∗ as elements of l
∞ (Γ)
∗
. Suppose not, then μ 6 ∈ Prob(Γ)
w∗ and since Prob(Γ) is convex
the Hahn-Banach separation theorem implies there is some f ∈ l
∞ (Γ) and t < s ∈ R such that
Re [ν(f )] < t < s < Re [μ(f )],
for all ν ∈ Prob(Γ). Upon replacing f with
f + f¯ 2
we obtain ν(f ) < t < s < μ(f ). Then replacing f with
f + ‖f ‖∞ (and t, s with t ′ = t + ‖f ‖∞, s ′ = s + ‖f ‖∞) ensures that f ≥ 0. Consequently sup{ν(f ) : ν ∈
Prob(Γ)} = ‖f ‖∞ and yet
‖f ‖∞ = sup{ν(f ) : ν ∈ Prob(Γ)} ≤ t < s < μ(f ) ≤ ‖f ‖∞,
a contradiction.
Hence we can find a net (μi) in Prob(Γ) which converges to μ in the weak
∗ topology. Thus for each s ∈ Γ
and f ∈ l
∞ (Γ) we know s.μi(f ) − μi(f ) → s.μ(f ) − μ(f ) = μ(f ) − μ(f ) = 0. But since the μi ∈ l
1 (Γ), this
is equivalent to saying they converge weakly to zero in l
1 (Γ). Thus for any finite E ⊂ Γ, the weak closure of
the convex subset
s∈E {s.μ − μ : μ ∈ Prob(Γ)} contains 0. As a convex set, the weak and norm closures
coincide by the Hahn-Banach theorem. Hence given > 0 we can find ν ∈ Prob(Γ) such that
∑
s∈E
‖s.ν − ν − 0 ‖ 1 < .
Hence we have an approximate invariant mean.
Definition 1.7. We say Γ satisfies the Følner condition if for any finite E ⊂ Γ and > 0, there exists a
finite subset F ⊂ Γ such that
max s∈E
|sF 4 F |
|F |
That is, the action of E does not move F around “too much.”
Furthermore, a sequence of finite sets Fn ⊂ Γ such that
|sFn 4 Fn|
|Fn|
is called a Følner sequence.
Proof of (2) ⇒ (3). Fix a finite subset E ⊂ Γ and > 0. Since we have an approximate invariant mean we
can find μ ∈ Prob(Γ) such that ∑
s∈E
‖s.μ − μ‖ 1 < .
4 BRENT NELSON
Let n = |F |, F = {t 1 ,... , tn}, and fix v = (z 1 ,... , zn) ∈ C
n
. It suffices to show
[ϕ(s
− 1 t)]v, v
≥ 0. We
compute
[〈λtξ, λsξ〉]v =
〈λt 1 ξ, λt 1 ξ〉 · · · 〈λt n ξ, λt 1 ξ〉
. . .
〈λt 1 ξ, λtn ξ〉 · · · 〈λtn ξ, λtn ξ〉
z 1
. . .
zn
〈λt 1 ξ, λt 1 ξ〉 z 1 + · · · + 〈λt n ξ, λt 1 ξ〉 zn
. . .
〈λt 1 ξ, λtn ξ〉 z 1 + · · · + 〈λtn ξ, λtn ξ〉 zn
〈z 1 λt 1 ξ, λt 1 ξ〉 + · · · + 〈znλtn ξ, λt 1 ξ〉
. . .
〈z 1 λt 1 ξ, λt n ξ〉 + · · · + 〈znλt n ξ, λt n ξ〉
∑n
i= 〈ziλt i ξ, λt 1 ξ〉
. . . ∑ n i= 〈ziλti ξ, λt 1 ξ〉
and so
〈[〈λtξ, λsξ〉]v, v〉 =
n i= 〈ziλti ξ, λt 1 ξ〉
. . . ∑n
i= 〈ziλt i ξ, λt 1 ξ〉
z 1
. . .
zn
n ∑
j=
n ∑
i=
ziλti ξ, λtj ξ
zj =
n ∑
j=
n ∑
i=
ziλti ξ, zj λtj ξ
n ∑
i=
ziλt i ξ,
n ∑
j=
zj λt j ξ
n ∑
i=
ziλt i ξ
2
Hence ϕ is positive definite. So letting (ξi) be the unit vectors from condition (4) and setting ϕi(s) :=
〈λsξi, ξi〉 we know ϕi(e) = 1 and from our above work that these functions are positive definite. From
condition (4) we also know that they converge pointwise to 1. In order to make them finitely supported we
need merely replace the ξi with finitely supported elements.
Starting with a discrete group Γ we can consider the group algebra C[Γ] = {
n i= αi · ti : n ∈ N, αt ∈
C, ti ∈ Γ} with addition and multiplication defined in the obvious ways and an involution defined by
( n ∑
i=
αi · ti
n ∑
i=
αi · t
− 1 i
We want to extend this into a C
∗ -algebra, but there are multiple norms we use. On the one hand we can
extend the left regular representation λ to a ∗-representation of C[Γ] on l
2 (Γ), still denoted by λ, by
λ
n ∑
i=
αi · ti
n ∑
i=
αiλti ∈ B(l
2 (Γ)).
The reduced C ∗ -algebra is then what we obtain by taking the closure of λ(C[Γ]) with respect to ‖ · ‖B(l (^2) (Γ));
we denote it by C
∗ λ (Γ). On the other hand, the left regular representation λ : Γ → U(l
2 (Γ)) is merely one
representation of our group. Hence we can consider the norm
‖x‖u = sup{‖π(x)‖B(H) : π : Γ → U(H) is a ∗-representation}.
This easily satisfies the C
∗ -identity. The full (or universal) C
∗ -algebra of Γ is the closure of C[Γ] with respect
to ‖ · ‖u and is denoted C
∗ (Γ).
Thus assuming (5) we’ll need to show that these two C
∗ -algebras coincide. But we first note that since
‖λ(x)‖ ≤ ‖x‖u for x ∈ C[Γ], λ extends to C
∗ (Γ) (which we still denote λ). It is clear that this is onto C
∗ λ(Γ)
(since C[Γ] ⊂ C
∗ (Γ)). We’ll need the following:
Definition 1.9. Let ϕ : Γ → C be a function. The associated multiplier mϕ : C[Γ] → C[Γ] is defined by
mϕ
t∈Γ
αt · t
t∈Γ
ϕ(t)αt · t.
AMENABLE GROUPS 5
We also define ˜mϕ : λ(C[Γ]) → λ(C[Γ]) by
m˜ϕ
λ
t∈Γ
αt · t
= ˜mϕ
t∈Γ
αtλt
t∈Γ
ϕ(t)αtλt
Lemma 1.10. Suppose ϕ is finitely supported, positive definite, and ϕ(e) = 1. Then mϕ extends to a
continuous map on C ∗ (Γ) and m˜ϕ extends to a continuous map on C ∗ λ (Γ) both with norm one.
Proof. First consider the case when ϕ = δe (i.e. ϕ(t) = 1 if t = e and ϕ(t) = 0 otherwise). Let τ (x) =
〈λ(x)δe, δe〉 l^2 (Γ) for x ∈ C ∗ (Γ), then τ is a tracial state. For x ∈ C[Γ] we compute:
‖mϕ(x)‖u = ‖τ (x) · e‖u = |τ (x)|‖e‖u = |τ (x)| ≤ ‖x‖u.
Hence we can extend mϕ to C
∗ (Γ) with norm one.
Let ˜τ (T ) = 〈T δe, δe〉 l^2 (Γ) for T ∈ B(l
2 (Γ)), then ˜τ is a tracial state. For x ∈ C[Γ] we compute:
‖ m˜ϕ(λ(x))‖ = ‖τ˜ (λ(x))λe‖ = |˜τ (λ(x))|‖λe‖ = |τ˜ (λ(x))| ≤ ‖λ(x)‖.
so that ˜mϕ extends to C ∗ λ (Γ) with norm one.
Next consider the case ϕ = δt for t ∈ Γ. Since
‖mϕ(x)‖u = ‖ 〈λ(x)δe, δt〉 · t‖u = ‖ 〈λ(x)δe, λtδe〉 · t‖u = ‖ 〈λt−^1 λ(x)δe, δe〉 · t‖u
= ‖τ (t
− 1 x) · t‖u = |τ (t
− 1 x)|‖t‖u ≤ ‖t
− 1 x‖u ≤ ‖x‖u,
we see that mϕ again extends to C
∗ (Γ) with norm one. A similar computation for ˜mϕ involving ˜τ yields an
extension in the reduced C
∗ -algebra case as well.
Thus for a finitely supported ϕ we can write ϕ =
t∈Γ ϕ(t)δt and so mϕ =
t∈Γ ϕ(t)mδ t
. Extending
each of the finitely many mδt yields an extension for mϕ. But since ϕ is positive definite, mϕ is positive and
hence attains its norm at the identity: ‖mϕ‖ = ‖mϕ(e)‖ = |ϕ(e)| = 1. A similar argument applies in the
reduced C
∗ -algebra case.
Proof of (5) ⇒ (6). By our previous comments, we know λ : C
∗ (Γ) → C
∗ λ (Γ) is onto and hence it remains to
show λ is injective.
Let (ϕi) be the net in condition (5). By the above lemma, we can define multipliers mϕi and ˜mϕi on C
∗ (Γ)
and C
∗ λ (Γ) respectively, each with norm one. We note that λ ◦ mϕi = ˜mϕi ◦ λ on C
∗ (Γ) since both functions
are continuous and agree on the dense subspace C[Γ]. Now, since ϕi → 1 pointwise on Γ, mϕi (x) → x for
x ∈ C[Γ]. Since the norms of the mϕi are uniformly bounded by one and C[Γ] is dense in C
∗ (Γ), this limit
holds for x ∈ C
∗ (Γ) as well.
Now, suppose x ∈ C
∗ (Γ) and λ(x) = 0. Then
λ(mϕ i (x)) = ˜mϕ i (λ(x)) = 0,
for every i. But since ϕi is finitely supported we know mϕ i (x) ∈ C[Γ] and hence λ(mϕ i (x)) = 0 implies
mϕ i (x) = 0. Hence x = limi mϕ i (x) = 0 and so λ is injective.
Proof of (6) ⇒ (7). C ∗ (Γ) always has a one-dimensional representation since the trivial representation C[Γ] 3 ∑
t αt · t 7 →
t αt ∈ C is always subordinate to ‖ · ‖u (as all ∗-representations are). Hence C ∗ λ
(Γ) = C
∗ (Γ
has a character.
We require a lemma:
Lemma 1.11. Let A be a unital C
∗ -algebra and ϕ : A → C a state. If x ∈ A satisfies ϕ(x
∗ x) = |ϕ(x)|
2 then
for all y ∈ A
ϕ(xy) = ϕ(x)ϕ(y) = ϕ(yx).
Proof. Let (πϕ, ξ, H) be a GNS representation. Then
‖πϕ(x)ξ‖ = 〈πϕ(x)ξ, πϕ(x)ξ〉 = 〈πϕ(x
∗ x)ξ, ξ〉 = ϕ(x
∗ x) = |ϕ(x)|
2 = | 〈πϕ(x)ξ, ξ〉 |
2 ≤ ‖πϕ(x)ξ‖
2 ,
where the last inequality comes from applying Cauchy-Schwarz. However, we in fact have equality and we
know that equality occurs in Cauchy-Schwarz only when the two vectors are scalar multiples of one another.
Hence πϕ(x)ξ = λξ for some λ ∈ C. In fact,
λ = 〈λξ, ξ〉 = 〈πϕ(x)ξ, ξ〉 = ϕ(x).
AMENABLE GROUPS 7
This is unitary since π is a unitary representation. We compute
U (λ ⊗ 1(s))
t
δt ⊗ ξt
= U
t
(λsδt) ⊗ ξt
= U
t
δst ⊗ ξt
= U
r
δr ⊗ ξs− (^1) r
r
δr ⊗ π(r)ξs− (^1) r ,
and
(λ ⊗ π(s))U
t
δt ⊗ ξt
= (λ ⊗ π(s))
t
δt ⊗ π(t)ξt
t
δst ⊗ π(st)ξt =
r
δr π(r)ξs−^1 r.
Proof (4) ⇒ (6). Let π : Γ → U(H) be a ∗-representation. Let (ξi) be the net from condition (4). For
x ∈ C[Γ] and η ∈ H a unit vector we have
〈π(x)η, η〉 = lim i
〈(π ⊗ λ(x))(η ⊗ ξi, η ⊗ ξi)〉 ≤ ‖π ⊗ λ(x)‖ = ‖ 1 ⊗ λ(x)‖ = ‖λ(x)‖,
where we have used Fell’s absorption principle in the second to last inequality. Thus
‖π(x)‖ = sup ‖η‖=
| 〈π(x)η, η〉 | ≤ ‖λ(x)‖.
As π was an arbitrary ∗-representation, this shows ‖x‖u ≤ ‖λ(x)‖.
References
[1] N.P. Brown and N. Ozawa; C ∗ -algebras and finite-dimensional approximations, Grad. Stud. Math., 88, Amer. Math. Soc.,
Providence, RI, 2008
