Some Embedding and Containment Results

F ∈ L²(X ×Y, µ×ν) is nonconstant Γ-invariant, X is embedded into a nontrivial subset of L²(Y, ν)by the map x7→F(x,·).

Deneτ_g(h) =f(g^h−1) =f(h⁻¹gh). We have

γ·τ_g(h) =τ_g(γ⁻¹h) = f(h⁻¹γgγ⁻¹h) =f(h⁻¹g^γh) =τ_g^γ(h)

and sincef is injective everywhere,τ_g1 =τ_g2 ⇐⇒ g₁ =g₂. Sog 7→τ_g is a continuous invariant embedding ofE_c^G into E_l^L2(G). Hence we have E_c^G vⁱ_c E_l^L2(G).

Deneλ_g(h) =f(g⁻¹hg). We have

λ^γ_g(h) =λ_g(h^γ−1) =λ_g(γ⁻¹hγ) =f(g⁻¹γ⁻¹hγg) =λ_γg(x).

λ_g1 = λ_g2 ⇐⇒ ∀^∗_µh ∈ G(g₁⁻¹hg⁻¹₁ = g⁻¹₂ hg⁻¹₂ ) ⇐⇒ g₂g₁⁻¹ ∈ Z(G), where Z(G) is the center ofG.

If Z(G) ⊆Γ, then g 7→ λ_g is a continuous Borel reduction of E_l^G into Ec^L2(G), so E_l^G ≤_B Ec^L2(G). It is an embedding if Z(G) ={1}.

Suppose a BorelΓspace X with invariant Borel probability is properly isometriz- able (see next section) with witnessd. Denef_x(y) =d(x, y). It is easy to check that f_x1 =f_x2 ⇐⇒ x₁ = x₂ and γ·f_x = f_γ·x. Thus E_Γ^X vⁱ_c E_Γ^L2(X). Since conjugations are isometries, E_c^G vⁱ_c Ec^L2(G) for a compact Polish group G.

We can also show that Ec^L2(G) vⁱ_c E_l^L2(G). Let Φ : L²(G) → L²(G) be the map dened by

Φ(f)(g) = Z

h∈G

f(ghg⁻¹)dµ(h),

wheref is any element inL²(G)(we no longer needf to be a xed injective function).

Since

Φ(f^γ)(g) = Z

h∈G

f(γ⁻¹ghg⁻¹γ)dµ(h) = Φ(f)(γ⁻¹g) = γ·Φ(f)(g),

Φ is a Γ-map. And

Φ(f₁) = Φ(f₂) ⇐⇒ ∀^∗_µg(

Z

h∈G

f₁(ghg⁻¹)dµ(h) = Z

h∈G

f₂(ghg⁻¹)dµ(h))

⇐⇒ ∀^∗_µg(

Z

h∈G

(f₁−f₂)(ghg⁻¹)dµ(h) = 0

⇐⇒ f₁ =f₂.

Combining the above results, we have

Proposition 3.4.1. Let G be a compact Polish group and Γ be a subgroup of G. We have E_c^G vⁱ_c Ec^L2(G) vⁱ_c E_l^L2(G) and E_l^G vⁱ_c E_l^L2(G). If Z(G) ≤ Γ, E_l^G ≤c Ec^L2(G)

and if Z(G) = {1}, E_l^G vⁱ_c Ec^L2(G).

Example 3.4.2. Let X = G be a simply connected compact semisimple Lie group with a trivial center, Γ ⊂ Aut(G) a countable subgroup acting on G by au- tomorphisms. Identify G and Inn(G) ≤ Aut(G), which is a Γ invariant nite in- dex normal subgroup. We have E_Γ^G vⁱ_c Ec^Aut(G). Therefore E_Γ^G vⁱ_c E_l^L2(Aut(G)) and E_Γ^G vⁱ_c Ec^L2(Aut(G)).

3.4.2. Embeddings related to hyperniteness.

Proposition 3.4.3. Let X be a Borel Γ-space with invariant Borel probability measure µ. Then E_Γ^L2(X) ∼=_B E_Γ^{(M ALGµ)N} ∼=_B E^L2(X)N.

Proof.

M ALG_µv_B L²(X),

so

(M ALG_µ)^N v_B L²_B(X)^N.

To see that

L²(X)v_B (M ALG_µ)^N,

let

α:L²(X)→(M ALG_µ)^N

be the map dened by

α(f) = (f⁻¹(A_n))n∈N,

where{A_n}n∈Nis an enumeration of basic open subsets ofC. Clearly αis an injective Borel Γ-map. Since

L²(X)v_B (M ALG_µ)^N,

we also have

L²(X)^Nv_B (M ALG_µ)^N×N.

Therefore, we have

E_Γ^L2(X)vⁱ_B E_Γ^{(M ALGµ)N} vⁱ_B E^L2(X)N vⁱ_B E_Γ^{(M ALGµ)N×N}.

Notice that

E_Γ^{(M ALGµ)N} ∼=E_Γ^{(M ALGµ)N×N}.

So

E_Γ^L2(X) ∼=B E_Γ^{(M ALGµ)N} ∼=B E^L2(X)N.

Let X be a Borel Γ-space with invariant Borel probability measure µ. Consider the following property:

(*) For every sequence of Borel subsets (A_i), there is an N ∈ N such that T

i≤NΓ_Ai =T

i∈NΓ_Ai.

Property (*) holds for mixing and mild mixing action. In fact, it is easy to check that the smoothness ofE_Γ^L2(X)implies property (*). IfΓacts freely onM ALG_µ\{∅, X}, i.e. , γ is µ-ergodic for all γ ∈ Γ, then it satises (*). If there exists an r < 1 such that

γ·A=A⇒µ(A)> r

for every A∈M ALG_µ and γ ∈Γ\{1}, then (*) holds. More generally, if

sup

γ∈Γ\{1}

|{A∈M ALGµ :γ·A =A}|<∞,

then (*) holds.

Corollary 3.4.4. Let X be a Borel Γ-space with invariant Borel probability mea- sure µ. If the above (*)-property holds, then E_Γ^L2(X) is hypernite ⇐⇒ E_Γ^{M ALGµ} is hypernite.

Proof. Let

P_N ={(A_n)∈(M ALG_µ)^N: \

i<N

Γ_Ai 6= \

i≤N

Γ_Ai = \

i∈N

Γ_Ai}

so that each P_N isΓ-invariant Borel and{P_N}is a partition of (M ALG_µ)^N. We only need to show that E_Γ^PN = E_Γ^{(M ALGµ)N}|P_N is hypernite for each N. Fix an arbitrary N. Notice that

E_Γ^{(M ALGµ)N} ⊆(E^{M ALGµ})^N

is hypernite. So

E_Γ^{(M ALGµ)N} =E_{<T >}

for some Borel automorphism T : (M ALG_µ)^N →(M ALG_µ)^N. Dene

S : (M ALG_µ)^N→(M ALG_µ)^N

by

S((A_n)n∈N) = (γ·A_n)n∈N

for some γ ∈Γ such that(γ·A_n)_n≤N =T((A_n)_n≤N). Notice that if

(γ1·An)n≤N = (γ2·An)n≤N,

then

γ1 ∈γ2(\

i≤N

ΓAi) =γ2(\

i∈N

ΓAi).

So clearlyS is well-dened. Since

E_Γ^{(M ALGµ)N} =E_{<T >},

for every γ₁ ∈Γ, we can nd an n ∈N so that

(γ1·An)n≤N =Tⁿ((An)n≤N).

It it easy to check that

Sⁿ((A_n)n∈N)|N =Tⁿ((A_n)n≤N),

so

Sⁿ((A_n)n∈N) = (γ₂·A_n)n∈N

for some γ₂ ∈γ₁(T

i≤NΓ_Ai). Using the condition

(\

i≤N

Γ_Ai) = (\

i∈N

Γ_Ai)

again, we have

Sⁿ((A_n)_n∈N) = (γ₁·A_n)_n∈N

and

E_<S> =E_Γ^PN.

Therefore,E_Γ^PN is hypernite.

Let X be a Γ-space and µ a quasi-invariant measure. E_Γ^X is amenable i E_Γ^X is hypernite on an invariant µ-conull subset (see [Zimmer] or [Kaimanovich]).

Recall that 1_Γ ≺ λ_Γ i Γ is amenable (see [BHV]). If an ergodic Γ-space (X, µ) is amenable (see [Zimmer] for denition), then π^X ≺ λΓ (see [Kuhn]). The converse

is in general not true (see [AD]). Let

λ^X =λ^X_Γ = Z ⊕

X

λ_Γ/Γxdµ(x).

We have in analogy to the amenability of E_Γ^X:

Theorem 3.4.5. Let X be a Borel Γ-space with quasi-invariant Borel probability measure µ. If E =E_Γ^X is amenable, then π^X ≺λ^X.

Proof. We can nd a sequence λⁿ:E →Rof non-negative Borel functions such that

(i)λⁿ_x ∈`¹([x]_E), where λⁿ_x(y) =λⁿ(x, y), forxEy; (ii) kλⁿ_xk₁ = 1; and

(iii) There is a µ-conull Borel E-invariant set A⊆ X , such that

λⁿ_x −λⁿ_y →0 for all x, y ∈A (see [KM] or [Kaimanovich]).

Letφⁿ=√

λⁿ. We have φⁿ_x ∈`²([x]E), kφⁿ_xk= 1 and

φⁿ_x −φⁿ_y

2 ≤

λⁿ_x−λⁿ_y 1 →0.

Fix an arbitrary f ∈ L²(X, µ). Let f_n : E → C , (x, y) 7→ φⁿ(x, y)f(x) and γ·(x, y) = (γ·x, y) for xEy. Note that f(x)φⁿ_x ∈ `²([x]_E)for every x∈ X. For any γ ∈Γ,

|hγ ·f_n, f_ni − hγ ·f, fi| = Z

(

φⁿ_γ−1·x, φⁿ_x

−1) s

d(γµ)

dµ (x)f(γ⁻¹·x)f(x)dµ(x)

≤ Z

φⁿ_γ−1·x−φⁿ_x

2

2

s d(γµ)

dµ (x)f(γ⁻¹ ·x)f(x)dµ(x) .

Since 4≥

φⁿ_γ−1·x−φⁿ_x

2

→0 a.e. and

(d(γµ)

dµ (x))¹²f(γ⁻¹·x)f(x)∈L¹(X, µ),

|hγ·f_n, f_ni − hγ·f, fi| → 0 for every γ ∈ Γ. So for any nite subset F ∈ Γ and ε >0, we can nd an n such that

|hγ·f_n, f_ni − hγ·f, fi|< ε

for all γ ∈F. Since [y]E ∼=B Γ/Γy, it is easy to check that

fn ∈ Z ⊕

y∈X

`²([y]E)dµ(y) = Z ⊕

y∈X

`²(Γ/Γx)dµ(y)

and theΓ action on E is Borel isomorphic to the left translation of Γ on

Z ⊕ y∈X

Γ/Γ_xdµ(y).

Therefore every positive denite function realized in π^X is the pointwise limit of a sequence of positive denite functions realized in λ^X. In particular, we have π^X ≺

λ^X.

Call a Γ-space X coamenable if Γ_x is amenable for everyx ∈X. A Γ-space X is said to be coamenable µ-a.e. if an invariant µ-conull subset of X is coamenable.

Corollary 3.4.6. Let X be a Borel Γ-space with quasi-invariant Borel probability measure µ. If X is coamenable µ-a.e. and E_Γ^X is amenable, then π^X ≺λ_Γ.

Proof. Assume that E_Γ^X is amenable and X is coamenable µ-a.e. Since Γ_x is amenable µ-a.e., we have λ_Γ/Γx ≺ λ_Γ µ-a.e. Notice that the condition π₁ ≺ π₂ for unitary representations π₁, π₂ is equivalent to kπ₁(a)k ≤ kπ₂(a)k for all a ∈ `¹(Γ), whereπ(a) =P

a_γπ(γ)for every unitary representationπ(see Section F.4 of [BHV]).

So we have

λ_Γ/Γx(a)

≤ kλ_Γ(a)k for every µ-a.e. x∈X and a ∈`¹(Γ). It is easy to see that

λ^X(a)·f, f

= X

γ∈Γ

a_γ Z

X

λ_Γ/Γx ·f_x, f_x dµ(x)

= Z

X

X

γ∈Γ

a_γ

λ_Γ/Γx ·f_x, f_x dµ(x)

= Z

X

λ^X(a)·f_x, f_x dµ(x)

≤ Z

X

kλ_Γk²kf_xk²dµ(x)

= kλ_Γk²kfk²

for every f ∈R⊕

x∈X`<sup>2</sup>(Γ/Γ<sub>x</sub>)dµ(x) and a ∈`¹(Γ). So

λ_Γ/Γx(a)

≤ kλ_Γ(a)k for every a∈`¹(Γ)and hence by Theorem 3.4.5,

π^X ≺λ^X ≺λ_Γ.

Corollary 3.4.7. Let X be a Borel Γ-space with invariant Borel probability mea- sure µ.

If ∃^∗_µx(Γ_x is amenable) and E_X^Γ is amenable, then Γ is amenable.

Proof. Since amenability is preserved under subsets, we may assume thatE_Γ^X is amenable andX is coamenableµ-a.e. (otherwise, replace X by an non-null invariant Borel subsetX⁰ ⊆X, which is coamenable µ|X⁰-a.e.)

By Corollary 3.4.6,

1≤π^X ≺λ^X ≺λ_Γ.

SoΓ is amenable.

Corollary 3.4.8. Let X be a Borel Γ-space and Γ a free nonabelian group. If X is coamenable, then E =E_Γ^X0 is hypernite and compressible, whereX⁰ is the nonfree part of X.

In particular, if Γ is a free nonabelian group and X is coamenable, then the Γ action is free µ-a.e. for every Γ-invariant Borel probability measure µ on X.

Proof. By replacing X with X⁰, we may assume Γ_x is amenable and nontrivial for every x∈ X. Thus by the Nielsen-Schreier theorem,Γ_x is cyclic for everyx. Fix a well-ordering on Γ. Let

ρ(γ) = min

α∈Γ{αγα⁻¹} and put γ₁ <_ργ₂ if

ρ(γ₁)≤ρ(γ₂)∨(ρ(γ₁) =ρ(γ₂)⇒γ₁ < γ₂).

<ρis also a well-ordering onΓ. Consider the assignment x7→ax ∈Γ, whereax is the smallest generator of Γ_x respect to <_ρ, i.e.,

haxi= Γx∧ax ≤ρa⁻¹_x .

This assignment is clearly Borel and a_x = a_y if Γ_x = Γ_y. Furthermore, if y ∈ [x]_E, then a_yE_c^Γa_x (recall that γ₁E_c^Γγ₂ i γ₁ = γγ₂γ⁻¹ for some γ ∈ Γ ). This is because γa_xγ⁻¹ is a generator ofΓ_y, for someγ ∈Γ such thatγ·x=y. If γa_xγ⁻¹ 6=a_y, then a_xE_c^Γa⁻¹_y . Notice that ρ is a selector of E_c^Γ. We have

ρ(a⁻¹_y ) = ρ(a_x)≤ρ(a⁻¹_x ) = ρ(a_y)≤ρ(a⁻¹_y ).

and hence a_yE_c^Γa_x. Let

Y ={y∈X :a_y =ρ(a_y)}.

Since

ρ(ρ(a_x)⁻¹) =ρ(a⁻¹_x )≤ρ(a_x)

and ρ(a_x) is a generator of Γγ·x for some γ ∈Γ, ρ(a_x) = a_y for some y ∈[x]_E. So Y is a complete section.

Also ifxEy, thenxE_c^Γy. Therefore ifx, y ∈Y andxEy, thena_y =ρ(a_x) = a_x. By a basic fact from combinatorial group theory, two elements of a free group commute if and only if they are powers of a common element, that is,γ₁γ₂ =γ₂γ₁ i γ₁, γ₂ ∈ hγi for some γ ∈ Γ. Fix an arbitrary x ∈ Y. It is easy to check that we can uniquely write a_x asa_x =αβⁿα⁻¹, wheren ∈N, α is reduced, β is cyclically reduced (i.e., ββ is a reduced word) and β is not a nontrivial power of other elements, i.e.,

∀γ ∈Γ∀m >0(γ^m 6=β).

Putb_x =αβα⁻¹. We have

x(E|Y)y ⇐⇒ ∃n ∈Z(x=bⁿ_x ·y).

LetY_γ ={x∈Y :b_x=γ}. We haveE|Y_γ =E_hγi^Yγ. So

E|Y =⊕γ∈ΓE|Y_γ

is hypernite. Since Y is a complete section of E, E is hypernite. By Corollary 3.4.7, there is no Γ-invariant Borel probability measure onX.

The following is an analog to a property of amenable groups (Corollary G.3.8, [BHV]) and amenable actions (Lemma 4.5.1, [AD]), which is related to Theorem 3.4.5.

Proposition 3.4.9. Let X be a Borel Γ-space with quasi-invariant Borel proba- bility measure µ. Suppose H ≤ Γ and 1_Γ ≺ λ_Γ/H. Then E_H^X is amenable i E_Γ^X is amenable.

Proof. E_H^X ⊆E_Γ^X. So the amenability of E_Γ^X implies the amenability ofE_H^X. So assume E_H^X is amenable from now on. Let m be a Γ-invariant mean on λ_Γ/H and {p_x} a set of H-invariant local means on E_H^X (see [Kaimanovich]) such that x7→p_x(F) is measurable for anyF ∈L^∞(X, µ).

Denegf,x and qx by

g_f,x(γ_iH) =p_γ⁻¹

i ·x(f|_[γ⁻¹

i ·x]

EXH

)

and q_x(f) =m(g_f,x), where {γ_i} is a representative of Γ/H for allf ∈`<sup>∞</sup>([x]<sub>E</sub>X Γ). It is straightforward to check that g<sub>f,x</sub> ∈ `^∞(Γ/H), q_x ∈ (`^∞)^∗[x]_EX

Γ and x 7→ q_x(F) is measurable for any F ∈L^∞(X, µ). Check

gf,γ·x(γ_iH) = p_(γ⁻¹_γi)⁻¹_·x(f|_[(γ⁻¹_γi)⁻¹_·x]

EXH

)

= p_h−1γ_j⁻¹·x(f|_[γ⁻¹

j ·x]

EXH

)

= p_γ⁻¹

j ·x(f|_[γ⁻¹

j ·x]

EXH

)

= g_f,x(γ_jH)

= gf,x(γ⁻¹γiH)

= γ·g_f,x(γ_iH),

where γ⁻¹γ_i =γ_jh for some h∈H and

qγ·x(f) =m(gf,γ·x) =m(γ·g_f,x) =m(g_f,x) =q_x(f).

Soq_x. is Γ-invariant. E_Γ^X is amenable.

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