The Furstenberg lemma characterizes amenability^{∗}

Greg Hjorth^{†}

April 29, 2005

Abstract

We characterize amenability in terms of the existence of equivariant assignments of measures for cocycles into the homeomorphism group of a single compact metric space.

1 Introduction

It is well known that every cocycle from an amenable equivalence relation to the homeomorphism group of a compact metric space or from an amenable action of a group to the homeomorphism group of a compact metric space admits a suitably equivariant assignment of measures. This theorem in some form dates back to [3], though an explicit discussion can be found in chapter 4 of [5].

In this note we obtain a converse using ideas suggested by the concept of ultrahomogeneous structure from [4].

It is important to note here that the metric space is fixed and metrizable. If we have the cocycle into the homeomorphism groups of different compact non-metrizable spaces, then the existence of an equivariant assignment of measures becomes closer to the definition of §4.3 [5] and the result is already known; see [1].

Theorem 1.1 Let E be a countable, measurable equivalence relation on a standard Borel probability space (S, μ) and suppose that for every measurable cocycle

α : E → Hom(K)

from E to the homeomorphism group of a compact metric space we have a measurable assignment

x ↦ μ_{x},
S → M(K),

from S to the probability measures on K, which is equivariant in the sense that

μ_{z} = α(z, x) · μ_{x}.

Then E is amenable.

∗Key words and phrases: Amenability, orbit equivalence, mean, homogeneous structure. 2000 Mathematics Subject Classification: Primary 03E15, 37A20. Secondary 28A60, 03C15

†NSF grant DMS 0140503

Recall that a cocycle from E to a group Γ consists in a function α : E → Γ with

α(x_{3}, x_{2})α(x_{2}, x_{1}) = α(x_{3}, x_{1}).

In the case that a group Λ acts on the space X a cocycle for the action is a function

α : X × Λ → Γ

such that

α(g, h · x)α(h, x) = α(gh, x).

(In the below we generally do not distinguish between something happening everywhere and almost everywere; the distinction will be irrelevant.)

We take the Connes-Feldman-Weiss definition of amenability for equivalence relations.

Definition An equivalence relation E on (S, μ) is amenable if there is a linear function

P : L^{∞}(E) → L^{∞}(S)

which is positive, sends the constant function 1 on E to the constant function 1 on S, and commutes with all projections and all morphisms included in the equivalence relation. Following [5], we say that the measurable action of a lcsc group Γ on a space X is amenable if the induced orbit equivalence relation is amenable and almost every point x has its stabilizer,

Γ_{x} = {g ∈ Γ : g · x = x}

amenable.

Notation For ψ ∈ L^{∞}(E) and f : A → B a measurable bijection included in the graph of E, we let ψ^{f} be defined by ψ^{f} (f(x), y) = ψ(x, y) for (f(x), y) ∈ E ∩ B × S and ψ^{f} (z, y) = 0 for z not in B. For ψ ∈ L^{∞}(S) we would let ψ^{f} (f(x)) = ψ(x) and ψ^{f} (z) = 0 for z outside B.

With this notation we can rephrase the commutativity requirement for P: Given any such f : A → B a measurable bijection with f(x)Ex all x ∈ A, and given an ψ ∈ L^{∞}(E),

P (ψ^{f}) = (P (ψ))^{f} .

The Connes-Feldman-Weiss definition of amenability is equivalent to the equivalence relation being measurably hyperfinite.

Theorem 1.2 Let Γ be a countable group acting measurably on a standard Borel probability space (S, μ) and suppose that for every measurable cocycle

α : S × Γ → Hom(K)

we have a measurable assignment

x ↦ μ_{x},
S → M(K),

from S to the probability measures on K, which is equivariant in the sense that

μ_{g·x} = α(x, g) · μ_{x}.

Then the action is amenable.

As a working definition, let us simply say that an action is amenable if the resulting equivalence relation is amenable and the stabilizers of a.e. point are amenable.

2 Proof

We need a number of specific definitions to support the construction ahead.

Definition A partition of an infinite set X consists in a set {P_{1}, P_{2}, ..., P_{ℓ}} such that P_{i} is disjoint from P_{j} for i ≠ j and

⋃_{i≤ℓ} P_{i} = X.

Given a finite sequence f⃗ = (f_{1}, f_{2}, ...f_{n}) of functions from X to {0, 1}, we say that P = {P_{1}, P_{2}, ..., P_{ℓ}} is the partition generated by f⃗ if each P_{i} equals

{a ∈ X : f_{1}(a) = k_{1}^{i}, f_{2}(a) = k_{2}^{i}, ...f_{n}(a) = k_{n}^{i}}

for some choice of k_{1}^{i}, ..., k_{n}^{i} ∈ {0, 1}.

Remark: The trivial partition P = {X} is generated by the empty sequence.

Definition Let X be a countably infinite set. A collection

F ⊂ {0, 1}^{X}

is said to be random if whenever (i) P = {P_{1}, P_{2}, ..., P_{ℓ}} is a partition generated by some finite sequence f⃗ from F, and (ii) we have cardinals κ_{i}, λ_{i} ∈ {0, 1, 2, ...ℵ_{0}}, each i ≤ n, with κ_{i} + λ_{i} always equal to |P_{i}|, and (iii) at each i we have finite subsets u⃗_{i} ⊂ P_{i}, v⃗_{i} ⊂ P_{i}, each of cardinality at most κ_{i} and λ_{i} respectively,

there is some g ∈ F, not appearing on the f⃗ sequence, with

|{n ∈ P_{i} : g(n) = 1}| = κ_{i},
|{n ∈ P_{i} : g(n) = 0}| = λ_{i},

and g assuming the value 1 on each element of u⃗_{i} and the value 0 on each element of v⃗_{i}.

Remarks: (i) Here |Y | denotes the cardinality of Y.

(ii) If one of κ_{i}, λ_{i} equals ℵ_{0} then κ_{i} + λ_{i} = ℵ_{0}. (ℵ_{0} + 1 = ℵ_{0} + 2 = ... = ℵ_{0} + ℵ_{0} = ℵ_{0}). (ii) The idea of the definition of a random collection F is this: Given any finite sequence f⃗

from F and finite a⃗ from X, we can find a new function g ∈ F which behaves relative to f⃗ and a⃗ in any previously prescribed manner. Anything which can happen does.

Lemma 2.1 If X is a countably infinite set, and F_{0} ⊂ {0, 1}^{X} is a countable collection of functions, then we can find a countable random collection F ⊂ {0, 1}^{X} which includes F_{0}.

Proof We build F in stages. We first take some countable F_{1} ⊃ F_{0} which includes enough functions to satisfy the definition of random for any choice of parameters from F_{0}. We then repeat, replacing F_{0} by F_{1} and obtain F_{2} ⊃ F_{1}. In this way we obtain an ascending chain

F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ ... ⊂ F_{n} ⊂ F_{n+1}...

and finish with

F = ⋃_{n∈ℕ}F_{n}.

□

Lemma 2.2 If X_{1}, X_{2} are countably infinite sets, F_{1} ⊂ {0, 1}^{X_{1}}, F_{2} ⊂ {0, 1}^{X_{2}} both random, then there is a bijection

π : X_{1} → X_{2}

such that the induced function

φ : {0, 1}^{X_{1}} → {0, 1}^{X_{2}}

defined by

φ(f)(a) = f(π^{−1}(a))

provides a bijection between F_{1} and F_{2}.

Proof We build π in a sequence of stages, with only a finite amount of π established at each stage. We simultaneosly build φ finite step by finite step. We guarantee an isomorphism at the very end stage by a “back and forth argument”, in the sense of §3.2 [4].

Let (f_{i})_{i∈ℕ}, (g_{i})_{i∈ℕ} enumerate F_{1}, F_{2} respectively, and let (a_{i})_{i∈ℕ}, (b_{i})_{i∈ℕ} enumerate X_{1}, X_{2} respectively. Using that X_{2} is random and considering the trivial partition {X_{2}} we can find some g ∈ F_{2} such that

|{b ∈ X_{2} : g(b) = 1}| = |{a ∈ X_{1} : f_{1}(a) = 1}|,
|{b ∈ X_{2} : g(b) = 0}| = |{a ∈ X_{1} : f_{1}(a) = 0}|.

Let φ(f_{1}) = g, choose b′ ∈ X_{2} with g(b′) = f_{1}(a_{1}) and let π(a_{1}) = b′.

Now consider g_{1}. We want to find suitable f ∈ F_{1} with φ(f) = g_{1}. First let us assume that g_{1} ≠ g. Then by considering the randomness of F_{1} we can find some f ∈ F_{1} with

f(a_{1}) = g_{1}(b′)

and

|{b ∈ X_{2} : g(b) = ℓ_{1}, g_{1}(b) = ℓ_{2}}| = |{a ∈ X_{1} : f_{1}(a) = ℓ_{1}, f(a) = ℓ_{2}}|,

any choice of ℓ_{1}, ℓ_{2} ∈ {0, 1}. We then let φ(f) = g_{1}, and go on to consider how to extend π by looking at b_{1}. If b_{1} = b′ then π(a_{1}) = b_{1}; otherwise we choose some a′ ≠ a_{1} with

f(a′) = g_{1}(b_{1}),
f_{1}(a′) = g(b_{1}),

and let π(a′) = b_{1}. (Note: We can find such an a′ since |{a : f(a) = g_{1}(b_{1}), f_{1}(a) = g(b_{1})}| equals |{a : g_{1}(a) = g_{1}(b_{1}), g(a) = g(b_{1})}|.)

Having done all this under the assumption g_{1} does not equal g we need to consider also the case g_{1} = g. But here there is really nothing to do, since we have already decided that φ(f) = g, and we simply go on to find some suitable choice of a′ with π(a′) = b_{1} as above.

Now we go back to the X_{1} side and try to find a suitable value for φ(f_{2}).

Again there is a split in cases. If f_{2} ∈ {f_{1}, f} then the value of φ(f_{2}) is already determined and the whole process is kind of trivial.

So suppose instead f_{2} ≠ f, f_{1}. Then we can choose g′ ∈ F_{2}, g′ ≠ {g_{1}, g} so that at each choice of ℓ_{1}, ℓ_{2}, ℓ_{3} we have

|{b ∈ X_{2} : g(b) = ℓ_{1}, g_{1}(b) = ℓ_{2}, g′(b) = ℓ_{3}}| = |{a ∈ X_{1} : f_{1}(a) = ℓ_{1}, f(a) = ℓ_{2}, f_{2}(a) = ℓ_{3}}|,

and

g′(b_{1}) = f_{2}(a′),
g′(b′) = f_{2}(a_{1}).

We again choose a suitable value of π(a_{2}) so that

g(π(a_{2})) = f_{1}(a_{2}),

g_{1}(π(a_{2})) = f(a_{2}),
g′(π(a_{2})) = f_{2}(a_{2}).

With only futher notational complications, we keep going in this way for infinitely many steps, back and forth between the two sides, and thereby enforce an isomorphism. □

Definition A measurable equivalence relation E on a standard Borel probability space (S, μ) is said to have the Furstenberg property if for every measurable cocycle

α : E → Hom(K)

from E to the homeomorphism group of a compact metric space we have a measurable assignment

x ↦ μ_{x},
S → M(K),

from S to the probability measures on K, which is equivariant in the sense that

μ_{z} = α(z, x) · μ_{x}.

A measurable action of a countable group Γ on standard Borel probability space (S, μ) is said to have the Furstenberg property if for every measurable cocycle

α : S × Γ → Hom(K)

we have a measurable assignment

x ↦ μ_{x},
S → M(K),

from S to the probability measures on K, which is equivariant in the sense that

μ_{g·x} = α(x, g) · μ_{x}.

Assume from now on that E on (S, μ) is an equivalence relation with (i) (S, μ) a standard Borel probability space; (ii) every E equivalence class is countably infinite; (iii) E is measurable (e.g. in the sense of being a measurable subset of (S^{2}, μ^{2})); (iv) E has the Furstenberg property.

We will work towards showing that E is amenable. After all that is said and finished, we will indicate how the argument modifies for the case of group actions.

Note that the collection of means (i.e. positive, linear, norm one, unit respecting)

P : L^{∞}(E) → L^{∞}(S)

is compact in the topology given by the functions

P ↦ ∫_{A} P (ψ)dμ

(ψ ∈ L^{∞}(E), A ⊂ X measurable). (The point here is that we can represent P by looking at its restriction to the unit ball (L^{∞}(E))_{1}; then each P (ψ) lands in (L^{∞}(S))_{1} which is compact in the weak-∗ topology.) Inside this collection, for any given ψ ∈ L^{∞}(E), f ⊂ E, the ones which appropriately commute, in the sense P (ψ^{f}) = (P (ψ))^{f} form a closed subset.

For purely notational convenience, let us fix G a group of measurable bijections of S whose orbit equivalence relation equals E. Appealing to the compactness of the space of means, it suffices to show that given a countable collection F_{0} ⊂ L^{∞}(E) consisting of functions assuming only the values 0 and 1 and closed under the G action, we may find

P : F_{0} → L^{∞}(S)

such that:

(i) P (ψ^{f}) = (P (ψ))^{f} all ψ ∈ F_{0} and f ∈ G;

(ii) if A ⊂ S measurable and ψ ∈ F_{0} has ψ(x, y) = 0 all x ∈ A, then likewise P (ψ)(x) = 0

all x ∈ A;
(iii) if ψ ≤ ψ′ both in F_{0}, then P (ψ) ≤ P (ψ′);
(iv) if ψ + ψ′ = ψ′′ all in F_{0}, then P (ψ′′) = P (ψ′) + P (ψ);
(v) P (1) = 1.

For each ψ ∈ F_{0} and x ∈ S we define ψ_{x} ∈ ℓ^{∞}([x]_{E}) by ψ_{x}(y) = ψ(x, y). After possibly expanding the set F_{0} we may assume that at each x ∈ S, the collection

{ψ_{x} : ψ ∈ F_{0}}

is random (in our earlier sense) for {0, 1}^{[x]_{E}}.

Now fix some arbitrary countably infinite set X and let F be a collection of random functions on X. Let K be the space of all means just on F; that is to say, all

m : F → [0, 1]

such that (a) if ψ ≤ ψ′ both in F, then m(ψ) ≤ m(ψ′);

(b) if ψ + ψ′ = ψ′′ all in F, then m(ψ′′) = m(ψ′) + m(ψ);
(c) m(1) = 1.

Since F is countable, K is compact, metric.

At each x ∈ S we may in a measurable manner assign a bijection

π_{x} : [x]_{E} → X

such that the induced map

ρ_{x} : ℓ^{∞}([x]_{E}) → ℓ^{∞}(X)

defined by

(ρ_{x}(ψ))(a) = ψ(π_{x}^{−1}(a))

provides a bijection of {ψ_{x} : ψ ∈ F_{0}} with F. (The existence of some such assignment, without worrying about measurability, follows by 2.2. Then to see it can be found measurably, we can either appeal to the Jankov von Neumann uniformization theorem for measurable selectors, or go back into the proof of 2.2 and see that it is so uniform as to actually allow a Borel choice of x ↦ ρ_{x}.)

We can then define a measurable cocyle

α : E → Hom(K)

by

(α(x, z))(m)(ψ) = m(ρ_{z}(ρ_{x}^{−1}(ψ))).

We apply the assumption of Furstenberg’s property, and get a measurable assignment

S → M(K)
x ↦ μ_{x}

with

μ_{x} = α(x, z) · μ_{z}.

We then let

(P (ψ))(x) = ∫_{K} m(ρ_{x}(ψ_{x}))dμ_{x}(m).

We want to check that the induced

P : L^{∞}(E) → L^{∞}(S)

satisfies (i)-(v) above, and then we will be done. (ii)-(v) follow at once from the definition, so we only need to deal with (i).

So fix some f : S → S in Γ and ψ ∈ F_{0}. Then

(P (ψ^{f}))(x) = ∫_{K} mρ_{x}((ψ^{f})_{x})dμ_{x}(m)

= ∫_{K} mρ_{x}(ψ_{f^{−1}(x)})dμ_{x}(m),

since

(ρ_{x}((ψ^{f})_{x})))π_{x}(y) = (ψ^{f})_{x}(y) = ψ^{f} (x, y)
= ψ(f^{−1}(x), y) = ψ_{f^{−1}(x)}(y) = ρ_{x}(ψ_{f^{−1}(x)})π_{x}(y).

On the other hand,

(P (ψ))^{f} (x) = (P (ψ))(f^{−1}(x))
= ∫_{K} mρ_{f^{−1}(x)}((ψ)_{f^{−1}(x)})dμ_{f^{−1}(x)}(m),

which in turn, by our assumption of α-equivariance, equals

∫_{K} mρ_{f^{−1}(x)}((ψ)_{f^{−1}(x)})dα(f^{−1}(x), x) · μ_{x}(m),

which by the definition of the cocycle unwinds as

∫_{K}(ρ_{x}(ρ_{f^{−1}(x)}^{−1}(ρ_{f^{−1}(x)}(ψ_{f^{−1}(x)}))))dμ_{x}(m) = ∫_{K}(ρ_{x}(ψ_{f^{−1}(x)})dμ_{x}(m),

as above and as required.

Thus we have established the amenability of E assuming the existence of equivariant cocycles for all the appropriate cocycles. It remains now to indicate the modifications will enable the same result to be proved for group actions.

From now on assume that Γ is a countable group acting measurably on a standard Borel (S, μ) and the action of Γ has the Furstenberg property. Given the earlier result, we know that the orbit equivalence relation E_{Γ} is amenable, and it suffices to show that a.e. the stabilizers are amenable. For this purpose it suffices to show that if Γ_{0} is a finitely generated subgroup of Γ and there is a non-null set of x ∈ S whose stabilizer includes Γ_{0} then Γ_{0} is non-null.

So fix such a Γ_{0}. Let {f_{i} : i ∈ ℕ} be a countable collection of functions

Γ_{0} → {0, 1}

closed under Γ_{0}-translation. By compactness again, it suffices to show that there is a Γ_{0}-invariant mean defined just on {f_{i} : i ∈ ℕ}.

Let (Γ_{0}γ_{i})_{i∈Λ} enumerate without repititions the left cosets of Γ_{0}. At each j ∈ ℕ define

f̂_{j} : Γ → {0, 1}

by

f̂_{j}(γγ_{i}) = f_{j}(γ)

for γ ∈ Γ_{0}; in other words, we multiply out f_{j} across the various cosets using our representatives (γ_{i})_{i∈Λ}.

Now take a countable collection

F ⊂ L^{∞}(S × Γ)

such that (i) each h ∈ F assumes only the values 0 and 1; (ii) if x ∈ S, j ∈ ℕ, then there is h ∈ F with

h_{x} = f̂_{j}

(where as usual we define h_{x} by h_{x}(γ) = h(x, γ));

(iii) at each x, {h_{x} : h ∈ F} is a random collection of functions from Γ to {0, 1}; (iv) if γ ∈ Γ, h ∈ F, then

γ · h ∈ L^{∞}(S × Γ)

(where as usual γ · h is defined by (γ · h)(y, γ_{1}) = h(γ^{−1} · y, γ^{−1}γ_{1})).

We again let X be some countably infinite space and let F be a random collection of functions from X to {0, 1}. At each x ∈ S we again measurably assign a corresponding bijection

π_{x} : Γ → X

so that the induced

ρ_{x} : ℓ^{∞}(Γ) → ℓ^{∞}(X)

provides a bijection. We again let K be the space of means on F.

Given x ∈ S, γ ∈ Γ, we obtain a continuous

α(x, γ) : K → K

defined by

(α(x, γ))(m)(ψ) = m(ρ_{x}(γ^{−1} · (ρ_{γ·x}^{−1}(ψ)))),

where Γ acts on ℓ^{∞}(Γ) by left translation in the usual way – that is to say, (γ·φ)(γ′) = φ(γ^{−1}γ′). It is routinely seen that this provides a measurable cocycle

α : X × Γ → Hom(K).

We apply the assumption of the Furstenberg property and get

S → M(K)
x ↦ μ_{x}

with

μ_{g·x} = α(x, g) · μ_{x}.

At each x we can pull back and obtain a mean ν_{x} defined on {φ_{x} : φ ∈ F} by

ν_{x}(φ) = ∫_{K} m(ρ_{x}(φ))dμ_{x}(m).

It is then immediate that if Γ_{0} is included in the stabilizer of x then ν_{x} is Γ_{0}-invariant. From this we obtain a Γ_{0}-invariant mean on {f_{i} : i ∈ ℕ} with

ν(f_{j}) = ν_{x}(f̂_{j}).

References

[1] T. Bates, MSc, Thesis, University of Ottawa, 1994.

[2] A. Connes, J. Feldman, B. Weiss, An amenable equivalence relation is generated by a single transformation, Journal of Ergodic Theory and Dynamical Systems, vol. 1(1981), pp. 431-450.

[3] H. Furstenberg, Boundary theory and stochastic processes on homogeneous spaces in Harmonic analysis on homogeneous spaces, in Symposia on pure and applied math, Williamstown, 1972, Proceedings vol. 26(1973), pp. 193-229

[4] W. Hodges, Model Theory, Cambridge University Press, Cambridge, 1993.

[5] R. Zimmer, Ergodic theory and semi-simple groups, Birkhauser, Basel, 1984.

6363 MSB Mathematics UCLA 405 Hilgard Avenue Los Angeles CA90095-1555 greg@math.ucla.edu www.math.ucla.edu/˜ greg