jessepeterson march17,2011 - vanderbilt university · 2016. 10. 4. · chapter 1 representations of...

Lecture notes on ergodic theory

Jesse Peterson

March 17, 2011

Contents

1 Representations of groups 5

1.1 Definitions and constructions . . . . . . . . . . . . . . . . . . . . 51.2 Functions of positive type . . . . . . . . . . . . . . . . . . . . . . 81.3 Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Induced representations . . . . . . . . . . . . . . . . . . . . . . . 121.5 Invariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6.1 Von Neumann’s Mean Ergodic Theorem . . . . . . . . . . 211.7 Mixing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.7.1 Compact representations . . . . . . . . . . . . . . . . . . . 241.7.2 Weak mixing for amenable groups . . . . . . . . . . . . . 26

1.8 Weak containment . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Group actions on measure spaces 31

2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 The Koopman representation . . . . . . . . . . . . . . . . . . . . 35

2.2.1 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2.2 Weak mixing . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.3 Compact actions . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 A remark about measure spaces . . . . . . . . . . . . . . . . . . . 412.4 Gaussian actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5 Ergodic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6 Recurrence theorems . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Uniform multiple recurrence 53

3.1 Multiple recurrence for compact actions . . . . . . . . . . . . . . 543.2 Multiple recurrence for weak mixing actions . . . . . . . . . . . . 553.3 Joinings and the basic construction . . . . . . . . . . . . . . . . . 58

3.3.1 Joinings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.2 The basic construction . . . . . . . . . . . . . . . . . . . . 59

3.4 Ergodic, weak mixing, and compact extensions . . . . . . . . . . 64

3

4 CONTENTS

Chapter 1

Representations of groups

1.1 Definitions and constructions

Definition 1.1.1. Let Γ be a group, a unitary representation (resp. anorthogonal representation) of Γ is a homomorphism π : Γ → U(H) (resp.π : Γ → O(H)) from Γ into the unitary (resp. orthogonal) group of a complex(resp. real) Hilbert space H.

Example 1.1.2. Let H be a Hilbert space and suppose Γ is a group. Thetrivial representation of Γ on H is given by the homomorphism which takesany group element to the identity operator on H.

Example 1.1.3. Let Γ be a group, the left-regular representation (resp.right-regular representation) is given by λ : Γ → U(ℓ2Γ), (resp. ρ : Γ →U(ℓ2Γ)) which is defined by the formula λ(γ)(Σx∈Γαxδx) = Σx∈Γαxδγx (resp.ρ(γ)(Σx∈Γαxδx) = Σx∈Γαxδxγ−1).

Example 1.1.4. More generally, if a group Γ acts on a set I then there is an as-sociated unitary representation on ℓ2I defined by λ(γ)(Σx∈Iαxδx) = Σx∈Iαxδγx.When Σ < Γ is a subgroup, then Γ acts by left multiplication on the quotientspace Γ/Σ and the associated unitary representation is the quasi-regular rep-

resentation.

Suppose Γ is a group and Σ < Γ is a subgroup. If π : Γ → U(H) is aunitary representation, then we obtain the reduced representation π|Σ of Σby restricting π to Σ.

Given a complex Hilbert space H the adjoint Hilbert space H is theHilbert space which agrees with H as a set, has scalar multiplication given by

λξ = λξ and whose inner-product is given by 〈ξ, η〉H = 〈ξ, η〉H.Given a unitary representation π : Γ → U(H) we define the adjoint rep-

resentation π : Γ → U(H) by setting π(γ)ξ = π(γ)ξ. Note that we have thenatural identification π = π.

If π : Γ → O(H) is an orthogonal representation where H is a real Hilbertspace then we will use the convention H = H and π = π.

5

6 CHAPTER 1. REPRESENTATIONS OF GROUPS

If π : Γ → U(H), and ρ : Γ → U(K) are two unitary representations, thenπ is contained in ρ if there is an isometry U : H → K such that for all γ ∈ Γwe have that Uπ(γ) = ρ(γ)U . If U can be chosen to be a unitary from H to Kthen the representations are isomorphic and we will write π ∼= ρ.

Given a family of unitary representations πι : Γ → U(Hι), with ι ∈ I, thedirect-sum representation is given by the map ⊕ι∈Iπι : Γ → U(⊕ι∈IHι)defined by

(⊕ι∈Iπι)(γ) = ⊕ι∈I(πι(γ)).

A unitary representation π : Γ → U(H) is reducible if it contains a non-trivial sub-representations, or equivalently, there exists a closed Γ-invariant sub-space K ⊂ H such that K 6= 0,H. In this case it is easy to see that K⊥ is alsoΓ-invariant and the natural isomorphism from K ⊕ K⊥ induces an equivalencebetween the representations π|K ⊕ π|K⊥ and π. If a unitary representation hasno non-trivial invariant subspace we say that it is irreducible.

A vector ξ ∈ H is a cyclic vector if the smallest Γ-invariant subspace of Hwhich contains ξ is H itself. Note that if π is irreducible then every non-zerovector is cyclic.

Lemma 1.1.5 (Schur’s Lemma). Let π : Γ → U(H), and ρ : Γ → U(K)be two irreducible unitary representations of a group Γ, if T ∈ B(H,K) is Γ-invariant then either T = 0, or else T is a scalar multiple of a unitary. Inparticular, B(H,K) has a non-zero Γ-invariant operator if and only if π and ρare isomorphic.

Proof. Let π and ρ be as above and suppose T ∈ B(H,K) is Γ-invariant. Thus,T ∗T ∈ B(H) is Γ-invariant and hence any spectral projection of T ∗T givesa Γ-invariant subspace. Since π is irreducible it then follows that T ∗T ∈ C.If T ∗T 6= 0 then my multiplying T by a scalar we may assume that T is anisometry. Hence, TT ∗ ∈ B(K) is a non-zero Γ-invariant projection, and since ρis irreducible it follows that TT ∗ = TT ∗ = 1.

If I is finite, then the tensor product representation is given by the map⊗ι∈Iπι : Γ → U(⊗ι∈IHι) defined by

(⊗ι∈Iπι)(γ) = ⊗ι∈I(πι(γ)).

Lemma 1.1.6 (Fell’s Absorption Principle). Let π : Γ → U(H) be a unitaryrepresentation of a group Γ, let 1H denote the trivial representation of Γ onH, and let λ : Γ → U(ℓ2Γ) denote the left-regular representation. Then therepresentations λ⊗ π and λ⊗ 1H are isomorphic.

Proof. Consider the unitary U ∈ U(ℓ2Γ⊗ H) determined by U(δx ⊗ ξ) = δx ⊗π(x)ξ, for all x ∈ Γ, ξ ∈ H. Then for all γ, x ∈ Γ, and ξ ∈ H we have that

U∗(λ⊗ π)(γ)U(δx ⊗ ξ) = U∗(λ⊗ π)(γ)(δx ⊗ π(x)ξ)

= U∗(δγx ⊗ π(γ)π(x)ξ)

= δγx ⊗ π((γx)−1)π(γ)π(x)ξ = (λ⊗ 1H)(γ)(δx ⊗ ξ).

1.1. DEFINITIONS AND CONSTRUCTIONS 7

Example 1.1.7 (Hilbert-Schmidt operators). Given two Hilbert spaces H andK with respective orthonormal bases ξi ⊂ H and ηj ⊂ K we define thespace of Hilbert-Schmidt operators from H to K by

HS(H,K) = S ∈ B(H,K) | ‖S‖2HS = Σi‖Sξi‖2 < ∞.

Parseval’s identity gives

Σi‖Sξi‖2 = Σi,j |〈Sξi, ηj〉|2 = Σj‖S∗ηj‖2,

hence we see that this sum is independent of the orthonormal bases chosen.Moreover, this shows that if A ∈ B(H0,H), B ∈ B(K,K0), and S ∈ HS(H,K)then BSA ∈ HS(H0,K0) and

‖BSA‖ ≤ ‖BSA‖HS ≤ ‖B‖‖S‖HS‖A‖.

The space of Hilbert-Schmidt operators is an inner-product space with inner-product given by

〈S, T 〉HS = Σi,j〈Sξi, ηj〉〈ηj , T ξi〉,which is well defined by the Cauchy-Schwarz inequality, and does not dependon the bases by applying Perseval’s identity.

The inner-product defined above turns HS(H,K) into a Hilbert space. Tosee this suppose Sn ∈ HS(H,K) is a sequence of Hilbert-Schmidt operators,which is Cauchy in the Hilbert-Schmidt norm, then Sn is also Cauchy in theoperator norm and hence converges in the operator norm to an operator S :H → K.

For any finite dimensional subspace K0 ⊂ K it is easy to see that

‖PK0S − PK0

Sn‖HS → 0,

and thus we have that

‖S‖HS = supK0⊂K, dim(K0)<∞

‖PK0S‖HS

≤ lim supn→∞

‖Sn‖HS < ∞.

Hence, S ∈ HS(H,K) and so all that remains to show is that ‖S − Sn‖HS → 0.Suppose ε > 0 is given. Since S−Sn is Cauchy in the Hilbert-Schmidt norm,

there exists some N > 0 such that for all n ≥ N we have that ‖SN − Sn‖HS <ε/2. Take K0 ⊂ K a finite dimensional subspace such that ‖PK⊥

0(S−SN )‖HS <

ε/2. Then for all n ≥ N we have

‖S − Sn‖HS

≤ ‖PK0S − PK0

Sn‖HS + ‖PK⊥0S − PK⊥

0SN‖HS + ‖PK⊥

0SN − PK⊥

0Sn‖HS.

The first term above tends to zero while the others are each at most ε/2, hence

lim supn→∞

‖S − Sn‖HS ≤ ε.


Since ε > 0 was arbitrary it follows that ‖S − Sn‖HS → 0.Note that because this does not depend on the choice of bases, we have that

unitaries U ∈ U(H) and V ∈ U(K) define a unitary on HS(H,K) by the mapS 7→ USV .

Suppose now that Γ is a group and π : Γ → U(H), ρ : Γ → U(K) aretwo unitary representations. We then have a unitary representation of Γ onHS(H,K) given by T 7→ ρ(γ)Tπ(γ−1), for each γ ∈ Γ.

Consider the linear map Ξ between K⊗H and HS(H,K) which takes a vectorζ ∈ K⊗H to the operator Ξ(ζ) determined by the formula

〈Ξ(ζ)ξ, η〉 = 〈ζ, η ⊗ ξ〉,

for all ξ ∈ H and η ∈ K.Then Ξ is a unitary since if ζ, ζ′ ∈ HS(H,K) we have that

〈Ξ(ζ),Ξ(ζ′)〉HS = Σi,j〈Ξ(ζ)ξi, ηj〉〈ηj ,Ξ(ζ′)ξi〉

= Σi,j〈ζ, ηj ⊗ ξi〉〈ηj ⊗ ξi, ζ′〉 = 〈ζ, ζ′〉.

It is easy to see that this isomorphism in turn gives an isomorphism betweenρ⊗ π and the representation of Γ on HS(H,K) described above.

1.2 Functions of positive type

Definition 1.2.1. Given a group Γ, a function of positive type on Γ is amap ϕ : Γ → C such that for all Σγ∈Γαγuγ ∈ CΓ we have

Σγ,λ∈Γαλαγϕ(λ−1γ) ≥ 0.

One way in which function of positive types appear is when we have a unitaryrepresentation π : Γ → U(K), together with a vector ξ ∈ K. Then functiondefined by:

ϕξ(γ) = 〈π(γ)ξ, ξ〉, for all γ ∈ Γ,

describes a function of positive type on Γ. Indeed, if Σγ∈Γαγuγ ∈ CΓ then

Σγ,λ∈Γαλαγϕ(λ−1γ) = Σγ,λ∈Γαλαγ〈π(λ−1γ)ξ, ξ〉

= ‖Σγ∈Γαγπ(γ)ξ‖2 ≥ 0.

Every function of positive type arrises in this way. Specifically, if ϕ : Γ → C

is a function of positive type then we may place a pseudo-inner product on CΓby the formula:

〈Σγ∈Γαγuγ ,Σλ∈Γβλuλ〉ϕ = Σγ,λ∈Γβλαγϕ(λ−1γ).

The positivity of this inner product is given by the fact that ϕ is of positivetype. After quotienting by the kernel and completion we obtain a Hilbert spaceK. Moreover, there is a natural unitary representation of Γ on K given by

πϕ(γ0)Σγ∈Γαγuγ = Σγ∈Γαγuγ0γ .

1.3. COCYCLES 9

The fact that this is well defined and describes a unitary follows from thefact that for all γ0 ∈ Γ, and Σγ∈Γαγuγ ∈ CΓ we have

‖Σγ∈Γαγuγ0γ‖2φ = Σγ,λ∈Γαλαγφ((γ0λ)−1(γ0γ))

= Σγ,λ∈Γαλαγφ(λ−1γ) = ‖Σγ∈Γαγuγ‖2φ.

One has to check that this action is well defined and preserves the innerproduct structure, and then the cyclic vector ξϕ = ue gives the formula above.

If we start with a unitary representation π : Γ → U(H), with a non-zerovector ξ ∈ H, consider the function of positive type ϕξ, and then considerthe representation πϕξ

. Then the map Σγ∈Γαγuγ 7→ Σγ∈Γαγπ(γ)ξ defines anisometry from K to H which is Γ equivariant. If ξ is a cyclic vector for π thenthis gives a natural isomorphism between πϕξ

and π.One can easily check that the correspondence described above satisfies the

following relationships:

functions of positive type pointed unitary representations

ϕ = 1 the trivial representationϕ = δe the left regular representation on ℓ2Γ, ξ = δe

ϕ = δΛ, Λ < Γ the quasi-regular representation on ℓ2(Γ/Λ), ξ = δΛϕ a character the one dimensional representation given by the character

addition direct sumproduct tensor product

1.3 Cocycles

Definition 1.3.1. Suppose ΓyX is an action of a group Γ on a set X , and Λis a group. A cocycle for the action into Λ is a map α : Γ×X → Λ such that

α(γ1γ2, x) = α(γ1, γ2x)α(γ2, x),

for all γ1, γ2 ∈ Γ, and x ∈ X . Two cocycles α, β : Γ×X → Λ are cohomologous

if there is a map ξ : X → Λ such that

α(γ, x) = ξ(γx)β(γ, x)ξ(x)−1 ,

for all γ ∈ Γ, x ∈ X . A cocycle α is trivial if it is cohomologous to thecocycle which takes constant value e ∈ Λ. A cocycle α untwists if there is ahomomorphism δ : Γ → Λ such that α is cohomologous to the cocycle given by(γ, x) 7→ δ(γ), for all γ ∈ Γ, and x ∈ X .

The set of all cocycles for the action ΓyX with values in Λ is denoted byZ1(ΓyX ; Λ), the set of trivial cocycles is denoted by B1(ΓyX ; Λ), and theset of equivalence classes of cohomologous cocycles is denoted by H1(ΓyX ; Λ).Note that if Λ is abelian then Z1(ΓyX ; Λ) is an abelian group under pointwisemultiplication, B1(ΓyX ; Λ) is a subgroup and H1(ΓyX ; Λ) is the quotientgroup.


Example 1.3.2. If α : Γ×X → Λ is a cocycle for an action ΓyX on a groupΛ, and if S ⊂ Γ is a generating set, then the cocycle relation implies that thecocycle is completely determined by its values on the set S ×X ⊂ Γ × X . Inparticular, for the group Z, a cocycle α : Z × X → Λ for an action ZyX iscompletely determined by the function x 7→ α(1, x). Moreover, any functionξ : X → Λ determines a cocycle α such that α(1, x) = ξ(x).

For instance if T : X → X is a bijection and ZyX by n · x = T n(x), and ifξ : X → R is a function, then the corresponding cocycle

α(n, x) = Σn−1k=0ξ T k(x),

for n > 0. Hence, if we denote by Sn : X → R the function given by

Sn(x) = α(n, T−n(x)),

then 1nSn is the average of the functions ξ T−k for 1 ≤ k ≤ n.

Example 1.3.3. Suppose Γ and Λ are two groups and consider the space

[Γ,Λ] = f : Γ → Λ | f(e) = e ⊂ ΛΓ.

We have an action of Γ on this set by

(γ · f)(x) = f(xγ)f(γ)−1.

Note that the fixed points of this action are precisely the set of homomor-phisms from Γ to Λ.

We then have a cocycle for this action α : Γ× [Γ,Λ] → Λ given by

α(γ, f) = f(γ).

Note that this action preserves the subsets of injective, surjective, and bijec-tive maps.

Exercise 1.3.4. Think about the previous example and verify all the claims.

Example 1.3.5. One natural way in which cocycles arise is if we have a set Xtogether with a pair of actions ΓyX , and ΛyX of groups Γ, and Λ.

If the action of Λ is free (if λ 6= e then λx 6= x for all x ∈ X), and if theorbits of Γ are contained in the orbits of Λ (Γx ⊂ Λx for each x ∈ X), then wecan define a cocycle α : Γ×X → Λ by setting α(γ, x) to be the unique elementin Λ such that

γx = α(γ, x)x.

We can verify that α is a cocycle since for each γ1, γ2 ∈ Γ and x ∈ X we have

α(γ1γ2, x)x = γ1γ2x

= α(γ1, γ2x)(γ1x) = α(γ1, γ2x)α(γ2, x)x.

Since the action ΛyX is free this then implies that

α(γ1γ2, x) = α(γ1, γ2x)α(γ2, x).

1.3. COCYCLES 11

Definition 1.3.6. If ΓyY is an action of a group Γ on a set Y then a fun-

damental domain for the action is a subset D ⊂ Y such that each orbit of Γcontains exactly one element in D, i.e., for all x ∈ Y we have |Γx ∩D| = 1.

Note that if D ⊂ Y is a fundamental domain for the action ΓyY then wehave that the map from Γ×D to Y given by (γ, x) 7→ γx is sujective. Moreover,if the action ΓyY is free then the map is also injective and hence the space Ydecomposes as a disjoint union of orbits

Y = ⊔x∈DΓx.

We thus obtain a bijection between the fundamental domain D and the orbitspace Γ\X given by x 7→ Γx.

Example 1.3.7. Another way in which cocycles arise is if Y is a set and we havea left action of Γ on Y and a right action of Λ such that the actions commute,i.e., for each γ ∈ Γ, λ ∈ Λ, and x ∈ Y we have

(γx)λ = γ(xλ).

As in Example 1.3.5 we will require the action of Λ to be free.Since the actions of Γ and Λ commute, the action of Γ on Y passes to the

space of Λ orbits Y/Λ. If D ⊂ Y is a fundamental domain for the action Y x Λthen as mentioned above the map from D to Y/Λ given by x 7→ xΛ is a bijection.We will denote by Φ : Y/Λ → D the inverse map.

We then obtain a cocycle α : Γ× (Y/Λ) → Λ by assigning to each γ ∈ Γ andxΛ ∈ Y/Λ the value of α(γ, xΛ) to be the unique element in Λ such that

γΦ(xΛ) = Φ(γxΛ)α(γ, xΛ).

If γ1, γ2 ∈ Γ and xΛ ∈ Y/Λ then we have

Φ(γ1γ2xΛ)α(γ1γ2, xΛ) = γ1γ2Φ(xΛ)

= γ1Φ(γ2xΛ)α(γ2, xΛ)

= Φ(γ1γ2xΛ)α(γ1, γ2xΛ)α(γ2, xΛ).

Since the action Y x Λ is free this shows that α is indeed a cocycle.Suppose D′ ⊂ Y is also a fundamental domain for the action Y x Λ and

β : Γ× (Y/Λ) → Λ is the corresponding cocycle. If we let Φ′ : Y/Λ → D′ be thecorresponding selection map for D′, and we define ξ : Y/Λ → Λ so that for allxΛ ∈ Y/Λ we have

ξ(xΛ) = Φ′(xΛ)Φ(xΛ)−1,

then it’s easy to see that for each γ ∈ Γ and xΛ ∈ Y/Λ we have

ξ(γxΛ)α(γ, xΛ)ξ(xΛ)−1 = β(γ, xΛ).

Hence, the cohomology class of α is independent of the fundamental domain D.


Example 1.3.8. A special case of the above example to keep in mind is whenΓ is a group and Σ < Γ is a subgroup. Then ΓyΓ x Σ is a pair of commutingactions given by group multiplication. Thus, we obtain a cocycle (unique up tocohomology) α : Γ× (Γ/Σ) → Σ.

Exercise 1.3.9. Show that in Example 1.3.8 if the action Γ x Σ has a fun-damental domain Γ0 ⊂ Γ which is a subgroup of Γ, then we have that Σ ⊳ Γis a normal subgroup and Γ splits as a semidirect product Γ = Γ0 ⋉ Σ. Also,compute the corresponding cocycle for this fundamental domain.

1.4 Induced representations

Given an action ΓyX of a group Γ on a set X , a cocycle α : Γ ×X → Λ intoa group Λ, and a unitary representation π : Λ → U(K), we obtain an inducedrepresentation IndαΛ π : Γ → U(ℓ2X⊗K) by linearly extending the formula

IndαΛ π(γ)(δx ⊗ η) = δγx ⊗ (π(α(γ, x))η).

We can easily check that this is a representation since for all γ1, γ2 ∈ Γ, x ∈ X ,and η ∈ K we have

IndαΛ π(γ1γ2)(δx ⊗ η) = δγ1γ2x ⊗ (π(α(γ1γ2, x))η)

= δγ1γ2x ⊗ (π(α(γ1, γ2x))π(α(γ2 , x))η)

= IndαΛ π(γ1)(δγ2x ⊗ (π(α(γ2, x))η)

= IndαΛ π(γ1) Ind

αΛ π(γ2)(δx ⊗ η).

If β : Γ×X → Λ is cocycle which is cohomologous to α, and ξ : X → Λ suchthat α(γ, x) = ξ(γx)β(γ, x)ξ(x)−1 , for all γ ∈ Γ, and x ∈ X , then we obtain aunitary Uξ ∈ U(ℓ2X⊗K), by linearly extending the formula

Uξ(δx ⊗ η) = δx ⊗ π(ξ(x)−1)η.

We then see easily that for all γ ∈ Γ, x ∈ X , and η ∈ K we have

U∗ξ IndβΛ π(γ)Uξ(δx ⊗ η) = U∗

ξ IndβΛ π(γ)(δx ⊗ π(ξ(x)−1)η)

= U∗ξ (δγx ⊗ π(β(γ, x))π(ξ(x)−1)η)

= δγx ⊗ π(ξ(γx))π(β(γ, x))π(ξ(x)−1 )η

= IndαΛ π(γ)(δx ⊗ η).

Hence, the representations IndβΛ π and Indα

Λ π are isomorphic.

Lemma 1.4.1. Let ΓyX be an action of a group Γ on a set X, let Λ be a group,and suppose α : Γ ×X → Λ is a cocycle. If π : Λ → U(H) and ρ : Λ → U(K)are unitary representations such that π ∼= ρ then IndαΛ π ∼= IndαΛ ρ.

1.4. INDUCED REPRESENTATIONS 13

Proof. If W : H → K is a unitary such that Wπ(γ) = ρ(γ)W for all γ ∈ Γ.Then id⊗W : ℓ2X ⊗H → ℓ2X ⊗K, and it follows easily that

(id⊗W ) IndαΛ π(γ) = IndαΛ ρ(γ)(id⊗W ).

Example 1.4.2. Consider a group Γ acting on itself by left multiplication,and let π : Γ → U(H) be a unitary representation. Then we obtain cocyclesα, β : Γ× Γ → Γ given by α(γ, x) = γ, and β(γ, x) = e.

By considering the function ξ(x) = x we see that these cocycles are coho-mologous, for every γ, x ∈ Γ we have

α(γ, x) = γ = (γx)x−1 = ξ(γx)β(γ, x)ξ(x)−1 .

We therefore recover Fell’s absorption principle by obtaining an isomorphismbetween the representations λ⊗ π = IndαΓ π and λ⊗ 1H = IndβΓ π.

If Γ is a group, and Σ < Γ is a subgroup, then we may consider the cocycleα : Γ×(Γ/Σ) → Σ associated to some fundamental domain for Σ as is describedin Example 1.3.8. Hence if π : Σ → U(H) is a unitary representation of Σ thenwe obtain the induced representation IndΓΣ π : Γ → U(ℓ2(Γ/Σ)⊗H) as theinduced representation associated to the cocycle α. Since the cohomology classof α does not depend on the fundamental domain, we have that the inducedrepresentation is well defined up to unitary equivalence.

Observe that if ξ0 ∈ H, then we may consider the vector ξ′ = δΣ ⊗ ξ0 ∈ℓ2(Γ/Σ)⊗H. We then have that the positive definite function φξ′ : Γ → C isgiven by

φξ′(γ) =

φξ(γ) if γ ∈ Σ;0 otherwise.

Also observe that if π = 1H : Σ → U(H) is the trivial representation thenIndΓΣ π = λΓ/Σ⊗1H is a multiple of the quasi-regular representation correspond-ing to Σ.

Remark 1.4.3. Suppose Y is a set and we have commuting left and rightactions ΓyY x Λ of groups Γ and Λ such that the action of Λ is free. If π : Λ →U(H) is a unitary representation, and α : Γ× (Y/Λ) → Λ is the cocycle comingfrom a fundamental domain D for Y x as explained in Example 1.3.7, thenwe obtain the induced representation IndαΛ π. Here we will explain an alternateway to obtain this representation which makes no reference to a fundamentaldomain.

Consider a function ξ : X → H which is Λ-equivariant, i.e., ξ(xλ−1) =π(λ)ξ(x), for all x ∈ X , and λ ∈ Λ. Because, ξ is Λ-equivariant we have thatthe function x 7→ ‖ξ(x)‖ is constant on the Λ-orbits. We may therefore considerthe well defined space

L2(Y ;H)Λ = ξ : Y → H | ξ is Λ− equivariant, and ΣxΛ∈Y/Λ‖ξ(x)‖2 < ∞.


Since this space consists of Λ-equivariant functions, this becomes a Hilbert spacewhen it is endowed with the well defined inner-product

〈ξ, η〉 = ΣxΛ∈Y/Λ〈ξ(x), η(x)〉.

We then obtain the induced representation of π on L2(Y ;H)Λ by requiringthat for all γ ∈ Γ and x ∈ Y we have

(IndΓΛ π(γ)ξ)(x) = ξ(γ−1x).

If D ⊂ Y is a fundamental domain for the action Y x Λ, then any func-tion ξ ∈ ℓ2D⊗H ∼= ℓ2(D;H) extends uniquely to a Λ-equivariant functionξ ∈ ℓ2(Y ;H)Λ by setting ξ(xλ−1) = π(λ)ξ(x) for each λ ∈ Λ and x ∈ D.By identifying Y/Λ with D by the map Φ we then obtain a unitary W :ℓ2(Y/Λ)⊗H → ℓ2(Y ;H). If α : Γ × (Y/Λ) → Λ is the cocycle from Exam-ple 1.3.7, then unwinding the definitions gives

W IndαΛ(γ) = IndΓΛ(γ)W,

for all γ ∈ Γ.

Lemma 1.4.4. Let Γ be a group and ∆ < Σ < Γ subgroups. If π : ∆ → U(H)is a unitary representation then

IndΓ∆ π ∼= IndΓΣ IndΣ∆ π.

Proof. Using the equivalence in the previous remark, we may consider the mapW : L2(Γ;L2(Σ;H)∆)Σ → L2(Γ;H)∆ given by

(Wf)(γ) = (f(γ))(e).

Note that if δ ∈ ∆ then we have

(Wf)(γδ−1) = (f(γδ−1))(e) = (IndΣ∆ π(δ)f(γ))(e)

= (f(γ))(δ−1)

= π(δ)((f(γ))(e)) = π(δ)((Wf)(γ)).

Also, if D ⊂ Σ is a set of coset representatives for ∆, and E ⊂ Γ is a set of cosetrepresentatives for Σ then ED ⊂ Γ is a set of coset representatives for ∆, hencejust as above we see that for all f ∈ L2(Γ;L2(Σ;H)∆)Σ we have

‖Wf‖2 = Σ(γ,λ)∈E×D‖(Wf)(γλ)‖2

= Σγ∈EΣλ∈D‖((Wf)(γ))(λ)‖2 = ‖f‖2.Thus, W is a well defined isometry, which is easy to see is a unitary. An easycalculation then shows that for all γ ∈ Γ we have

IndΓ∆ π(γ)W = W IndΓΣ IndΣ∆ π(γ).

1.4. INDUCED REPRESENTATIONS 15

Proposition 1.4.5. If π : Γ → U(H) is a unitary representation of Γ andρ : Σ → U(K) is a unitary representation of Σ, then

π ⊗ IndΓΣ ρ ∼= IndΓΣ(π|Σ ⊗ ρ).

Proof. Consider the map W : H⊗L2(Γ;K)Σ → L2(Γ;H⊗K)Σ such that for eachγ ∈ Γ, ξ ∈ H, and f ∈ L2(Γ;K)Σ we have

(W (ξ ⊗ f))(γ) = (π(γ−1)ξ)⊗ f(γ).

Then, if σ ∈ Σ we have

(W (ξ ⊗ f))(γσ−1) = (π(σγ)ξ) ⊗ f(γσ−1)

= (π ⊗ ρ)(σ)(W (ξ ⊗ f)).

Hence, it follows easily that W is a well defined unitary operator. A routinecheck then shows that

IndΓΣ(π|Σ ⊗ ρ)(γ)W = W (π ⊗ IndΓΣ ρ)(γ),

for all γ ∈ Γ.

If Γ, Λ, and Υ are groups, ΓyX , and ΛyY are actions, and α : Γ×X → Λ,and β : Λ× Y → Υ are cocycles then just as we induced representations abovewe may induce the action ΓyX to an action ΓyX × Y by the formula

γ(x, y) = (γx, α(γ, x)y).

We then may define the composition of cocycles βα : Γ× (X × Y ) → Υ bythe formula

βα(γ, (x, y)) = β(α(γ, x), y).

We can verify that this is indeed a cocycle since for all γ1, γ2 ∈ Γ, and (x, y) ∈X × Y we have

βα(γ1γ2, (x, y)) = β(α(γ1, γ2x)α(γ2, x), y)

= β(α(γ1, γ2x), α(γ2, x)y)β(α(γ2, x), y)

= βα(γ1, (γ2x, α(γ2, x)y))βα(γ2 , (x, y))

= βα(γ1, γ2(x, y))βα(γ2, (x, y)).

Exercise 1.4.6. If we have an inclusion of groups ∆ < Σ < Γ and we considerthe cocycles αΣ<Γ, α∆<Σ, and α∆<Γ as described in Example 1.3.8, then showthat by identifying the sets (Γ/Σ)× (Σ/∆) and Γ/∆ we obtain an identificationof α∆<ΣαΣ<Γ and α∆<Γ.

In light of the previous exercise we then have that the following lemma is anextension of Lemma 1.4.4.


Lemma 1.4.7. Let Γ, Λ, and Υ be groups, ΓyX, and ΛyY be actions, andα : Γ×X → Λ, and β : Λ × Y → Υ be cocycles. If π : Υ → U(H) is a unitaryrepresentation, then

IndβαΥ π ∼= IndβΥ IndαΛ π.

Proof. If we consider the natural identification V : ℓ2X ⊗ ℓ2Y → ℓ2(X × Y ),then it is easy to see that V ⊗ idH implements an equivalence between therepresentations IndβΥ IndαΛ π and IndβαΥ π.

Lemma 1.4.8. Let ΓyX be a action of a group Γ on a set X, let Λ be a group,and let α : Γ × X → Λ be a cocycle, if πi : Λ → U(Hi), i ∈ I is a family ofunitary representations, then

IndαΛ(⊕iπi) ∼= ⊕i∈I Ind

αΛ(πi).

Proof. This follows easily by considering the natural unitary from ⊕i∈I(ℓ2X ⊗

Hi) to ℓ2X ⊗ (⊕i∈IHi).

1.5 Invariant vectors

Definition 1.5.1. Let Γ be a group, a unitary representation π : Γ → U(H)contains invariant vectors if there exists a non-zero vector ξ ∈ H such thatπ(γ)ξ = ξ for all γ ∈ Γ. The representation contains almost invariant vectors

if for each F ⊂ Γ, and ε > 0, there exists ξ ∈ H, such that

‖π(γ)ξ − ξ‖ < ε‖ξ‖, for all γ ∈ F.

We will also say that a representation π : Γ → U(H) is ergodic if it doesnot contain invariant vectors. In general, if we denote by H0 ⊂ H the subspaceof Γ-invariant vectors, then we say the representation π has spectral gap ifH0 6= H and the sub-representation π|H⊥

0does not contain almost invariant

vectors.

Proposition 1.5.2. Let Γ be a group, and π : Γ → U(H) a unitary represen-tation. If there exists ξ ∈ H and c > 0 such that ℜ(〈π(γ)ξ, ξ〉) > c‖ξ‖2 for allγ ∈ Γ, then π contains an invariant vector.

Proof. Let K be the closed convex hull of the orbit π(Γ)ξ. We therefore havethat K is Γ-invariant and ℜ(〈η, ξ〉) ≥ c‖ξ‖2 for every η ∈ K. Let ξ0 ∈ K be theunique element of minimal norm, then since Γ acts isometrically we have thatfor each γ ∈ Γ, π(γ)ξ0 is the unique element of minimal norm for π(γ)K = K,and hence π(γ)ξ0 = ξ0 for each γ ∈ Γ. Since ξ0 ∈ K we have that ℜ(〈ξ0, ξ〉) 6= 0,and hence ξ0 6= 0.

Corollary 1.5.3. Let Γ be a group, and π : Γ → U(H) a unitary representation.If there exists ξ ∈ H and c <

√2 such that ‖π(γ)ξ − ξ‖ < c‖ξ‖ for all γ ∈ Γ,

then π contains an invariant vector.

1.5. INVARIANT VECTORS 17

Proof. For each γ ∈ Γ we have

2ℜ(〈π(γ)ξ, ξ〉) = 2‖ξ‖2 − ‖π(γ)ξ − ξ‖2 ≥ (2− c2)‖ξ‖2.

Hence, we may apply Proposition 1.5.2.

Lemma 1.5.4. Let Γ be a group, and π : Γ → U(H) a unitary representation.Then π contains almost invariant vectors if and only if π⊕n contains almostinvariant vector, where n ≥ 1 is any cardinal number.

Proof. If π does not contain almost invariant vectors then there exists c > 0,and S ⊂ Γ finite, such that for all ξ ∈ H we have

c‖ξ‖2 ≤ Σγ∈S‖π(γ)ξ − ξ‖2.

If I is a set |I| = n, and ξi ∈ H for i ∈ I, such that Σi∈I‖ξi‖2 < ∞, then

c‖ ⊕i∈I ξi‖2 = Σi∈Ic‖ξi‖2

≤ Σi∈IΣγ∈S‖π(γ)ξi − ξi‖2 = Σγ∈S‖π⊕n(γ)(⊕i∈Iξi)−⊕i∈Iξi‖2.Hence, π⊕n does not contain almost invariant vectors. The converse is trivialsince π is contained in π⊕∞.

Proposition 1.5.5. Let Γ be a group, and π : Γ → U(H) a unitary represen-tation. Then π contains almost invariant vectors if and only if for any finitesymmetric set S ⊂ Γ the operator TS = 1

|S|Σγ∈Sπ(γ) satisfies ‖TS‖ = 1.

Proof. If π contains almost invariant vectors then the triangle inequality easilyimplies that 1 is in the spectrum of TS, for each finite symmetric set S ⊂ Γ.

Conversely, if S ⊂ Γ is a finite symmetric set with e ∈ S and ‖TS‖ = 1, thensince TS is self-adjoint either 1 or −1 is contained in its spectrum, however sincee ∈ S it is easy to see that −1 6∈ σ(TS), hence for any ε > 0 there exists ξ ∈ H,‖ξ‖ = 1 such that

|1− 〈TSξ, ξ〉| < ε2/2|S|.We then have that for each γ ∈ S

‖ξ − π(γ)ξ‖2 = 2|1−ℜ〈π(γ)ξ, ξ〉|

≤ 2|S||1− 〈TSξ, ξ〉| < ε2.

Since, ε and S were arbitrary this shows that π contains almost invariant vectors.

Proposition 1.5.6. Let Γ be a group, and π : Γ → U(H) a unitary representa-tion. The following are equivalent:

(1). The representation π ⊗ π contains invariant vectors.

(2). The representation π ⊗ λ contains invariant vectors for some unitaryrepresentation λ : Γ → U(K).

(3). The representation π contains a finite dimensional sub-representation.


Proof. (1) =⇒ (2) is obvious.To show (2) =⇒ (3) suppose λ : Γ → U(K) is a unitary representation such

that π⊗λ contains invariant vectors. IdentifyingH⊗K with the space of Hilbert-Schmidt operators HS(K,H) we then have that there exists T ∈ HS(K,H),non-zero, such that π(γ)Tρ(γ−1) = T , for all γ ∈ Γ. Then TT ∗ ∈ B(H,H)is positive, non-zero, compact, and π(γ)TT ∗π(γ−1) = TT ∗, for all γ ∈ Γ. Bytaking the range of a non-trivial spectral projection of TT ∗ we then obtain afinite dimensional invariant subspace of π.

(3) =⇒ (1) follows because if L ⊂ H is a finite dimensional invariantsubspace then ProjL is a finite rank projection such that π(γ) ProjL π(γ−1) =ProjL, for all γ ∈ Γ. By identifying HS(H,H) with H⊗H, we then obtain anon-zero invariant vector for π ⊗ π.

1.6 Amenability

Much of the material of this section has been taken from Section 2.6 in the bookof Brown and Ozawa [BO08].

Definition 1.6.1. Let Γ be a group, a Følner net is a net of non-empty finitesubsets Fi ⊂ Γ such that |Fi∆γFi|/|Fi| → 0, for all γ ∈ Γ.

Note that we do not require that Γ = ∪iFi, nor do we require that Fi areincreasing, however, if |Γ| = ∞ then it is easy to see that any Følner net Fiimust satisfy |Fi| → ∞.

Exercise 1.6.2. Let Γ be a group, show that if Γ has a Følner net, then foreach finite set E ⊂ Γ, and ε > 0, there exists a finite set F ⊂ Γ such thatE ⊂ F , F = F−1, and |F∆γF |/|F |, |F∆Fγ|/|F | < ε.

Definition 1.6.3. A mean m on a non-empty set X is a finitely additiveprobability measure on 2X , i.e., m : 2X → [0, 1] such that m(X) = 1, and ifA1, . . . , An ⊂ X are disjoint then m(∪n

j=1An) = Σnj=1m(An).

Given a mean m on X it is possible to define an integral over X just as inthe case if m were a measure. We therefore obtain a state φm ∈ (ℓ∞X)∗ by theformula φm(f) =

∫Xf dm. Conversely, if φ ∈ (ℓ∞X)∗ is a state, then restricting

φ to characteristic functions defines a corresponding mean.If ΓyX is an action, then an invariant mean m on X is a mean such that

m(γA) = m(A) for all A ⊂ X . An approximately invariant mean is a net ofprobability measures µi ∈ Prob(X), such that ‖γ∗µi − µi‖1 → 0, for all γ ∈ Γ.

Definition 1.6.4. Let Γ be a group, Γ is amenable if it Γ has a mean whichis invariant under the action of left multiplication, or equivalently, if there is astate on ℓ∞Γ which is invariant under the action of left multiplication.

Amenable groups were first introduced by von Neumann [vN29]. The termamenable was coined by M. M. Day.

1.6. AMENABILITY 19

Theorem 1.6.5. Let Γ be a group. The following conditions are equivalent.

(1). Γ is amenable.

(2). Γ has an approximate invariant mean under the action of left multipli-cation.

(3). Γ has a Følner net.

(4). The left regular representation λ : Γ → U(ℓ2Γ) contains almost invariantvectors.

(5). For any finite symmetric set S ⊂ Γ the operator TS = 1|S|Σγ∈Sλ(γ)

satisfies ‖TS‖ = 1.

(6). There exists a state Φ ∈ (B(ℓ2Γ))∗ such that Φ(λ(γ)T ) = Φ(Tλ(γ)) forall γ ∈ Γ, T ∈ B(ℓ2Γ).

(7). The continuous action of Γ on its Stone-Cech compactification βΓ whichis induced by left-multiplication admits an invariant Radon probability measure.

(8). Any continuous action ΓyK on a compact Hausdorff space K admitsan invariant Radon probability measure.

Proof. We show (1) =⇒ (2) using the method of Day [Day57]. Since ℓ∞Γ =(ℓ1Γ)∗, the unit ball in ℓ1Γ is wk∗-dense in the unit ball of (ℓ∞Γ)∗ = (ℓ1Γ)∗∗.It follows that Prob(Γ) ⊂ ℓ1Γ is wk∗-dense in the state space of ℓ∞Γ.

Let S ⊂ Γ, be finite and let K ⊂ ⊕γ∈Sℓ1Γ be the wk-closure of the set

⊕γ∈S(γ∗µ − µ) | µ ∈ Prob(Γ). From the remarks above, since Γ has a leftinvariant state on ℓ∞Γ, we have that 0 ∈ K. However, K is convex and so bythe Hahn-Banach Separation Theorem the wk-closure coincides with the normclosure. Thus, for any ε > 0 there exists µ ∈ Prob(Γ) such that

Σγ∈S‖γ∗µ− µ‖1 < ε.

We show (2) =⇒ (3) using the method of Namioka [Nam64]. Let S ⊂ Γ bea finite set, and denote by Er the characteristic function on the set (r,∞). Ifµ ∈ Prob(Γ) then we have

Σγ∈S‖γ∗µ− µ‖1 = Σγ∈SΣx∈Γ|γ∗µ(x)− µ(x)|

= Σγ∈SΣx∈Γ

∫

R≥0

|Er(γ∗µ(x)) − Er(µ(x))| dr

= Σγ∈S

∫

R≥0

Σx∈Γ|Er(γ∗µ(x)) − Er(µ(x))| dr

= Σγ∈S

∫

R≥0

‖Er(γ∗µ)− Er(µ)‖1 dr.


By hypothesis, if ε > 0 then there exists µ ∈ Prob(Γ) such that Σγ∈S‖γ∗µ−µ‖1 < ε, and hence for this µ we have

Σγ∈S

∫

R≥0

‖Er(γ∗µ)− Er(µ)‖1 dr < ε = ε

∫

R≥0

‖Er(µ)‖1 dr.

Hence, if we denote by Fr ⊂ Γ the (finite) support of Er(µ), then for someparticular r > 0 we must have

Σγ∈S |γFr∆Fr | = Σγ∈S‖Er(γ∗µ)− Er(µ)‖1 < ε‖Er(µ)‖1 = ε|Fr|.

For (3) =⇒ (4) just notice that if Fi ⊂ Γ is a Følner net, then 1|Fi|1/2

Σx∈Fiδx ∈ℓ2Γ is a net of almost invariant vectors.

(4) ⇐⇒ (5) follows from Proposition 1.5.5.For (4) =⇒ (6) let ξi ∈ ℓ2Γ be a net of almost invariant vectors for λ.

We define states Φi on B(ℓ2Γ) by Φi(T ) = 〈Tξi, ξi〉. By wk-compactness of thestate space, we may take a subnet and assume that this converges in the weaktopology to Φ ∈ B(ℓ2Γ)∗. We then have that for all T ∈ B(ℓ2Γ) and γ ∈ Γ,

|Φ(λ(γ)T − Tλ(γ))| = limi|〈(λ(γ)T − Tλ(γ))ξi, ξi〉|

= limi|〈Tξi, λ(γ−1)ξi〉 − 〈Tλ(γ)ξi, ξi〉|

≤ limi‖T ‖(‖λ(γ−1)ξi − ξi‖+ ‖λ(γ)ξi − ξi‖) = 0.

For (6) =⇒ (1), we have a natural embedding M : ℓ∞Γ → B(ℓ2Γ) as“diagonal matrices”, i.e., for a function f ∈ ℓ∞Γ we have Mf (Σx∈Γαxδx) =Σx∈Γαxf(x)δx. Moreover, for f ∈ ℓ∞Γ and γ ∈ Γ we have λ(γ)Mfλ(γ

−1) =Mγ·f . Thus, if Φ ∈ B(ℓ2Γ)∗ is a state which is invariant under the conjugationby λ(γ), then restricting this state to ℓ∞Γ gives a state on ℓ∞Γ which is Γ-invariant.

For (1) =⇒ (7), the map β : ℓ∞Γ → C(βΓ) which takes a bounded functionon Γ to its unique continuous extension on βΓ, is a C∗-algebra isomorphism,which is Γ-equivariant. Hence amenability of Γ implies the existence of a Γ-invariant state on C(βΓ). The Riesz Representation Theorem then gives aninvariant probability measure on βΓ.

For (7) ⇐⇒ (8), suppose Γ acts continuously on a compact Hausdorff spaceK, and fix a point x0 ∈ K. Then the map f(γ) = γx0 on Γ extends uniquelyto a continuous map βf : βΓ → K, moreover since f is Γ-equivariant, so is βf .If µ is an invariant Radon probability measure for the action on βΓ then weobtain the invariant Radon probability measure f∗µ on K. Since βΓ itself iscompact, the converse is trivial.

For (7) =⇒ (1), if there is a Γ-invariant Radon probability measure µ onβΓ, then we obtain an invariant mean m on Γ my setting m(A) = µ(A).

Example 1.6.6. Any finite group is amenable, and any group which is locallyamenable (each finitely generated subgroup is amenable) is also amenable. The

1.6. AMENABILITY 21

group of integers Z is amenable (consider the Følner sequence Fn = 1, . . . , nfor example). From this it follows easily that all finitely generated abelian groupsare amenable, and hence all abelian groups are.

It is also easy to see from the definition that subgroups of amenable groupsare amenable. A bit more difficult is to show that if 1 → Σ → Γ → Λ → 1 isan exact sequence of groups then Γ is amenable if and only if both Σ and Λ areamenable. Thus all solvable groups, and even all nilpotent groups are amenable.

There are also finitely generated amenable groups which cannot be con-structed from finite, and abelian groups using only the operations above [Gri84].

Example 1.6.7. Let F2 be the free group on two generators a, and b. Let A+

be the set of all elements in F2 whose leftmost entry in reduced form is a, letA− be the set of all elements in F2 whose leftmost entry in reduced form is a−1,let B+, and B− be defined analogously, and consider C = e, b, b2, . . .. Thenwe have that

F2 = A+ ⊔ A− ⊔ (B+ \C) ⊔ (B− ∪ C)

= A+ ⊔ aA−

= b−1(B+ \ C) ⊔ (B− ∪ C).

If m were a left-invariant mean on F2 then we would have

m(F2) = m(A+) +m(A−) +m(B+ \ C) +m(B− ∪ C)

= m(A+) +m(aA−) +m(b−1(B+ \ C)) +m(B− ∪C)

= m(A+ ⊔ aA−) +m(b−1(B+ \ C) ⊔ (B− ∪ C)) = 2m(F2).

Hence, F2 and also any group containing F2 is not-amenable. There are alsofinitely generated nonamenable groups which do not contain F2[Ols80], and evenfinitely presented nonamenable groups which do not contain F2 [OS02].

1.6.1 Von Neumann’s Mean Ergodic Theorem

Amenable groups allow for nice averaging properties, we give such an examplehere.

Theorem 1.6.8. Let Γ be an amenable group with Følner net Fi ⊂ Γ, and letπ : Γ → U(H) be a unitary representation. Let P0 ∈ B(H) be the projectiononto the subspace of Γ invariant vectors. Then for each ξ ∈ H we have that

‖ 1

|Fi|Σγ∈F−1

iπ(γ)ξ − P0(ξ)‖ → 0.

Proof. By considering the vector ξ−P0(ξ) instead, we may assume that P0(ξ) =0.

Note that if γ ∈ Γ, and η ∈ H then η − π(γ)η is orthogonal to the space ofinvariant vectors. Indeed, if ζ ∈ H is an invariant vector, then 〈η − π(γ)η, ζ〉 =〈η, ζ − π(γ−1)ζ〉 = 0.


Denote by L the closure of the subspace of H spanned by vectors of the formη − π(γ)η for η ∈ H, γ ∈ Γ. Then if ζ ∈ L⊥ we have that 0 = 〈ζ, η − π(γ)η〉 =〈ζ − π(γ−1)ζ, η〉 for all γ ∈ Γ, and η ∈ H, thus ζ − π(γ)ζ = 0, for all γ ∈ Γ.Therefore we have shown that L⊥ is precisely the space of invariant vectors.

Fix γ0 ∈ Γ, and η ∈ H, then if ξ = η − π(γ0)η we have that

‖ 1

|Fi|Σγ∈F−1

iπ(γ)ξ‖ = ‖ 1

|Fi|Σγ∈F−1

i(π(γ)η − π(γγ0)η)‖

≤ (|Fi∆γ0Fi|/|Fi|)‖η‖ → 0.

Since 1|Fi|

Σγ∈F−1i

π(γ) ∈ B(H) is always a contraction we may then pass to the

closure of the span to conclude that for all ξ ∈ L we have

‖ 1

|Fi|Σγ∈F−1

iπ(γ)ξ‖ → 0.

In the case when the representation π is ergodic, Γ = Z, and the we considerthe Følner sequence Fn = 0,−1, . . . ,−n + 1, the above theorem then givesthe following corollary.

Corollary 1.6.9. Let π : Z → U(H) be a unitary representation. If π is ergodic,then for each ξ ∈ H we have that

limn→∞

‖ 1nΣn−1

k=0π(k)ξ‖ = 0.

We remark that from the perspective of Example 1.3.2 another possiblegeneralization of Corollary 1.6.9 could be given in terms of cocycles. This per-spective would then lead to Corollary 3.3 in [dCTV07] the proof of which isquite similar to the proof of von Neumann’s Ergodic Theorem given above.

1.7 Mixing properties

Definition 1.7.1. Let Γ be a group, a unitary representation π : Γ → U(H) isweak mixing if for each finite set F ⊂ H, and ε > 0 there exists γ ∈ Γ suchthat

|〈π(γ)ξ, ξ〉| < ε,

for all ξ ∈ F .

The representation π is (strong) mixing if |Γ| = ∞, and for each finite setF ⊂ H, we have

limγ→∞

|〈π(γ)ξ, ξ〉| = 0.

1.7. MIXING PROPERTIES 23

These definitions should be compared with the characterization of ergodicityfound in Proposition 1.5.2. Hence we see that mixing implies weak mixing, whichin turn implies ergodic. It is also easy to see that if π : Γ → U(H) is mixing(resp. weak mixing) then so is π⊕∞, and if π is mixing then so is π ⊗ ρ for anyrepresentation ρ. We’ll see below in Corollary 1.7.6 that weak mixing is alsostable under tensoring.

Exercise 1.7.2. Let Γ be a group and Σ < Γ a subgroup. Show that thequasi-regular representation of Γ on ℓ2(Γ/Σ) is weak mixing if and only if it isergodic, if and only if [Γ : Σ] = ∞. Show that it is mixing if and only if |Γ| = ∞and |Σ| < ∞.

Exercise 1.7.3. Let ΓyX be an action of a group Γ on a set X , let Λ bea group and let α : Γ × X → Λ be a cocycle. Suppose π : Λ → U(H) is aunitary representation. Find necessary and sufficient conditions for the inducedrepresentation of Section 1.4 to be mixing.

Lemma 1.7.4. Let Γ be a group, a unitary representation π : Γ → U(H) isweak mixing if and only if for each finite set F ⊂ H, and ε > 0 there existsγ ∈ Γ such that

|〈π(γ)ξ, η〉| < ε,

for all ξ, η ∈ F .The representation π is mixing if |Γ| = ∞ and for each finite set F ⊂ H, we

havelimγ→∞

|〈π(γ)ξ, η〉| = 0,

for all ξ, η ∈ F .

Proof. This follows from the polarization identity. For each ξ, η ∈ H, and γ ∈ Γwe have

〈π(γ)ξ, η〉 = 1

4Σ3

k=0ik〈π(γ)(ξ + ikη), (ξ + ikη)〉.

We now add another equivalent condition to Proposition 1.5.6.

Proposition 1.7.5. Let Γ be a group, and let π : Γ → U(H) be a unitary repre-sentation. Then π is weak mixing if and only if π contains no finite dimensionalsup-representations.

Proof. If π is weak mixing then if L ⊂ H is a non-trivial, finite dimensional sub-space with orthonormal basis F ⊂ H, there exists γ ∈ Γ such that |〈π(γ)ξ, η〉| <1/

√dim(L), for all ξ, η ∈ F . Hence, if ξ ∈ F then ‖ProjL(π(γ)ξ)‖ < 1 = ‖ξ‖.

Thus, L is not an invariant subspace.Conversely, If π has no finite dimensional invariant subspaces, L ⊂ H is a

finite dimensional subspace, and ε > 0, then there exists γ ∈ Γ such that for allξ ∈ L we have that

‖ProjL(π(γ)ξ)‖ < ε.


Indeed, if this is not the case then we would have that there exists c > 0 suchthat

〈π(γ−1) ProjL π(γ),ProjL〉HS ≥ supξ∈L,‖ξ‖=1

‖ProjL(π(γ)ξ)‖2 ≥ c.

It would then follow Proposition 1.5.2 that there is a non-zero Hilbert-Schmidtoperator T such that π(γ)Tπ(γ−1) = T , for all γ ∈ Γ. Proposition 1.5.6 wouldthen give a contradiction.

Thus, if F ⊂ H is finite such that ‖ξ‖ ≤ 1 for each ξ ∈ F , then by consideringthe finite dimensional subspace L spanned by F we have shown that there existsγ ∈ Γ such that for all ξ, η ∈ F we have

|〈π(γ)ξ, η〉| ≤ ‖ProjL(π(γ)ξ)‖‖η‖ < ε.

From the above proposition together with the equivalence between conditions2) and 3) in Proposition 1.5.6 we obtain the following.

Corollary 1.7.6. Let Γ be a group and let π : Γ → U(H) be a unitary repre-sentation. Then π is weak mixing if and only if π ⊗ π is weak mixing, if andonly if π⊗ρ is weak mixing for all unitary representations ρ.

Corollary 1.7.7. Let Γ be a group and let π : Γ → U(H) be a weak mixingunitary representation. If Σ < Γ is a finite index subgroup then π|Σ is also weakmixing.

Proof. Let D ⊂ Γ be a set of coset representatives for Σ. If π|Σ is not mixing,then by Proposition 1.7.5 there is a finite dimensional subspace L ⊂ H whichis Σ-invariant. We then have that Σγ∈Dπ(γ)(L) ⊂ H is finite dimensional andΓ-invariant. Hence, again by Proposition 1.7.5, π is not weak mixing.

1.7.1 Compact representations

If H is a Hilbert space the strong operator topology on B(H) is such thatlimi Ti = T if and only if limi ‖(Ti−T )ξ‖ = 0, for all ξ ∈ H. The weak operatortopology on B(H) is such that limi Ti = T if and only if limi〈(Ti−T )ξ, η〉 = 0, forall ξ, η ∈ H. The unitary group U(H) then becomes a topological group whenendowed with the strong operator topology. We also note that if Ui ∈ U(H)is a net of unitaries such that Ui → U ∈ U(H) in the weak operator topology,then we also have that Ui → U in the strong operator topology. Indeed, for anyξ ∈ H we have

‖(Ui − U)ξ‖2 = 2‖ξ‖2 − 2ℜ(〈Uiξ, Uξ〉) → 0.

Definition 1.7.8. Let Γ be a group, a unitary representation π : Γ → U(H) iscompact if π(Γ) ⊂ U(H) is pre-compact in the strong operator topology.


Lemma 1.7.9. Let (X, d) be a compact metric space, then Isom(X, d) with thetopology of point-wise convergence is compact.

Proof. By Tychonoff’s Theorem XX is compact, thus we need only to showthat Isom(X, d) ⊂ XX is closed. Suppose g ∈ Isom(X, d), for all x, y ∈ X wehave that f ∈ XX | d(f(x), f(y)) = d(x, y) is closed and contains Isom(X, d),hence g is isometric.

For each x ∈ X denote by dx the distance from x to g(X). Then we havethat for all m ∈ N, d(x, gm(x)) ≥ dx, hence for all n,m ∈ N, n < m we haved(gn(x), gm(x)) = d(x, gm−n(x)) ≥ dx. Since X is compact, it must be totallybounded and hence we must have that dx = 0. Hence g is surjective and thusIsom(X, d) is compact.

Lemma 1.7.10. Let π : Γ → U(H) be a unitary representation of a group Γ.Then π is compact if and only if for each ξ ∈ H, the orbit π(Γ)ξ is pre-compactin H.

Proof. If π(Γ) is pre-compact in the strong operator topology and G = π(Γ),then for each ξ ∈ H we have that π(Γ)ξ = Gξ is compact, being the continuousimage of the compact set G.

Conversely, suppose that each orbit π(Γ)ξ ⊂ H is pre-compact. By Zorn’sLemma we can find a collection of vectors J ⊂ H such that H = ⊕ξ∈J spπ(Γ)ξ.

We therefore have a strong operator topology continuous embedding of π(Γ)into the compact space Πξ∈J Isom(π(Γ)ξ, dH) ⊂ U(⊕ξ∈J spπ(Γ)ξ), hence π(Γ)is pre-compact.

The following is part of the Peter-Weyl Theorem.

Theorem 1.7.11. Let G be a compact group, and let π : G → U(H) be a strongoperator topology continuous unitary representation. Then π decomposes as adirect sum of finite dimensional representations.

Proof. Let λ denote the Haar measure on G. Since every representation decom-poses into a direct sum of cyclic representations we may assume that the rep-resentation π has a cyclic vector ζ ∈ H. We then define an operator K ∈ B(H)such that for all ξ, η ∈ H we have

〈Kξ, η〉 =∫

G

〈π(g)ξ, ζ〉〈ζ, π(g)η〉 dλ(g).

First, note that from left invariance of the Haar measure we have thatπ(g)Kπ(g−1) = K, for all g ∈ G. Also, if ξ ∈ H, then we have

〈Kξ, ξ〉 =∫

G

|〈π(g)ξ, ζ〉|2 dλ(g) ≥ 0.

Thus, K is positive.Moreover, if 〈Kξ, ξ〉 = 0 then we have that 〈π(g)ξ, ζ〉 = 〈ξ, π(g−1)ζ〉 = 0,

for all g ∈ G. This then implies that ξ = 0 since ζ is a cyclic vector. Thus K isstrictly positive.


If ξn ∈ H is a bounded sequence which weakly converges to 0, then

limn→∞

‖Kξn‖2 = limn→∞

∫

G

〈π(g)ξn, ζ〉〈ζ, π(g)Kξn〉 dλ(g)

= limn→∞

∫

G

∫

G

〈π(g)ξn, ζ〉〈ζ, π(h)ξn〉〈π(h)π(g−1)ζ, ζ〉 dλ(g) dλ(h) = 0.

We therefore have shown that K is a compact operator. Hence, H decom-poses as a direct sum of the (finite dimensional) eigenspaces of K, each of whichis G-invariant.

Lemma 1.7.10 and Theorem 1.7.11 then give us a structural result for com-pact representations.

Corollary 1.7.12. Let π : Γ → U(H) be a representation of a group Γ. Thenπ is compact if and only if π decomposes as a direct sum of finite dimensionalrepresentations.

Proposition 1.7.13. Let Γ be a group and let π : Γ → U(H) be a unitaryrepresentation. Then there is a unique Γ-invariant closed subspace K ⊂ H suchthat π|K is compact and π|K⊥ is weak mixing.

Proof. Let Z be the set of all orthonormal sets J ⊂ H such that sp(π(Γ)ξ) isfinite dimensional for all ξ ∈ J and sp(π(Γ)ξ) ⊥ sp(π(Γ)η) for all ξ, η ∈ J ,ξ 6= η. If we order Z by inclusion then it is easy to see that the union of anyincreasing chain in Z is again in Z.

If π is not weakly mixing then by Proposition 1.7.5 we have that Z 6= ∅, henceby Zorn’s Lemma there is a maximal element J ∈ Z. Let K = Σξ∈J sp(π(Γ)ξ) ⊂H, then π|K

∼= ⊕ξ∈J π|sp(π(Γ)ξ) is compact, and by maximality of J we have thatπ|K⊥ contains no finite dimensional sub-representation, and hence is weaklymixing.

If K0 ⊂ H is a finite dimensional Γ-invariant subspace, then ProjK⊥(K0) ⊂K⊥ is also a finite dimensional Γ-invariant subspace. Since π|K⊥ is weak mixingit then follows that ProjK⊥(K0) = 0 and hence K0 ⊂ K. This then impliesuniqueness of the decomposition.

1.7.2 Weak mixing for amenable groups

Proposition 1.7.14. Let Γ be an amenable group with a Følner net Fi ⊂ Γ,and let π : Γ → U(H) be a unitary representation. Then π is weak mixing ifand only if for each ξ, η ∈ H we have

1

|Fi|Σγ∈F−1

i|〈π(γ)ξ, η〉|2 → 0.

Proof. If π is not weak mixing, then there exists a finite set F ⊂ H, and ε > 0such that Σξ,η∈F |〈π(γ)ξ, η〉|2 > ε, for all γ ∈ Γ. It then follows that

lim infi→∞

1

|Fi|Σγ∈F−1

iΣξ,η∈F |〈π(γ)ξ, η〉|2 ≥ ε.


Hence, by the pigeonhole principle, for some ξ, η ∈ F we have that

lim supi→∞

1

|Fi|Σγ∈F−1

i|〈π(γ)ξ, η〉|2 ≥ ε/|F|2 > 0.

Conversely, if π is weak mixing then by Propositions 1.5.6 and 1.7.5 we havethat π⊗π is ergodic. Hence by von Neumann’s Ergodic Theorem the operators1

|Fi|Σγ∈F−1

i(π ⊗ π)(γ) converge in the strong operator topology (and hence also

in the weak operator topology) to 0. If ξ, η ∈ H we then have

1

|Fi|Σγ∈F−1

i|〈π(γ)ξ, η〉|2 =

1

|Fi|Σγ∈F−1

i〈(π ⊗ π)(γ)(ξ ⊗ ξ), η ⊗ η〉 → 0.

Lemma 1.7.15 (Koopman, von Neumann 1932). Let X be a set, and let Fi ⊂X, i ∈ I be a net of non-empty finite subsets of X. Suppose a ∈ ℓ∞X, then

1

|Fi|Σx∈Fi |a(x)| → 0

if and only if1

|Fi|Σx∈Fi |a(x)|2 → 0.

Proof. By the Cauchy-Schwarz inequality we have

1

|Fi|Σx∈Fi |a(x)| ≤ (

1

|Fi|Σx∈Fi |a(x)|2)1/2 ≤ ‖a‖1/2∞ (

1

|Fi|Σx∈Fi |a(x)|)1/2.

Corollary 1.7.16. Let Γ be an amenable group with a Følner net Fi ⊂ Γ, andlet π : Γ → U(H) be a unitary representation. Then π is weak mixing if andonly if for each ξ, η ∈ H we have

1

|Fi|Σγ∈F−1

i|〈π(γ)ξ, η〉| → 0.

Specializing to representations of the integers we also have the followingspectral characterization of weak mixing.

Proposition 1.7.17. Let π : Z → U(H) be a unitary representation. Then πis weak mixing if and only if π(1) has no eigenvalues.

Proof. If π is weak mixing then by Proposition 1.7.5 there is no finite dimen-sional subspace of H which is Z invariant. In particular, this shows that π(1)has no eigenvalues.

Conversely, suppose π(1) has no eigenvalues. Then π has no 1-dimensionalinvariant subspaces. However, since any unitary matrix in Mn(C) has an eigen-value, it then follows that π has no finite dimensional invariant subspaces.Hence, π is weak mixing by Proposition 1.7.5.


1.8 Weak containment

Definition 1.8.1. Let Γ be a group, and π : Γ → U(H), ρ : Γ → U(K) twounitary representations. The representation π is weakly contained in therepresentation ρ (written π ≺ ρ) if for each ξ ∈ H, F ⊂ Γ finite, and ε > 0,there exists η1, . . . , ηn ∈ K such that

|〈π(γ)ξ, ξ〉 − Σnj=1〈ρ(γ)ηj , ηj〉| < ε,

for all γ ∈ F . The representations π and ρ are weakly equivalent (written π ∼ ρ)if π ≺ ρ and ρ ≺ π.

It follows easily from the definition that π ∼ π⊕∞ for any representation π.Also, it is easy to see that containment implies weak containment, and weakcontainment is a partial order. We also have that if πi, and ρi, i ∈ I are familiesof representations such that πi ≺ ρi for each i ∈ I, then ⊕i∈Iπi ≺ ⊕i∈Iρi.

Exercise 1.8.2. Show that if π1, π2, ρ1, and ρ2 are unitary representations ofa group Γ such that πi ≺ ρi, for i ∈ 1, 2, then π1 ⊗ π2 ≺ ρ1 ⊗ ρ2.

Example 1.8.3. If a representation π : Γ → U(H) weakly contains the trivialrepresentation then for any finite symmetric set S ⊂ Γ and ε > 0 there existsη1, . . . , ηn ∈ H such that

|1− Σnj=1〈π(γ)ηj , ηj〉| < ε,

for all γ ∈ S ∪e. Hence, if we denote by η = ⊕nj=1ηj ∈ H⊕n ⊂ H⊕∞, then we

have |1− ‖η‖2| < ε, and hence for each γ ∈ S we have

‖η − π⊕∞(γ)η‖2 = 2(‖η‖2 −ℜ(〈π⊕∞(γ)η, η〉)) < 4ε.

It follows that π contains almost invariant vectors by Lemma 1.5.4.Conversely, if π contains almost invariant vectors, then it is easy to see that

π weakly contains the trivial representation.

The following lemma from [Fel63] is a useful tool for checking if one repre-sentation is weakly contained in another.

Lemma 1.8.4. Let π : Γ → U(H) and ρ : Γ → U(K) be two unitary representa-tions of a group Γ. Let L ⊂ H be a set such that spπ(Γ)L = H. Then π ≺ ρ ifand only if for each ξ ∈ L, F ⊂ Γ finite, and ε > 0, there exists η1, . . . , ηn ∈ Ksuch that

|〈π(γ)ξ, ξ〉 − Σnj=1〈ρ(γ)ηj , ηj〉| < ε,

for all γ ∈ F .

Proof. Suppose L ⊂ H is as above, and consider X ⊂ H the set of vectors ξ ∈ Hsuch that the positive definite function γ 7→ 〈π(γ)ξ, ξ〉 can be approximatedarbitrarily well on finite sets by sums of positive definite functions associated toρ. By hypothesis L ⊂ X , and we need to show that X = H.

1.8. WEAK CONTAINMENT 29

If ξ ∈ X , η ∈ K⊕∞, and Σx∈Γαxux ∈ CΓ, then from the formula

|〈π(γ)(Σx∈Γαxπ(x)ξ),Σx∈Γαxπ(x)ξ〉 − 〈ρ(γ)(Σx∈Γαxπ(x)η),Σx∈Γαxπ(x)η〉|

≤ Σx,y∈Γ|αyαx||〈π(y−1γx)ξ, ξ〉 − 〈ρ(y−1γx)η, η〉|,we see that Σx∈Γαxπ(x)ξ ∈ X . In particular, X is Γ-invariant.

It is also easy to see that X is a closed set. Moreover, if ξ, ξ′ ∈ X are suchthat sp(π(Γ)ξ) ⊥ sp(π(Γ)ξ′), then it is easy to see that ξ + ξ′ ∈ X .

In general, we then have that if ξ, ξ′ ∈ X then

ξ + ξ′ = (ξ + Projsp(π(Γ)ξ)(ξ′)) + (ξ′ − Projsp(π(Γ)ξ)(ξ

′)) ∈ X .

We therefore have shown that X is a closed Γ-invariant subspace which containsL and hence X = H.

If ϕ : Γ → C is a function of positive type and πϕ : Γ → U(Hϕ) is thecorresponding representation described in Section 1.2, then πϕ is generated bya single vector. We therefore obtain the following corollary.

Corollary 1.8.5. If ρ : Γ → U(K) is a representation of a group Γ, ϕ : Γ →C is a function of positive type, and πϕ : Γ → U(Hϕ) is the correspondingrepresentation. Then πϕ ≺ ρ if and only if F ⊂ Γ finite, and ε > 0, there existsη1, . . . , ηn ∈ K such that

|φ(γ)− Σnj=1〈ρ(γ)ηj , ηj〉| < ε,

for all γ ∈ F .

Exercise 1.8.6. Suppose ΓyX is an action of a group Γ on a set X , andα : Γ×X → Λ is a cocycle into a group Λ. If π : Λ → U(H), and ρ : Λ → U(K)are two representations such that π ≺ ρ, then show that IndαΛ π ≺ IndαΛ ρ.

Conclude that if Σ < Γ is an amenable subgroup of a group Γ then λΓ/Σ ≺λΓ.

For further properties of weak containment a good place to look is AppendixF in [BdlHV08].

Definition 1.8.7. [Bek90] Let π : Γ → U(H) be a unitary representation ofa group Γ, then π is amenable if there exists a tate Φ ∈ (B(H))∗ such thatΦ(π(γ)T ) = Φ(Tπ(γ)) for all γ ∈ Γ, T ∈ B(H).

Note that Γ is amenable if and only if the left-regular representation isamenable. We also have an analogue of Theorem 1.6.5 for amenable represen-tations. The proof is similar, however we will not present it here.

Theorem 1.8.8. [Bek90] Let π : Γ → U(H) be a unitary representation of agroup Γ, then the following conditions are equivalent.

(1). π is amenable.


(2). There exists a net of trace class operators Ti ∈ B(H) such that ‖Ti‖Tr =1, and ‖Tiπ(γ)− π(γ)Ti‖Tr → 0, for all γ ∈ Γ.

(3). There exists a net of finite rank projections Pi ∈ B(H) such that1

‖Pi‖HS‖Piπ(γ)− π(γ)Pi‖HS → 0, for all γ ∈ Γ.

(4). π ⊗ π contains almost invariant vectors.

(5). For any finite symmetric set S ⊂ Γ the operator TS = 1|S|Σγ∈Sπ⊗ π(γ)

satisfies ‖TS‖ = 1.

Chapter 2

Group actions on measure

spaces

2.1 Examples

Definition 2.1.1. Let (X,B, υ) be a σ-finite measure space, and let Γ be acountable group, an action ΓyX such that γ−1(B) = B, for all γ ∈ Γ is mea-sure preserving if for each measurable set A ∈ B we have υ(γ−1A) = υ(A).Alternately, we will say that υ is an invariant measure for the action Γy(X,B).

We’ll say that Γ preserves the measure class of υ, or alternately, υ is a quasi-invariant measure, if for each γ ∈ Γ, and A ∈ B we have that υ(A) = 0 if andonly if υ(γ−1A) = 0, i.e., for each γ ∈ Γ, the push-forward measure γ∗υ definedby γ∗υ(A) = υ(γ−1A) for all A ∈ B is absolutely continuous with respect to υ.

Note, that since the action ΓyX is measurable we obtain an action σ :Γ → Aut(M(X,B)) of Γ on the B-measurable functions by the formula σγ(f) =f γ−1.

If Γy(X,B, υ) preserves the measure class of υ, then for each γ ∈ Γ thereexists the Radon-Nikodym derivative dγ∗υ

dυ : X → [0,∞), such that for eachmeasurable set A ∈ B we have

υ(γ−1A) =

∫1A

dγ∗υ

dυdυ.

The Radon-Nikodym derivative is unique up to measure zero for υ.

If γ1, γ2 ∈ Γ, and A ∈ B then we have

υ((γ1γ2)−1A) =

∫1γ−1

1 A

dγ2∗υ

dυdυ

=

∫1Aσγ1

(dγ2∗υ

dυ) dγ1∗υ =

∫1Aσγ1

(dγ2∗υ

dυ)dγ1∗υ

dυdυ.

31

32 CHAPTER 2. GROUP ACTIONS ON MEASURE SPACES

Hence, we have almost everywhere

d(γ1γ2)∗υ

dυ= σγ1

(dγ2∗υ

dυ)dγ1∗υ

dυ.

Or, in other words, the map (γ, x) 7→ dγ∗υdυ (γx) ∈ R× is a cocycle almost every-

where.

Definition 2.1.2. Let Γy(X,B, υ), and Γy(Y,A, ν) be measure class preserv-ing actions of a countable group Γ on σ-finite measure spaces Γy(X,B, υ), andΓy(Y,A, ν). The actions are isomorphic (or conjugate) if there exists anisomorphism of measure spaces1 such that for all γ ∈ Γ and almost every x ∈ Xwe have

θ(γx) = γθ(x).

Example 2.1.3. Any action ΓyX of a countable group Γ on a countable set Xcan be viewed as a measure preserving action on X with the counting measure.

Example 2.1.4. Consider the torus T with the Borel σ-algebra, and Lebesguemeasure, if a ∈ [0, 1) then we obtain a measure preserving transformation T :T → T by T (eiθ) = ei(θ+2πa). This then induces a measure preserving action ofZ.

Example 2.1.5 (The odometer action). Consider the space 0, 1, with theuniform probability measure, and consider X = 0, 1N with the product mea-sure. We obtain a measure preserving transformation T : X → X given by“adding one”. That is to say that T applied to a sequence a1a2a3 · · · will bethe sequence 000 · · ·01an+1an+2 · · · where an is the first position in which a 0occurs in a1a2a3 · · · . Then T induces a probability preserving action of Z on0, 1N.

Example 2.1.6 (Bernoulli shift). Let (X0,B0, µ0) be a probability space, let Γbe a countable group, and consider X = XΓ

0 with the product measure. Thenwe have a measure preserving action of Γ on X by γx = xγ−1 for each x ∈ XΓ

0 .

Example 2.1.7 (Generalized Bernoulli shift). Let (X0,B0, µ0) be a probabilityspace, let ΓyI be an action of a countable group Γ on a non-empty countableset I, and consider X = XI

0 with the product measure. Then just as in the caseof the Bernoulli shift we have a measure preserving action of Γ on X given byγx = x γ−1 for each x ∈ XΓ

0 .

Exercise 2.1.8 (“The baker’s map”). Let X = [0, 1] × [0, 1] with Lebesguemeasure, consider the map T : X → X defined by

T (x, y) =

(2x, y

2 ), 0 ≤ x ≤ 12 ;

(2x− 1, y+12 ), 1

2 < x ≤ 1.

1A map θ : X → Y is an isomorphism of measure spaces if θ is almost everywhere abijection such that θ and also θ−1 is measure preserving.

2.1. EXAMPLES 33

Then T and T−1 are both measure preserving and hence give a measure pre-serving action of Z on X .

Show that the map θ : 0, 1Z → [0, 1]× [0, 1] given by

θ(x) = (Σn≤0x(n)2−(n+1),Σn>0x(n)2

n)

is an isomorphism of measure spaces which implements a conjugacy betweenthe Bernoulli shift Zy0, 1Z and the baker’s action Zy[0, 1]× [0, 1].

Example 2.1.9. Let G be a σ-compact, locally compact group, and let Γ be acountable group. Then any homomorphism from Γ to G induces a Haar measurepreserving action of Γ on G by left multiplication.

If G is compact, then we obtain a measure preserving action of Γ on aprobability space.

Example 2.1.10. Let G be a σ-compact, locally compact group, let H < G bea closed subgroup, and let Γ < G be a countable subgroup. There always existsa G-quasi-invariant measure on the homogeneous space G/H , (see for exampleSection 2.6 in [Fol95]). Thus we always obtain a measure class preserving actionof Γ on G/H .

A lattice in G is a discrete subgroup ∆ such that G/∆ has an invariantprobability measure. In this case we obtain a probability measure preservingaction of Γ on G/∆.

Example 2.1.11. Suppose A is an abelian group, and Γ is a group of automor-phisms of A, then not only does Γ preserve the Haar measure of A, but also Γhas a Haar measure preserving action of the dual group A given by γx = xγ−1.As an example, we can consider the action of SLnZ on Zn given by matrix mul-tiplication, this then induces a probability measure preserving action of SLnZ

on the dual group Tn.

Example 2.1.12 (Uniform ordering [MOP79, Kie75]). Let Γ be a countablegroup, and consider O(Γ) the set of all total orders of Γ. By interpreting totalorders <t on Γ with functions from Γ × Γ to 0, 1 which take the value 1 at(γ1, γ2) if and only if γ1 <t γ2, we may consider O(Γ) as a subset of 0, 1Γ×Γ,and we consider the corresponding σ-algebra.

We have an action of Γ on O(Γ) by requiring that x <γt y if and only ifγx <t γy. We may place a probability measure µ on O(Γ) by requiring that foreach pairwise distinct x1, x2, . . . , xn ∈ Γ we have

µ(<t∈ O(Γ) | x1 <t x2 <t · · · <t xn) =1

n!.

By Caratheodory’s Extension Theorem, this extends to an invariant measurefor the action of Γ. If ΓyX is an action of Γ on a countable set X , then we canof course also consider the corresponding probability measure preserving actionof Γ on O(X).


Example 2.1.13 (Furstenberg’s Correspondence Principle [Fur77]). Let Γ bea countable amenable group and let Fn ⊂ Γ be a Følner sequence. Then eachset Fn gives rise to a probability measure µn ∈ Prob(Γ) ⊂ Prob(βΓ), given asthe uniform probability measure on Fn. Since the Stone-Cech compactificationβΓ is compact, the Arzela-Ascoli Theorem implies that the sequence µn has acluster point µ ∈ Prob(βΓ).

Since Fn is a Følner sequence we have that µ is an invariant probabilitymeasure for the action of Γ on βΓ. Moreover, if A ⊂ Γ ⊂ βΓ then we have that

lim infn→∞

|A ∩ Fn||Fn|

≤ µ(A) ≤ lim supn→∞

|A ∩ Fn||Fn|

.

We also remark that if A1, A2, . . . , Ak ⊂ Γ then it is easy to see that

∩kj=1Aj = ∩k

j=1Aj .

Since βΓ is not second countable L2(βΓ, µ) will not be separable in general.However, since Γ is countable, if A ⊂ Γ then we can consider the Γ-invariantsub-σ-algebra generated by A ⊂ βΓ, this then gives a separable Hilbert space.

Example 2.1.14. Similar to Furstenberg’s Correspondence Principle, supposeG is a locally compact group. The Baire σ-algebra Baire on G is the σ-algebragenerated by the Gδ sets which are compact. Alexanderoff showed that thereis a Banach space isomorphism between Cb(G)∗ and the space ba(G,Baire) ofregular, bounded, finitely additive means m on Baire with norm given by totalvaluation (see for example Theorem IV.6.2 in [DS88]).

Hence if m is a regular, finitely additive mean on the Baire σ-algebra of Gwhich is invariant under the action of G. Then m gives a state on Cb(G) =C(βG) which by the Riesz Representation Theorem gives a Radon probabilitymeasure µ on βG, hence, if Γ is a countable group and we have a homomorphismfrom Γ to G then left multiplication induces a µ-probability measure preservingaction of Γ on βG.

Example 2.1.15 (Randomorphisms [Mon06]). Let Γ and Λ be two countablegroups and consider the action of Γ on the space [Γ,Λ] = f ∈ ΛΓ | f(e) = eas described in Example 1.3.3. We consider ΛΓ with the Polish space structuregiven by the product topology where Λ is discrete, we then endow ΛΓ with theBorel σ-algebra. A randomorphism from Γ to Λ is a Γ-invariant probabilitymeasure µ on ΛΓ on this σ-algebra. Note that homomorphism from Γ to Λ isjust a Γ fixed point in ΛΓ, hence the Dirac measure at such a point gives arandom homomorphism.

If a random homomorphism µ is supported on the space of maps which areinjective then we say that µ is a randembedding. There is also of course thecorresponding notion of a random surjection, and a random bijection.

Notice that we can identify the space of bijections in [Γ,Λ] with the space ofbijections in [Λ,Γ] by the inverse map. Under this identification we then obtainan action of Λ on the space of bijections in [Γ,Λ] given by

(λf)(x) = f(xf−1(λ))λ−1.

2.2. THE KOOPMAN REPRESENTATION 35

Example 2.1.16. Let Γy(X,B, υ), and Γy(Y,A, ν) be measure class preserv-ing actions of a countable group Γ on σ-finite measure spaces (X,B, υ), and(Y,A, ν). Then we obtain a diagonal action Γy(X × Y,B ⊗A, υ × ν) given by

γ(x, y) = (γx, γy).

If actions Γy(X,B, υ), and Γy(Y,A, ν) are measure preserving, then so is thediagonal action.

2.2 The Koopman representation

Definition 2.2.1 ([Koo31]). Let Γy(X,B, υ) be an action on a measure spacewhich preserves the measure class υ. The Koopman representation of Γassociated to this action is the unitary representation π : Γ → U(L2(X, υ))given by

(π(γ)f)(x) = f(γ−1x)(dγ∗υ

dυ)1/2(x).

Note that for all γ1, γ2 ∈ Γ, and f ∈ L2(X, υ) we have

π(γ1γ2)f = σγ1γ2(f)(

d(γ1γ2)∗υ

dυ)1/2

= σγ1(σγ2

(f)(dγ2∗υ

dυ)1/2)(

dγ1∗υ

dυ)1/2 = π(γ1)(π(γ2)f).

Also, for all γ ∈ Γ, and f ∈ L2(X, υ) we have

‖π(γ)f‖22 =∫

|σγ(f)|2dγ∗υ

dυdυ

=

∫|σγ(f)|2 dγ∗υ = ‖f‖22.

Hence, π is indeed a unitary representation.If (X,B, υ) is a finite measure space, and Γy(X,B, υ) is measure preserving,

then 1 ∈ L2(X, υ) and π(γ)(1) = 1 is an invariant vector. For this reason, inthis setting the Koopman representation usually denotes the restriction of theabove representation to the orthogonal complement

L20(X, υ) = f ∈ L2(X, υ) |

∫f dυ = 0.

Exercise 2.2.2. Let Γy(X,B, υ) be an action on a measure space which pre-serves the measure class υ. Show that the Koopman representation π is isomor-phic to its conjugate representation π.

Exercise 2.2.3. Let Γy(X,B, µ), and Γy(Y,A, ν) be measure preserving ac-tions of a countable group Γ on probability spaces (X,B, υ), and (Y,A, ν). Showthat the Koopman representation πX×Y for the product action decomposes asπX×Y

∼= πX ⊕ πY ⊕ (πX ⊗ πY ) where πX and πY are the Koopman representa-tions for Γy(X,B, µ), and Γy(Y,A, ν).


2.2.1 Ergodicity

Definition 2.2.4. Let Γy(X,B, υ) be a measure class preserving action of acountable group Γ on a σ-finite measure space (X,B, υ). The action Γy(X,B, υ)is ergodic, if for any E ∈ B which is Γ-invariant, we have that either υ(E) = 0or υ(X \ E) = 0.

Lemma 2.2.5. Let Γy(X,B, µ) be a measure preserving action of a countablegroup Γ on a probability space (X,B, µ). The action Γy(X,B, µ) is ergodic ifand only if the Koopman representation is ergodic.

Proof. If E ∈ B is Γ-invariant then χE − µ(E) ∈ L20(X,µ) is also Γ-invariant,

which is non-zero if µ(E) 6= 0, 1. On the other hand, if ξ ∈ L20(X,µ) is a non-

zero Γ-invariant function then Et = x ∈ X | |ξ(x)| < t is Γ-invariant for allt > 0, and we must have µ(Et) 6= 0, 1 for some t > 0.

Theorem 1.6.5 can be adapted to the setting of actions on measure spacesas follows.

Theorem 2.2.6. Let Γy(X,B, υ) be a measure preserving action of a count-able group Γ on an infinite, σ-finite measure space (X,B, υ). The followingconditions are equivalent.

(1). There exists a state ϕ ∈ (L∞(X,B, υ))∗ such that for all γ ∈ Γ, andf ∈ L∞(X,B, υ) we have ϕ(σ(γ)(f)) = ϕ(f).

(2). For every ε > 0, and F ⊂ Γ finite, there exists ν ∈ Prob(X,B), suchthat ν is absolutely continuous with respect to υ and

‖dγ∗νdυ

− dν

dυ‖1 < ε.

(3). For every ε > 0, and F ⊂ Γ finite, there exists a measurable set A ⊂ Xsuch that υ(A) < ∞, and

υ(A∆γA) < ευ(A).

(4). The Koopman representation π : Γ → U(L2(X,B, υ)) has almost in-variant vectors.

(5). The Koopman representation π : Γ → U(L2(X,B, υ)) is amenable.

Exercise 2.2.7. Adapt the proof of Theorem 1.6.5 to prove Theorem 2.2.6.

If Γ acts by measure preserving transformations on a probability space, thenthe conditions above are trivially satisfied, however, we still have the following,non-trivial, adaptation.


Theorem 2.2.8. Let Γy(X,B, µ) be a measure preserving action of a countablegroup Γ on a probability space (X,B, µ). The following conditions are equivalent.

(1). There exists a state ϕ ∈ (L∞(X,B, µ))∗, different than f 7→∫f dµ,

such that for all γ ∈ Γ, and f ∈ L∞(X,B, µ) we have ϕ(σ(γ)(f)) = ϕ(f).

(2). There exits A ⊂ X measurable such that µ(A) < 1, and for everyε > 0, and F ⊂ Γ finite, there exists ν ∈ Prob(X,B), such that ν is absolutelycontinuous with respect to µ and

ν(A) > 1− ε, and ‖dγ∗νdµ

− dν

dµ‖1 < ε.

(3). For every ε > 0, and F ⊂ Γ finite, there exists a measurable set A ⊂ Xsuch that µ(A) ≤ 1/2, and

µ(A∆γA) < εµ(A).

(4). The Koopman representation π : Γ → U(L20(X,B, µ)) has almost in-

variant vectors.

(5). The Koopman representation π : Γ → U(L20(X,B, µ)) is amenable.

If Γy(X,B, µ) is an ergodic measure preserving action of a countable groupon a probability space (X,B, µ) such that none of the conditions of the previoustheorem hold, then we say that the action Γy(X,B, µ) has spectral gap.

2.2.2 Weak mixing

Definition 2.2.9. Let Γy(X,B, µ) be a measure preserving action of a count-able group on a probability space (X,B, µ). A sequence En ∈ B of measurablesubsets is an asymptotically invariant sequence if µ(En∆γEn) → 0, forall γ ∈ Γ. Such a sequence is said to be non-trivial if lim infn→∞ µ(En)(1 −µ(En)) > 0. The action Γy(X,B, µ) is strongly ergodic if there does notexist a non-trivial asymptotically invariant sequence.

Note that if Γy(X,B, µ) has a non-trivial invariant set E ∈ B, then the con-stant sequence En = E is a non-trivial asymptotically invariant sequence, hencestrongly ergodic implies ergodic. Also, a non-trivial asymptotically invariantsequence will show that condition (3) in Theorem 2.2.8 is satisfied, hence anergodic action with spectral gap must be strongly ergodic.

Definition 2.2.10. Let Γy(X,B, µ) be a measure preserving action of a count-able group Γ on a probability space (X,B, µ). The action Γy(X,B, µ) is weak

mixing, if |Γ| = ∞, and for all E ⊂ B finite, we have

lim infγ→∞

ΣA,B∈E |µ(A ∩ γB)− µ(A)µ(B)| = 0.


Proposition 2.2.11. Let Γy(X,B, µ) be a measure preserving action of acountable group Γ on a probability space (X,B, µ). The following conditionsare equivalent:

(1). The action Γy(X,B, µ) is weak mixing.

(2). The Koopman representation π : Γ → U(L20(X,B, µ)) is weak mixing.

(3). The diagonal action Γy(X ×X,B ⊗ B, µ× µ) is ergodic.

(4). For any ergodic, measure preserving action Γy(Y,A, ν) on a probabilityspace, the diagonal action Γy(X × Y,B ⊗A, µ× ν) is ergodic.

(5). The diagonal action Γy(X ×X,B ⊗ B, µ× µ) is weak mixing.

(6). For any weak mixing, measure preserving action Γy(Y,A, ν) on a prob-ability space, the diagonal action Γy(X × Y,B ⊗A, µ× ν) is weak mixing.

Proof. For (1) ⇐⇒ (2), if E ⊂ B is a finite set then we can consider the finiteset F = χE − µ(E) | E ∈ E ⊂ L2

0(X,B, µ). For each γ ∈ Γ we therefore have

Σξ,η∈F |〈π(γ)ξ, η〉| = ΣA,B∈E |〈σγ(χB)− µ(B), χA − µ(A)〉|

= ΣA,B∈E |〈χγB − µ(B), χA − µ(A)〉|

= ΣA,B∈E |µ(A ∩ γB)− µ(A)µ(B)|.

If the Koopman representation is weak mixing then from this we see imme-diately that the action is weak mixing. A similar calculation shows that if theaction is weak mixing and F ⊂ L2

0(X,B, µ) is a finite set of simple functions,then

lim infγ→∞

Σξ,η∈F |〈π(γ)ξ, η〉| = 0.

Since simple functions are dense in L20(X,B, µ) this shows that the Koopman

representation is weak mixing.

The remaining equivalences then easily follow from Exercise 2.2.3, togetherwith the corresponding properties of weak mixing for unitary representations inSections 1.5 and 1.7.

Corollary 1.7.7 then shows that weak mixing is preserved under taking finiteindex.

Corollary 2.2.12. Let Γy(X,B, µ) be a weak mixing, measure preserving ac-tion of a countable group Γ on a probability space (X,B, µ). If Σ < Γ is a finiteindex subgroup, then the restriction of the action of Γ to Σ is also weak mixing.

Corollary 1.7.16 gives the following result for weak mixing actions of amenablegroups.


Corollary 2.2.13. Let Γy(X,B, µ) be a measure preserving action of a count-able amenable group Γ on a probability space (X,B, µ). Let Fn ⊂ Γ, be a Følnersequence for Γ. The action Γy(X,B, µ) is weak mixing if and only if for allA,B ∈ B, we have

1

|Fn|Σγ∈Fn |µ(A ∩ γB)− µ(A)µ(B)| → 0.

Definition 2.2.14. Let Γy(X,B, µ) be a measure preserving action of a count-able group Γ on a probability space (X,B, µ). The action Γy(X,B, µ) is(strong) mixing, if |Γ| = ∞, and for any A,B ⊂ X measurable we have

limγ→∞

|µ(A ∩ γB)− µ(A)µ(B)| = 0.

The same proof as in Proposition 2.2.11 yields the following proposition formixing actions.

Proposition 2.2.15. Let Γy(X,B, µ) be a measure preserving action of acountable group Γ on a probability space (X,B, µ). The action Γy(X,B, µ)is mixing if and only if the Koopman representation is mixing.

2.2.3 Compact actions

Definition 2.2.16. Let (X,B, µ) be a probability space. We denote by Aut(X,B, µ)the group of automorphisms of (X,B, µ), where we identify two automorphismsif they agree almost everywhere. The weak topology on Aut(X,B, µ) is thesmallest topology such that the maps T 7→ µ(T (A)∆B) are continuous for allA,B ∈ B.

Exercise 2.2.17. Show that the weak topology endows Aut(X,B, µ) with atopological group structure.

Exercise 2.2.18. The Koopman representation for Aut(X,B, µ) is defined asbefore, i.e., π : Aut(X,B, µ) → U(L2

0(X,B, µ)) is defined by π(T )(f) = f T−1. Show that the image of π is closed and that π is homeomorphism fromAut(X,B, µ) with the the weak topology onto π(Aut(X,B, µ)) ⊂ U(L2

0(X,B, µ))with the strong operator topology.

Definition 2.2.19. Let Γy(X,B, µ) be a measure preserving action of a count-able group Γ on a probability space (X,B, µ). The action Γy(X,B, µ) is com-

pact if the image of Γ in Aut(X,B, µ) is precompact in the weak topology.

An immediate consequence of Exercise 2.2.18 is the following.

Proposition 2.2.20. Let Γy(X,B, µ) be a measure preserving action of acountable group Γ on a probability space (X,B, µ). Then Γy(X,B, µ) is compactif and only if the Koopman representation π : Γ → U(L2

0(X,B, µ)) is compact.


Definition 2.2.21. Let Γy(X,B, µ) be a measure preserving action of a count-able group Γ on a probability space (X,B, µ). A function f ∈ L2(X,B, µ) isalmost periodic if the Γ orbit σγ(f) | γ ∈ Γ is pre-compact in the ‖ · ‖2-topology.

Proposition 2.2.22. Let Γy(X,B, µ) be a measure preserving action of a dis-crete group Γ on a standard probability space (X,B, µ). Let A ⊂ B be theσ-algebra generated by all almost periodic functions in L2(X,B, µ). Then Ais Γ-invariant, and f ∈ L2(X,B, µ) is almost periodic if and only if f is Ameasurable.

Proof. For each f ∈ L2(X,B, µ) denote by Sf the Γ-orbit of f , and considerthe set AP (L2(X,B, µ)) of all almost periodic functions. Note that it is obviousthat AP (L2(X,B, µ)) is Γ-invariant, contains the scalars, and is closed underscalar multiplication.

Since addition is continuous with respect to ‖ · ‖2, if f, g ∈ AP (L2(X,B, µ))we have that Sf + Sg is compact being the continuous image of the compactset Sf × Sg. We therefore have that Sf+g ⊂ Sf + Sg is precompact, henceAP (L2(X,B, µ)) is closed under addition. Similarly, if f, g ∈ AP (L2(X,B, µ))then we have that

|f |, f ,max|f |, |g|,min|f |, |g| ∈ AP (L2(X,B, µ)).

Also, if we also have that g ∈ L∞(X,B, µ) then fg ∈ AP (L∞(X,B, µ)).If fn ∈ AP (L2(X,B, µ)) and f ∈ L2(X,B, µ) such that ‖fn − f‖2 → 0, then

fix ε > 0, and take n ∈ N such that ‖fn−f‖2 < ε/2. Since fn is almost periodicwe have that Sfn is totally bounded, hence there exists a finite set ⊂ L2(X,B, µ)such that infh∈F ‖σγ(fn) − h‖2 < ε/2 for all γ ∈ Γ. By the triangle inequalitywe then have infh∈F ‖σγ(f) − h‖2 < ε. This shows that Sf is totally boundedand hence f ∈ AP (L2(X,B, µ)).

The operations above then generate all A-measurable functions and so weobtain the result.

Definition 2.2.23. Let Γy(X,B, µ) be a measure preserving action of a dis-crete group Γ on a standard probability space (X,B, µ). Suppose A ⊂ B is aΓ-invariant σ-algebra, then we say that the action Γy(X,A, µ) is a factor ofthe action Γy(X,B, µ).

Note that the Koopman representation of a factor Γy(X,A, µ) is a sub-representation of the Koopman representation of Γy(X,B, µ). Hence, Proposi-tion 2.2.11 together with Proposition 2.2.22 gives the following.

Proposition 2.2.24. Let Γy(X,B, µ) be a measure preserving action of a dis-crete group Γ on a standard probability space (X,B, µ). Then there exists aunique maximal factor Γy(X,A, µ) which is compact. Moreover, Γy(X,B, µ)is weak mixing if and only if A is trivial, i.e., when A consists only of null orco-null sets.

2.3. A REMARK ABOUT MEASURE SPACES 41

2.3 A remark about measure spaces

So far we have been considering actions of a countable group on a generalprobability space (X,B, µ). For most aspects of ergodic theory this is the propersetting. However, occasionally this level of generality can be problematic. Forexample, consider the action of Z by rotation on the circle T. We can considerT equipped with either the Borel σ-algebra, or the Lebesgue σ-algebra. If weare not concerned with sets of measure 0, then both of these systems containthe same information, however the identity map from the Borel σ-algebra to theLebesgue σ-algebra is not measurable, and these two systems are not isomorphicunder our notion of isomorphism.

For another example, consider the case when Γy(X,B, µ) and Γy(Y,A, ν)are measure preserving actions. Then the diagonal action Γy(X×Y,B⊗A, µ×ν)has the factor Γy(X×Y,B⊗Y, ∅, µ×ν), which we would like to identify withthe action Γy(X,B, µ). However, a problem arises because the projection mapfrom X×Y to X is not almost everywhere 1-1 and hence is not an isomorphismof actions.

One way to overcome this problem is to restrict ourselves to only consideractions on nice measure spaces. Specifically, one considers the class of stan-dard probability spaces. A standard probability space (X,B, µ) is a proba-bility space such that the underlying σ-algebra space (X,B) is isomorphic (asσ-algebra spaces) to a Polish2 topological space with its Borel σ-algebra. Inthis setting we do not allow actions on T with the Lebesgue measure, or actionson the space (X × Y,B⊗ Y, ∅, µ× ν) since these are not standard probabilityspaces, and so the problems above do not arise.

An alternate approach, which we will take here, is to continue to allow gen-eral probability spaces, but instead generalize our notion of equivalence so thatthe spaces above are equivalent under this more general notion. Thus, from nowon we will say that two probability spaces (X,B, µ) and (Y,A, ν) are isomor-phic if there is an integral preserving unital ∗-isomorphism from L∞(Y,A, ν) toL∞(X,B, µ). We will also say that two measure preserving actions Γy(X,B, µ)and Γy(Y,A, ν) are isomorphic (or conjugate) if there exists a ∗-isomorphism

θ from L∞(Y,A, ν) to L∞(X,B, µ) such that θ σγ = σγ θ for all γ ∈ Γ. Notethat if θ : X → Y is a bijection such that θ and θ−1 is measure preserving suchthat θ γ = γ θ for all γ ∈ Γ, then θ : L∞(Y,A, ν) → L∞(X,B, µ) given by

θ(f) = f θ−1 implements an isomorphism between the two actions.We also have a process (essentially the GNS-construction) which takes us

from a general action on a separable measure space to an isomorphic action ona standard Borel space.

Proposition 2.3.1. Let A be a unital ∗-algebra with a state3 φ such that forall a ∈ A there exists K > 0 such that φ(a∗abb∗) ≤ Kφ(bb∗) for all b ∈ A. Let

2A topological space is Polish if it is separable and has a complete metric which inducesthe topology

3A state on a unital ∗-algebra is a linear map φ : A → C such that φ(1) = 1, and φ(a∗a) ≥ 0for all a ∈ A.


Γ be a countable group and suppose σ : Γ → Aut(A) is an action of Γ on A byunital ∗-homomorphisms such that φ(σγ(a)) = φ(a) for all a ∈ A, and γ ∈ Γ.

Then there exists a compact Hausdorff space X, a continuous action ΓyX,a Γ-invariant Radon probability measure µ on X, and a unital ∗-homomorphismπ : A → L∞(X,µ) such that π σγ = σγ π for all γ ∈ Γ, and

∫π(a) dµ = φ(a),

for all a ∈ A.Moreover, if A is countable generated as an algebra, then X is separable (and

hence a Polish space).

Proof. We endow A with the inner-product 〈a, b〉φ = φ(b∗a), by quotienting outby the kernel of this inner-product and then taking the completion, we obtaina Hilbert space L2(A, φ). Given a ∈ A we denote by a the equivalence class ofa in L2(A, φ). If a, b ∈ A, then we may consider a acting on L2(A, φ) by leftmultiplication. The formula

‖ab‖2φ = φ(a∗abb∗) ≤ K‖b‖2φshows that left multiplication is well defined and bounded, we therefore obtaina ∗-homomorphism π0 : A → B(L2(A, φ)).

We denote by C∗(A) the abelian C∗-algebra generated by π0(A) in B(L2(A, φ)),and we denote by X the Gelfand spectrum of C∗(A). By Gelfand’s Theoremwe have that X is a compact Hausdorff space and we obtain a ∗-isomorphismfrom C∗(A) to C(X). We therefore obtain a ∗-homomorphism π : A → C(X)by applying the Gelfand transform to the image π0(A).

On C(X) ∼= C∗(A) we may consider the state φ given by φ(x) = 〈x1, 1〉φ. Bythe Riesz Representation Theorem the state φ corresponds to a Radon measureµ on X such that φ(x) =

∫x dµ for all x ∈ C(X). We therefore have that for

all a ∈ A ∫π(a) dµ = φ(π(a))

= 〈π0(a)1, 1〉φ = 〈a, 1〉φ = φ(a).

Since σ : Γ → Aut(A) preserves the state φ, we have that for all γ ∈ Γ, anda, b ∈ A

‖σγ(a)b‖2φ = φ(σγ(a∗a)bb∗) = φ(a∗aσγ−1(bb∗)) = ‖ aσγ−1(b)‖2φ.

Hence, if we define σγ on π0(A) by σγ(π0(a)) = π0(σγ(a)) then this is welldefined and preserves the operator norm, hence extends to a map (which westill denote by σ) from Γ to Aut(C∗(A)). The Gelfand transform then gives acontinuous action of Γ on X , such that σγ(x) = x γ−1 for all x ∈ C(X).

An easy calculation then shows that π σγ = σγ π for all γ ∈ Γ, and wehave that for all γ ∈ Γ, and x ∈ C(X)

∫x dγ∗µ =

∫x γ−1 dµ = φ(σγ(x)) = φ(x) =

∫x dµ.

Hence µ is Γ-invariant.Finally, if A is countably generated as an algebra then C∗(A) is a separable

C∗-algebra and hence the Gelfand spectrum X is separable.

2.4. GAUSSIAN ACTIONS 43

Corollary 2.3.2. Suppose Γy(X,B, µ) is a measure preserving action of acountable group Γ on a probability space (X,B, µ) such that L2(X,B, µ) is sep-arable. Then the action Γy(X,B, µ) is isomorphic to a measure preservingaction of Γ on a standard probability space (Y,A, ν).

Proof. Since L2(X,B, µ) is separable there exists a countably generated ∗-subalgebraA ⊂ L∞(X,B, µ) ⊂ L2(X,B, µ) which is dense in L2(X,B, µ). Since Γ is count-able we may also assume that A is Γ-invariant. By Proposition 2.3.1 there existsa measure preserving action of Γ on a standard probability space (Y,A, ν) andhomomorphism π : A → L∞(Y,A, ν) which is Γ-equivariant and preserves theintegrals.

Since π preserves the integrals we may extend it to a unitary U : L2(X,B, µ) →L2(Y,A, ν) which also preserves the integrals, and is again Γ-equivariant. Themap f 7→ U∗fU then gives a ∗-isomorphism between L∞(Y,A, ν) and L∞(X,B, ν)which preserves the integrals and is again Γ-equivariant.

2.4 Gaussian actions

Let π : Γ → O(H) be an orthogonal representation of a countable group Γ. Theaim of this section, which is taken from [PS], is to describe the construction ofa measure-preserving action of Γ on a non-atomic standard probability space(X,µ) such that H is realized as a subspace of L2

R(X,µ) and π is contained in

the Koopman representation ΓyL20(X,µ). The action Γy(X,µ) is referred to

as the Gaussian action associated to π.Given a Hilbert space H, the n-symmetric tensor H⊙n is the subspace of

H⊗n fixed by the action of the symmetric group Sn by permuting the indices.For ξ1, . . . , ξn ∈ H, we define their symmetric tensor product ξ1⊙· · ·⊙ξn ∈ H⊙n

to be 1n!

∑σ∈Sn

ξσ(1) ⊗ · · · ⊗ ξσ(n). Denote

S(H) = (RΩ⊕∞⊕

n=1

H⊙n)⊗R C,

with renormalized inner product such that ‖ξ‖2S(H) = n!‖ξ‖2, for ξ ∈ H⊙n.

For ξ ∈ H let xξ be the symmetric creation operator,

xξ(Ω) = ξ, xξ(η1 ⊙ · · · ⊙ ηk) = ξ ⊙ η1 ⊙ · · · ⊙ ηk,

and its adjoint,

x∗ξ(Ω) = 0, x∗

ξ(η1 ⊙ · · · ⊙ ηk) =

k∑

i=1

〈ηi, ξ〉η1 ⊙ · · · ⊙ ηi ⊙ · · · ⊙ ηk.

Let

s(ξ) =1

2(xξ + x∗

ξ),

and note that it is an unbounded, self-adjoint operator on S(H).


The moment generating function M(t) for the Gaussian distribution is de-fined to be

M(t) =1√2π

∫ ∞

−∞

exp(tx) exp(−x2/2)dx = exp(t2/2).

It is easy to check that if ‖ξ‖ = 1 then

〈s(ξ)n Ω,Ω〉 = M (n)(0) =(2k)!

2kk!,

if n = 2k and 0 if n is odd. Hence, s(ξ) may be regarded as a Gaussian randomvariable. Note that if ξ, η ∈ H then s(ξ) and s(η) commute, moreover, if ξ ⊥ η,then

〈s(ξ)ms(η)nΩ,Ω〉 = 〈s(ξ)mΩ,Ω〉〈s(η)nΩ,Ω〉,for all m,n ∈ N; thus, s(ξ) and s(η) are independent random variables.

From now on we will use the convention ξ1ξ2 · · · ξk to denote the symmetrictensor ξ1 ⊙ ξ2 ⊙ · · · ⊙ ξk. Let Ξ be a basis for H and

S(Ξ) = Ω ∪ s(ξ1)s(ξ2) · · · s(ξk)Ω | ξ1, ξ2, . . . , ξk ∈ Ξ.

Lemma 2.4.1. The set S(Ξ) is a (non-orthonormal) basis of S(H).

Proof. We will show that ξ1 · · · ξk ∈ sp(S(Ξ)), for all ξ1, . . . , ξk ∈ H. We haveΩ ∈ sp(S(Ξ)). Also, since s(ξ)Ω = ξ, H ⊂ sp(S(Ξ)). Now as s(ξ1) · · · s(ξk)Ω =P (ξ1, . . . , ξk) is a polynomial in ξ1, . . . , ξk of degree k with top term ξ1 · · · ξk,the result follows by induction on k.

Let u(ξ1, . . . , ξk) = exp(πis(ξ1) · · · s(ξk)) and u(ξ1, . . . , ξk)t = exp(πits(ξ1) · · · s(ξk)).

Denote by A the von Neumann algebra generated by all such u(ξ1, . . . , ξk), whichis the same as the von Neumann algebra generated by the spectral projectionsof the unbounded operators s(ξ1) · · · s(ξk).

Theorem 2.4.2. We have that L2(A, τ) ∼= S(H), and A is a maximal abelian∗-subalgebra of B(S(H)) with faithful state τ = 〈·Ω,Ω〉.

Proof. By Lemma 2.4.1, A 7→ AΩ is an embedding of A into S(H). By Stone’sTheorem

limt→0

u(ξ1, . . . , ξk)t − 1

πitΩ = s(ξ1) · · · s(ξk)Ω;

hence, AΩ is dense inS(H). This implies that A is maximal abelian in B(S(H)).

There is a natural strong operator topology continuous embedding O(H) →U(S(H)) given by

T 7→ TS = 1⊕∞⊕

n=1

T⊙n.

2.5. ERGODIC THEOREMS 45

It follows that there is an embedding O(H) → Aut(A, τ), T 7→ σT , which canbe identified on the unitaries u(ξ1, . . . , ξk) by

σT (u(ξ1, . . . , ξk)) = Ad(TS)(u(ξ1, . . . , ξk)) = u(T (ξ1), . . . , T (ξk)).

Thus for an orthogonal representation π : Γ → O(H), there is a natural ac-tion σπ : Γ → Aut(A, τ) given by σπ

γ (u(ξ1, . . . , ξk)) = u(πγ(ξ1), . . . , πγ(ξk)) =

Ad(πSγ )(u(ξ1, . . . , ξk)). Applying Proposition 2.3.1 we then obtain a measure

preserving probability space (X,B, µ) and an action of Γ on this space so thatwe can identify A with L∞(X,µ) in a state preserving Γ-equivariant manor.The action of Γ on (X,B, µ) is the Gaussian action associated to π.

We have an explicit description of the Koopman representation of Γy(X,B, µ)given by πS(γ) = (π(γ))S ⊖ 1. Hence, we have that ergodic properties whichremain stable with respect to tensor products transfer from π to σπ.

Proposition 2.4.3. Let π : Γ → O(H) be an orthogonal representation of acountable group Γ. Then the Gaussian action Γy(X,B, µ) is ergodic, if andonly if it is weak mixing, if and only if π is weak mixing.

Proof. If π is not weak mixing, then there exists ξ ∈ H⊗2 such that for allγ ∈ Γ, π⊗2(γ)(ξ) = ξ. Viewing ξ as a Hilbert-Schmidt operator on H, let|ξ| = (ξξ∗)1/2. Since the map ξ ⊗ η 7→ η ⊗ ξ is the same as taking the adjointof the corresponding Hilbert-Schmidt operator, we have that |ξ| ∈ H⊙2 andπ⊙2(γ)(|ξ|) = |ξ|. Since π⊙2 embeds into the Koopman representation of theGaussian action, it follows from Lemma 2.2.5 that Γy(X,B, µ) is not ergodic.

Conversely, if Γy(X,B, µ) is not ergodic then neither is the Koopman rep-resentation πS. But πS is a sub-representation of

⊕∞n=1 π

⊗n = π ⊗ (1 ⊕⊕∞n=1 π

⊗n). Hence if πS has non-trivial invariant vectors then π is not weakmixing by Proposition 1.5.6.

2.5 Ergodic theorems

Using the Koopman representation, von Neumann’s Ergodic Theorem for Hilbertspaces can be rephrased for actions.

Theorem 2.5.1 (Von Neumann’s Ergodic Theorem [vN32]). Let Γy(X,µ) bea measure preserving action of a countable amenable group Γ on a probabilityspace (X,B, µ). Let EI ∈ B(L2(X,B, µ)) be the projection onto the subspaceof Γ-invariant functions. Let Fn ⊂ Γ be a Følner sequence, then for each f ∈L2(X,B, µ) we have that

‖ 1

|Fn|Σγ∈F−1

nσγ(f)− EI(f)‖2 → 0.

To state Birkoff’s Ergodic Theorem we first need to discuss conditional ex-pectations. Note that if (X,B, µ) is a probability space, then we may considera function f ∈ L∞(X,B, µ) as a bounded operator on L2(X,B, µ) by multipli-cation, i.e., g 7→ fg. To avoid confusion in the next lemmas, when we consider


a function f ∈ L∞(X,B, µ) as an element in L2(X,B, µ) we will continue touse the notation f , however, when we consider f as a bounded operator we willdenote this operator by Mf .

Lemma 2.5.2. Let (X,B, µ) be a probability space, then L∞(X,B, µ) ⊂ B(L2(X,B, µ))is a maximal abelian self-adjoint subalgebra.

Proof. Suppose T ∈ B(L2(X,B, µ)) commutes with L∞(X,B, µ), we want toshow that there exists some f ∈ L2(X,B, µ) such that T (g) = fg for all g ∈L2(X,B, µ).

Let f = T (1) ∈ L2(X,B, µ), if g ∈ L∞(X,B, µ) ⊂ L2(X,B, µ) then we have

T (g) = TMg(1) = MgT (1) = Mg(f) = fg.

Hence, if we consider the functions ft = 1[0,t)(|f |)f ∈ L∞(X,B, µ), then wehave

‖ft‖∞ = supg∈L2(X,B,µ),‖g‖2=1

‖ftg‖2

= supg∈L2(X,B,µ),‖g‖2=1

‖T (1[0,t)(|f |)g)‖2

≤ supg∈L2(X,B,µ),‖g‖2=1

‖T ‖‖1[0,t)(|f |)g‖2 ≤ ‖T ‖.

Therefore, ‖f‖∞ = limt→∞ ‖ft‖∞ ≤ ‖T ‖, and so f ∈ L∞(X,B, µ). Moreover,since L∞(X,B, µ) ⊂ L2(X,B, µ) is dense, we have that T (g) = fg for all g ∈L2(X,B, µ).

If A ⊂ B is σ-subalgebra we will denote by EA ∈ B(L2(X,B, µ)) the orthog-onal projection onto L2(X,A, µ). Note that if g ∈ L∞(X,A, µ) then we canthink of Mg as a bounded operator on L2(X,A, µ) or on L2(X,B, µ), and sincethe product of two A-measurable functions is again A-measurable it follows that

MgEA = EAMg.

Then for each f ∈ L∞(X,B, µ) and g ∈ L∞(X,A, µ) we have that

(EAMfEA)Mg = EAMfMgEA = Mg(EAMfEA).

By Lemma 2.5.2 there exists a function f0 ∈ L∞(X,B, µ) such that EAMfE∗A =

Mf0 . It also follows from the proof of Lemma 2.5.2 that actually f0 = EA(f)and so we see that EA(L

∞(X,B, µ)) ⊂ L∞(X,A, µ), and ‖EA(f)‖∞ ≤ ‖f‖∞for all f ∈ L∞(X,B, µ).

The map EA : L∞(X,B, µ) → L∞(X,A, µ) is called the conditional ex-

pectation with respect to A.

Lemma 2.5.3. Let (X,B, µ) be a probability space and let A ⊂ B be a σ-subalgebra, then the conditional expectation EA : L∞(X,B, µ) → L∞(X,A, µ)satisfies the following properties:


(1). EA preserves the integral, i.e.,∫EA(f) dµ =

∫f dµ, for all f ∈

L∞(X,B, µ).

(2). EA preserves positivity and for each f ∈ L∞(X,B, µ) we have |EA(f)| ≤EA(|f |).

(3). EA is L∞(X,A, µ)-bimodular, i.e., if f ∈ L∞(X,B, µ) and g ∈ L∞(X,A, µ)then EA(fg) = EA(f)g.

(4). EA extends continuously to a contraction on Lp(X,B, µ) for all 1 ≤p ≤ ∞.

Proof. (1) follows because if f ∈ L∞(X,B, µ) then we have

∫EA(f) dµ = 〈EA(f), 1〉L2(X,B,µ) = 〈f, EA(1)〉L2(X,B,µ) =

∫f dµ.

We have (2) because for all f ∈ L∞(X,B, µ) and g ∈ L2(X,A, µ) we have

〈EAM|f |EAg, g〉 = 〈M|f |EAg, EAg〉 ≥ 0,

and

〈((M|EA(f)|)2 − (EAM|f |EA)

2)g, g〉 = 〈(EAM|f |(1− EA)M|f |EA)g, g〉

= 〈(1 − EA)|f |g, |f |g〉 ≥ 0.

Hence EA(|f |) ≥ 0, and |EA(f)|2 ≥ EA(|f |)2.(3) follows because as noted above the product of two A-measurable func-

tions is again A-measurable.

Finally, we have (4) because if 1 ≤ p < ∞, then for every f ∈ Lp(X,B, µ)and g ∈ L∞(X,A, µ) we have

∫EA(f)g dµ =

∫EA(fg) dµ =

∫fg dµ,

hence if 1 < q ≤ ∞ such that 1p + 1

q = 1 then

‖EA(f)‖p = supg∈L∞(X,A,µ),‖g‖q≤1

|∫

EA(f)g dµ|

≤ supg∈L∞(X,B,µ),‖g‖q≤1

|∫

fg dµ| = ‖f‖p.

Note that because EA extends continuously to a contraction on Lp(X,B, µ)for any 1 ≤ p ≤ ∞ we will use the same notation EA for any of these extensions.


Exercise 2.5.4. If Γy(X,B, µ) is a measure preserving action, and A is Γ-invariant, then show that

σγ(EA(f)) = EA(σγ(f)),

for all γ ∈ Γ, and f ∈ Lp(X,B, µ), 1 ≤ p ≤ ∞.

Theorem 2.5.5 (Birkoff’s Ergodic Theorem [Bir31]). Let ZyT (X,B, µ) be ameasure preserving action of Z on a probability space (X,B, µ). Let I ⊂ Bdenote the σ-subalgebra of Γ-invariant sets. Then for each f ∈ L1(X,B, µ) wehave ‖ · ‖1, and almost everywhere convergence

1

NΣN−1

n=0 σn(f) → EI(f).

We will model the proof of this theorem by the proof of von Neumann’sErgodic Theorem (this is the approach taken in [PY98], that is to say we willprove the theorem for functions of the form g−σn(g) which are easier to handle,and we will also show that we can approximate an arbitrary function f withEI(f) = 0 by some linear combination of functions of the above type.

Lemma 2.5.6. Let ZyT (X,B, µ) be a measure preserving action of Z on aprobability space (X,B, µ). Let I ⊂ B denote the σ-subalgebra of Γ-invariantsets. Then for each ε > 0, and f ∈ L1(X,B, µ) with EI(f) = 0, there existsg ∈ L∞(X,B, µ) such that ‖f − (σ1(g)− g)‖1 < ε.

Proof. Let B be the ‖ ·‖1 closure of the subspace σ1(g)−g | g ∈ L∞(X,B, µ).If h ∈ L∞(X,B, µ) such that

∫(σ1(g)− g)h dµ = 0 for all g ∈ L∞(X,B, µ) then

we have ∫σ1(g)(h− σ1(h)) dµ =

∫(σ1(g)− g)h dµ = 0.

Thus σ1(h) = h and hence σn(h) = h for all n ∈ Z. We therefore have thath ∈ L∞(X, I, µ) and hence

∫fh dµ =

∫EI(fh) dµ =

∫EI(f)h dµ = 0.

Since L∞(X,B, µ) = (L1(X,B, µ))∗ by the Hahn-Banach Theorem this showsthat f ∈ B.

For any f ∈ L1(X,B, µ) and ε > 0, we will denote

Eε(f) = x ∈ X | lim supN→∞

1

N|ΣN−1

n=0 σn(f)(x)| ≥ ε.

Lemma 2.5.7. Using the notation above, for each f ∈ L1(X,B, µ) we haveµ(E2ε(f)) ≤ 1

ε

∫|f | dµ.


Proof. Suppose f ≥ 0, and for each M ≥ 1 consider

EMε (f) = x ∈ X | sup

1≤N≤M

1

NΣN−1

n=0 σn(f)(x) ≥ ε.

We will first show that if P > M then we have the inequality

ΣP−1n=0 σn(f)(x) ≥ εΣP−M−1

n=0 σn(1EMε (f))(x).

Indeed, if x ∈ X , then we may consider the first 0 ≤ n0 ≤ P −M such thatx ∈ T n0(EM

ε (f)), we then have that there exists some n0+1 ≤ N0 ≤ n0+M−1,such that ΣN0

n=n0σn(f)(x) ≥ ε(N0 − n0). Hence, we have that

ΣP−1n=0 σn(f)(x) ≥ σn0

(f) + ΣP−1n=N0

σn(f)(x)

= (εΣn0−1n=0 σn(1EM

ε (f))(x)) + σn0(f)(x) + ΣP−1

n=N0σn(f)(x)

≥ (εΣN0−1n=0 σn(1EM

ε (f))(x)) + ΣP−1n=N0

σn(f)(x).

We may now consider the first N0 ≤ n1 ≤ P −M such that x ∈ T n1(EMε (f))

and the corresponding n1+1 ≤ N1 ≤ n1+M−1, the same argument then gives

ΣP−1n=0 σn(f)(x) ≥ (εΣN1−1

n=0 σn(1EMε (f))(x)) + ΣP−1

n=N1σn(f).

Continuing in this manor we obtain the inequality

ΣP−1n=0 σn(f)(x) ≥ εΣP−M−1

n=0 σn(1EMε (f))(x),

for all x ∈ X , and P > M .Integrating this inequality we obtain

P

∫f dµ =

∫ΣP−1

n=0 σn(f)(x) dµ

≥ ε

∫ΣP−M−1

n=0 σn(1EMε (f))(x) dµ = ε(P −M)µ(EM

ε (f)).

Dividing by P and taking the limit as P → ∞ we have that∫

f dµ ≥ εµ(EMε (f)).

for all M ≥ 1.Since EM

ε (f) are increasing as M increases, and since Eε(f) ⊂ ∪∞M=1E

Mε (f),

we have that ∫f dµ ≥ εµ(Eε(f)).

For general f ∈ L1(X,B, µ) we may consider f = f+− f− where f+, f− ≥ 0,and f+f− = 0. We then have

∫|f | dµ ≥ ε(µ(Eε(f+)) + µ(Eε(f−))) ≥ εµ(E2ε(f)).


Proof of Birkoff’s Ergodic Theorem. Let f ∈ L1(X,B, µ) be given, and notethat by subtracting EI(f) we may assume that EI(f) = 0.

Let δ, ε > 0, and consider Eε(f) as defined above. By Lemma 2.5.6 thereexists g ∈ L∞(X,B, µ) such that ‖f− (σ1(g)−g)‖1 < δε/4 < δ. For each x ∈ Xwe have that

lim supN→∞

1

N|ΣN−1

n=0 σn(σ1(g)− g)(x)| ≤ lim supN→∞

1

N|(σN (g)− g)(x)|

≤ lim supN→∞

2‖g‖∞N

= 0.

Hence Eε/2(σ1(g)− g) = ∅.By Lemma 2.5.7 we therefore have that

µ(Eε(f)) ≤ µ(Eε/2(σ1(g)− g)) + µ(Eε/2(f − (σ1(g)− g)))

≤ 4

ε‖f − (σ1(g)− g)‖1 < δ,

and also

lim supN→∞

‖ 1

NΣN−1

n=1 σn(f)‖1

≤ lim supN→∞

‖ 1

NΣN−1

n=1 (σ1(g)− g)‖1 + ‖f − (σ1(g)− g)‖1 < δ.

Since δ and ε are arbitrary this shows that 1NΣN−1

n=1 σn(f) → 0 almost every-where and in ‖ · ‖1.

2.6 Recurrence theorems

Theorem 2.6.1 (Poincare’s Recurrence Theorem [Poi90]). Let ZyT (X,B, µ)be a measure preserving action on a probability space (X,B, µ). If A ⊂ X ismeasurable, such that µ(A) > 0 then for almost every point x ∈ A, the orbit Zxreturns to A infinitely often.

Proof. Let F ⊂ A be the set of points x such that T n(x) 6∈ A, for all n >0. Then F = A \ (∪n∈NT

n(A)) is measurable and if m > n then we haveTm(F ) ∩ T n(F ) = ∅. Indeed, if T−m(x) ∈ F ⊂ A then by the definition of Fwe have that T−n(x) = Tm−n(T−m(x)) 6∈ A. We therefore have that for anyN ∈ N

µ(F ) =1

NΣN

n=1µ(Tm(F )) =

1

Nµ(∪N

n=1Tm(F )) ≤ 1

N.

Hence µ(F ) = 0 and the theorem follows easily.

If ZyT (X,B, µ) is a measure preserving action on a probability space (X,B, µ),and A ⊂ X is measurable, such that µ(A) > 0, then for each x ∈ A we letnA(x) ∈ N ∪ ∞ be the smallest natural number such that T nA(x)(x) ∈ A. It

2.6. RECURRENCE THEOREMS 51

is easy to see that nA is measurable and by Poincare’s Recurrence Theorem, nA

is almost everywhere finite.For n ∈ N we let An ⊂ A be the set of points x ∈ A such that T n(x) ∈ A

but T k(x) 6∈ A for all 0 < k < n, so that nA = Σn∈Nn1An . If 0 ≤ k1 < n1 < ∞,and 0 ≤ k2 < n2 < ∞, then we have that T k1(An1

) ∩ T k2(An2) = ∅ unless

n1 = n2 and k1 = k2. Indeed, if k1 < k2 and x ∈ T k1(An1) ∩ T k2(An2

),then T−k2(x) ∈ An2

and T k2−k1(T−k2(x)) = T−k1(x) ∈ An1⊂ A, hence k2 ≥

k2 − k1 ≥ n2 > k2 which cannot happen. On the other hand, if k1 = k2, andn1 6= n2 then T k1(An1

) ∩ T k2(An2) = T k1(An1

∩ An2) = T k1(∅) = ∅.

There is a nice picture to go along with this disjoint decomposition known asthe Kakutani tower, however I am not TEX savvy enough to draw it, so insteadI will refer to page 45 in [Pet83].

Theorem 2.6.2 ([Kac47]). Let ZyT (X,B, µ) be a measure preserving actionon a probability space (X,B, µ). If A ⊂ X is measurable, such that µ(A) > 0,then ∫

A

nA dµ ≤ 1,

with equality when the action is ergodic.

Proof. Using the notation above, we have that

∫

A

nA dµ = Σn∈Nnµ(An)

= Σn∈NΣk = 0n−1µ(T k(An)) = µ(∪n∈N ∪n−1k=0 T k(An)) ≤ 1.

The set ∪n∈N ∪n−1k=0 T k(An) is a Z-invariant set which contains A, hence if

the action is ergodic then we have

∫

A

nA dµ = µ(∪n∈N ∪n−1k=0 T k(An)) = 1.

Theorem 2.6.3 ([Roh48]). Let ZyT (X,B, µ) be an ergodic measure preservingaction on a probability space (X,B, µ). Then for any N ∈ N, and ε > 0, thereexists A ⊂ X measurable, such that A, T (A), . . . , TN−1(A) are pairwise disjoint,and µ(∪N−1

j=0 T j(A)) > 1− ε, and µ(A∆TN (A)) < ε.

Proof. This theorem is an easy exercise if (X,B, µ) is not a diffuse subspace,hence we will assume that it is diffuse. Given B ⊂ X with µ(B) > 0 then we canconsider the Kakutani tower T k(Bn)n∈N,0≤k<n associated to B. If we thenconsiderA = ∪n≥N∪0≤k<n/NT kN (Bn) then we have that A, T (A), . . . , TN−1(A)are pairwise disjoint. Since the action is ergodic, we also have that

µ(∪N−1j=0 T j(A)) ≥ 1− µ(∪n∈N ∪maxn−N,0≤k<n T k(Bn))

≥ 1−NΣn∈Nµ(Bn) = 1−Nµ(B),


andµ(A∆TN (A)) ≤ 2Σn≥Nµ(Bn) ≤ 2µ(B).

Hence, if we choose B ⊂ X measurable, such that 0 < µ(B) < ε/N (which ispossible since (X,B, µ) is diffuse), then we obtain a set A which satisfies thedesired properties.

Theorem 2.6.4 ([Khi35]). Let Γy(X,µ) be a measure preserving action ofa countable amenable group Γ on a probability space (X,B, µ). Suppose f ∈L2(X,µ), f > 0. Then for each ε > 0, there exists F ⊂ Γ finite such that forall γ0 ∈ Γ we have that

γ0F ∩ γ ∈ Γ |∫

σγ(f)f dµ ≥ ‖f‖21 − ε 6= ∅.

Proof. Let Fn ⊂ Γ be a Følner sequence for Γ. By von Neumann’s ErgodicTheorem there exists n ∈ N, such that

‖ 1

|Fn|Σγ∈F−1

nσγ(f)− EI(f)‖2 < ε/(1 + ‖f‖2).

Therefore, we have that

|〈 1

|Fn|Σγ∈γ0F

−1n

σγ(f)− EI(f), f〉| = |〈 1

|Fn|Σγ∈F−1

nσγ(f)− EI(f), σγ−1

0(f)〉|

≤ ‖ 1

|Fn|Σγ∈F−1

nσγ(f)− EI(f)‖2‖f‖2 < ε.

Hence,

〈 1

|Fn|Σγ∈γ0F

−1n

σγ(f), f〉 ≥ 〈EI(f), f〉 − ε

= ‖EI(f)‖2 − ε ≥ |〈EI(f), 1〉|2 − ε = ‖f‖21 − ε.

Corollary 2.6.5. Let ZyT (X,B, µ) be a measure preserving action on a prob-ability space (X,B, µ), then for each A ⊂ X measurable, µ(A) > 0, and ε > 0,there exists K ∈ N such that for all j ∈ N we have

j, j + 1, . . . , j +K ∩ k ∈ N | µ(T k(E) ∩ E) ≥ µ(E)2 − ε 6= ∅.

Proof. This follows by applying Theorem 2.6.4 to the Følner sequence Fn =−1,−2, . . . ,−n for Z, and the function f = χE .

Chapter 3

Uniform multiple

recurrence

Furstenberg’s Recurrence Theorem [Fur77] states that if ZyT (X,B, µ) is a mea-sure preserving action on a probability space (X,B, µ) and A ⊂ X is a measur-able subset such that µ(A) > 0, then for each k ≥ 1 there exists n ≥ 1 suchthat

µ(A ∩ T n(A) ∩ · · · ∩ T kn(A)) > 0.

This was then generalized by Furstenberg and Katznelson [FK78] where it isshown that if k ∈ N, and ZyT (X,B, µ) is a measure preserving action on aprobability space (X,B, µ), then for each measurable set A ⊂ X , such thatµ(A) > 0, we actually have uniform multiple recurrence

lim infN→∞

1

NΣN−1

n=0 µ(Tn(A) ∩ T 2n(A) ∩ · · · ∩ T kn(A)) > 0.

A further generalization of these theorems which is conjectured by Bergelsonis the following.

Conjecture 3.0.6 ([Ber96]). Let Γ be a countable amenable group with a

Følner sequence Fn ⊂ Γ. Suppose k ∈ N and Γyαj

(X,B, µ) are measurepreserving actions of Γ on a probability space (X,B, µ), for 0 ≤ j < k. Assumemoreover that the actions pairwise commute, i.e., αi

γ αjλ = αj

λ αiγ , for all

i 6= j, and γ, λ ∈ Γ. Then for each measurable subset A ⊂ X with µ(A) > 0 wehave

lim infn→∞

1

|Fn|Σγ∈F−1

nµ(α0

γ(A) ∩ α0γα

1γ(A) ∩ · · · ∩ α0

γ · · ·αk−1γ (A)) > 0.

The purpose of the next two sections is to establish this conjecture in theextreme cases when either all the actions involved are compact or they are allweak mixing.

53

54 CHAPTER 3. UNIFORM MULTIPLE RECURRENCE

3.1 Multiple recurrence for compact actions

We will first introduce a bit of notation. If (X,B, µ) is a probability space,we will denote by Xk = Πk−1

j=0X , B⊗k = ⊗k−1j=0B, and µk = Πk−1

j=0µ. Givenf0, f1, . . . , fk−1 ∈ L∞(X,B, µ) we denote by f0 ⊗ f1 ⊗ · · · ⊗ fk−1 the functionon Xk given by

(f0 ⊗ · · · ⊗ fk−1)(x0, . . . , xk−1) = f0(x0)f1(x1) · · · fk−1(xk−1).

We also denote by L∞(X,B, µ)⊗algk ⊂ L∞(Xk,B⊗k, µk) the algebra generatedby all functions of the form f0 ⊗ · · · ⊗ fk−1. Note that this extends to a unitarybetween L2(X,B, µ)⊗k and L2(Xk,B⊗k, µk), hence we will identify these twospaces.

Theorem 3.1.1. Let Γ be a countable amenable group with a Følner sequenceFn ⊂ Γ. Suppose k ∈ N and Γyαj

(X,B, µ) are compact, measure preservingactions of Γ on a probability space (X,B, µ), for 0 ≤ j < k. Suppose f ∈L∞(X,B, µ), f 6= 0. Then we have

lim infn→∞

1

|Fn|Σγ∈F−1

n‖σ0

γ(f)σ1γ(f) · · ·σk−1

γ (f)‖22 > 0.

Proof. Suppose f ∈ L∞(X,B, µ) as above, note that we may assume that f ≥ 0,and ‖f‖2 = 1. Let f = f ⊗ f ⊗ · · · ⊗ f ∈ L∞(Xk,B⊗k, µk), and fix δ > 0to be chosen later. Since each action αj is compact, we have that αj(Γ) ⊂Aut(X,B, µ) is precompact in the weak topology. Thus the diagonal actionΓyα(Xk,B⊗k, µk) is also compact since α(Γ) ⊂ Aut(Xk,B⊗k, µk) is containedin a product of compact groups. Hence by Lemma 1.7.10 there exists a finiteset Eδ ⊂ Γ such that infγ0∈Eδ

‖σγ(f)− σγ0(f)‖2 < δ, for all γ ∈ Γ.

Therefore by the pigeon hole principle, for each finite set F ⊂ Γ there existsγ0 ∈ Eδ such that Φ(F, γ0) = γ ∈ F | ‖σγ(f)− σg0 (f)‖2 < δ satisfies

|Φ(F, γ0)| ≥ |F |/|Eδ|.

Note that if γ ∈ γ−10 Φ(F, γ0) then for each 0 ≤ i < k we have

2〈σiγ(f), f〉 ≥ 2Πk−1

j=0 〈σjγ(f), f〉

= 2− ‖σγ(f)− f‖22 ≥ 2− δ2,

and hence‖f − σi

γ(f)‖22 = 2− 2〈σγ(f), f〉 ≤ δ2.

We therefore have‖σ0

γ(f)σ1γ(f) · · ·σk−1

γ (f)‖2≥ ‖fk‖2 − Σk−1

j=0‖f‖k−1∞ ‖σj

γ(f)− f‖2≥ ‖fk‖2 − k‖f‖k−1

∞ δ.

3.2. MULTIPLE RECURRENCE FOR WEAK MIXING ACTIONS 55

Thus, if we set δ = ‖fk‖2/2k‖f‖k−1∞ then we have

lim infn→∞

1

|Fn|Σγ∈F−1

n‖σ0

γ(f) · · ·σk−1γ (f)‖22

≥ lim infn→∞

supγ0∈Eδ

|γ−10 Φ(F, γ0) ∩ Fn|

|Fn|‖fk‖22/4

= lim infn→∞

supγ0∈Eδ

|γ−10 Φ(F, γ0) ∩ γ−1

0 Fn||Fn|

‖fk‖22/4

≥ ‖fk‖22/4|Eδ| > 0.

Note that if αi is compact for all 0 ≤ i < k and the actions pairwise commute,then an easy exercise shows that the actions α0α1 · · ·αi are also compact, henceby considering f = 1A, the above theorem verifies Conjecture 3.0.6 in this case.

3.2 Multiple recurrence for weak mixing actions

Definition 3.2.1. Suppose Γy(X,B, µ) is a measure preserving action of acountable amenable group Γ on a probability space (X,B, µ). Let Fn ⊂ Γ bea Følner sequence, and suppose A ⊂ L∞(X,B, µ) is a Γ-invariant unital, self-adjoint subalgebra. A state ϕ on A is said to be generic with respect to Fn,and µ if

limn→∞

1

|Fn|Σγ∈F−1

nϕ(σγ(f)) =

∫f dµ,

for all f ∈ A.

The example to keep in mind is when Zy(T,Borel, µ) is the rotation actionof Z on the circle by an irrational (modulo 2π) angle. Where A is the algebraof continuous functions on T, and ϕ ∈ A∗ is given by evaluation at some pointz0 ∈ T. Then it follows from Birkoff’s Ergodic Theorem that for almost allchoices of z0, this state is generic. This justifies the terminology.

Lemma 3.2.2. Suppose Γy(X,B, µ) is an ergodic, measure preserving actionof a countable amenable group Γ on a probability space (X,B, µ). Let Fn ⊂ Γbe a Følner sequence, and suppose A ⊂ L∞(X,B, µ) is a Γ-invariant unital,self-adjoint subalgebra. If ϕ is a state on A which is generic with respect to Fn,and µ then

limn→∞

1

|Fn|Σγ∈F−1

nσγ(f) =

∫f dµ,

for all f ∈ A, where the convergence is in L2(A,ϕ).


Proof. Note, that when ϕ is given by ϕ(f) =∫f dµ then this is just a restate-

ment of von Neumann’s Ergodic Theorem.Let ε > 0 be given, and suppose that f ∈ A. Note that by subtracting the

integral we may assume that∫f dµ = 0. Since the action is ergodic it follows

from von Neumann’s Ergodic Theorem that there exists n0 ∈ N such that

‖ 1

|Fn0|Σγ0∈F−1

n0σγ0

(f)‖2L2(X,B,µ) < ε.

Using the triangle, followed by the Cauchy-Schwartz inequality ((| 1NΣN−1

n=0 an|)2 ≤( 1NΣN−1

n=0 |an|)2 ≤ 1NΣN−1

n=0 |an|2), and then using the fact that ϕ is generic wethen have

lim supn→∞

ϕ(| 1

|Fn|Σγ∈F−1

n

1

|Fn0|Σγ0∈F−1

n0

σγγ0(f)|2)

≤ lim supn→∞

1

|Fn|Σγ∈F−1

nϕ(σγ(|

1

|Fn0|Σγ0∈F−1

n0

σγ0(f)|2))

= ‖ 1

|Fn0|Σγ0∈F−1

n0σγ0

(f)‖2L2(X,B,µ) < ε.

Also, for each γ0 ∈ Fn0we have that

lim supn→∞

‖ 1

|Fn|Σγ∈F−1

n(σγ(f)− σγγ0

(g))‖∞ ≤ lim supn→∞

|Fn∆γ−10 Fn|

|Fn|= 0.

Hence, combining this with the above inequality we have

lim supn→∞

ϕ(| 1

|Fn|Σγ∈F−1

nσγ(f)|2) < ε.

Since ε > 0 was arbitrary this shows that

lim supn→∞

‖ 1

|Fn|Σγ∈F−1

nσγ(f)‖2L2(A,ϕ) = 0.

Exercise 3.2.3. Generalize the above lemma to arbitrary measure preservingactions. That is to say, suppose Γy(X,B, µ) is a measure preserving action ofa countable amenable group Γ on a probability space (X,B, µ). Let Fn ⊂ Γbe a Følner sequence, and suppose A ⊂ L∞(X,B, µ) is a Γ-invariant unital,self-adjoint subalgebra such that if L∞(X, I, µ) is the algebra of bounded Γ-invariant functions then L∞(X, I, µ) ∩ A is dense in L2(X, I, µ). Suppose ϕ isa state on A such that for all f ∈ A we have

limn→∞

1

|Fn|Σγ∈F−1

nϕ(σγ(f)) =

∫f dµ.

Then show that for all f ∈ A we have

limn→∞

1

|Fn|Σγ∈F−1

nσγ(f) = EI(f),

where the convergence is in L2(A,ϕ).

3.2. MULTIPLE RECURRENCE FOR WEAK MIXING ACTIONS 57

Theorem 3.2.4. Let Γ be a countable amenable group with Følner sequenceFn ⊂ Γ. Suppose k ∈ N and Γyαj

(X,B, µ) are weak mixing, measure preservingactions of Γ on a probability space (X,B, µ), for 0 ≤ j < k. Assume moreoverthat the actions pairwise commute, i.e., αi

γ αjλ = αj

λ αiγ , for all i 6= j,

and γ, λ ∈ Γ, and that the actions γ 7→ αiγα

i+1γ · · ·αj

γ are weak mixing for all0 ≤ i ≤ j < k. Suppose f0, . . . , fk−1 ∈ L∞(X,B, µ). Then we have

limn→∞

1

|Fn|Σγ∈F−1

nσ0γ(f0)(σ

0γσ

1γ(f1)) · · · (σ0

γ · · ·σk−1γ (fk−1)) = Πk−1

j=0 (

∫fj dµ),

where the convergence is in L2(X,B, µ).

Proof. Consider the action Γy(Xk,B⊗k, µk) given by γ(x0, x1, . . . , xk−1) =(α0

γx0, α0γα

1γx1, . . . , α

0γ · · ·αk−1

γ xk−1).

Denote by ν the diagonal measure onXk given by ν(A) = µ(x ∈ X | (x, x, . . . , x) ∈A). Then we obtain a well defined state ϕ on A = L∞(X,B, µ)⊗algk by

ϕ(f ) =

∫f dν,

for all f ∈ A.By Lemma 3.2.2, in order to prove the theorem it is enough to show that ϕ

is generic with respect to Fn, and µk. We will do by induction on k.Note that k = 1 is trivial. If this holds for k ≥ 1, and f0, f1, . . . , fk ∈

L∞(X,B, µ), then by Lemma 3.2.2 we have that

limn→∞

1

|Fn|Σγ∈F−1

nσ1γ(f1) · · · (σ1

γ · · ·σkγ (fk)) = Πk

j=1(

∫fj dµ),

in L2(X,B, µ).Multiplying this by f0 and integrating we obtain

limn→∞

1

|Fn|Σγ∈F−1

n

∫f0σ

1γ(f1) · · · (σ1

γ · · ·σkγ(fk)) dµ = Πk

j=0(

∫fj dµ).

Applying σ0γ then gives the result.

Corollary 3.2.5. Let ZyT (X,B, µ) be a weak mixing, measure preserving ac-tion on a probability space (X,B, µ). Then for each k ∈ N, and f0, f1, . . . , fk−1 ∈L∞(X,B, µ) we have

limN→∞

1

NΣN−1

n=0 σn(f0)σ2n(f1) · · ·σkn(fk−1) = Πk−1j=0 (

∫fj dµ),

where the convergence is in L2(X,B, µ).

Proof. If we consider the action αi : Z → Aut(X,B, µ) which takes the generatorof Z to T i, then it follows from Corollary 1.7.7 that these actions satisfy thehypotheses of the above theorem.


3.3 Relatively independent joinings and the ba-

sic construction

Having established Furstenberg’s multiple recurrence for the cases of compactand weak mixing actions of Z, it follows from Proposition ?? that every probabil-ity measure preserving action Zy(X,B, µ) has a non-trivial factor Zy(X,A, µ)for which Furstenberg’s multiple recurrence holds. To establish this principlein the general case we must show how to extend this result to larger factors.The strategy for doing this is to replace C with L∞(X,A, µ) and to mimic theprevious constructions and proofs.

3.3.1 Joinings

Let Γy(X,B, µ), and Γy(Y,A, ν) be two measure preserving action of a count-able group Γ on probability spaces (X,B, µ), and (Y,A, ν). For the moment letus denote by B = L∞(X,B, µ) and A = L∞(Y,A, ν). Suppose φ : B → A isa map which is positive (φ(f) ≥ 0, whenever f ≥ 0), unital (φ(1) = 1), pre-serves the integrals (

∫φ(f) dν =

∫f dµ, for all f ∈ B), and is Γ-equivariant

(φ(σγ(f)) = σγ(φ(f)), for all γ ∈ Γ, and f ∈ B).Then on the algebra B⊗algA we obtain a state τ such that for Σn

j=1bj⊗aj ∈B ⊗alg A we have

τ(Σnj=1bj ⊗ aj) =

∫Σn

j=1φ(bj)aj dµ.

Since φ is Γ-equivariant, it follows from Proposition 2.3.1 that there exists aprobability space (Z, C, η) with a measure preserving action of Γ, and thereexists a ∗-homomorphism π : B⊗alg A → L∞(Z, C, η) such that π σγ = σγ π,for all γ ∈ Γ, and

∫π(x) dη = τ(x) for all a ∈ B ⊗alg A.

Because φ is unital, and preserves the integral, it follows that π|B⊗1 andπ|1⊗A injective and integral preserving. Thus, L∞(Z, C, η) contains isomorphiccopies of L∞(X,B, µ) and L∞(Y,A, ν). Specifically, if we denote by CX ⊂ C(resp. CY ⊂ C) the σ-subalgebra generated by the image of π|B⊗1 (resp. π|1⊗A)then CX , and CY are Γ-invariant, and we have that π|B⊗1, and π|1⊗A give Γ-eqivariant isomrophism of L∞(X,B, µ) onto L∞(Z, CX , η), and L∞(Y,A, ν) ontoL∞(Z, CY , η).

Moreover, after this identification, we recover the map φ by the formulaφ(f) = ECY (f) ∈ L∞(Z, CY , η), for all f ∈ L∞(Z, CX , η). Indeed, this followssince if b ∈ B, and a ∈ A, then we have

∫φ(b)a dν = τ(b ⊗ a)

=

∫π(b)π(a) dη =

∫ECY (π(b))π(a) dη.

Given b ∈ B, and a ∈ A, we will often abuse notation and denote by b ⊗ athe function π(b ⊗ a) ∈ L∞(Z, C, η). One should be careful however with thisabuse of notation because π need not be faithful in general.

3.3. JOININGS AND THE BASIC CONSTRUCTION 59

Definition 3.3.1. Let Γy(X,B, µ), and Γy(Y,A, ν) be two measure preserv-ing action of a countable group Γ on probability spaces (X,B, µ), and (Y,A, ν).A joining of these two actions is a measure preserving action Γy(Z, C, η) to-gether with Γ-equivariant, integral preserving embeddings of L∞(X,B, µ) andL∞(Y,A, ν) in to L∞(Z, C, η), such that C is the σ-algebra generated by theseembeddings.

If Γy(Y,A, ν) is the same action as Γy(X,B, µ) then we say that a joiningis a self joining of the action Γy(X,B, µ).

From the discussion above, joinings are in 1-1 correspondence with Γ-equivariant,unital, integral preserving, positive maps φ : L∞(X,B, µ) → L∞(Y,A, ν).

If Γy(Z0, C, η) is a probability measure preserving action, and we have twoprobability measure preserving actions Γy(X,B, µ) and Γy(Y,A, ν), and Γ-equivariant embeddings α : L∞(Z0, C, η) → L∞(X,B, µ) and β : L∞(Z0, C, η) →L∞(Y,A, ν), then we obtain a Γ-equivariant, positive, unital, integral preserv-ing map from L∞(X,B, µ) to L∞(Y,A, ν) by first taking the conditional expec-tion from L∞(X,B, µ) to α(L∞(Z0, C, η)) and then applying the isomorphismβ α−1. The joining corresponding to this map is called the relatively inde-

pendent joining over (Z0, C, η). We denote the new space on which Γ acts byX ×α(C)=β(C) Y , or simply by X ×C Y if the embeddings α and β are clear from

the context. We also denote by L∞(X,B, µ)⊗algα(C)=β(C) L

∞(X,B, µ) the vector

space generated by functions of the type b⊗ a where b ∈ B, and a ∈ A.

A special case to consider is when Γy(X,B, µ) contains a Γ-invariant σ-subalgebra A, and we have (Z0, C, η) = (X,A, µ), then we may consider therelatively independent self joining over (X,A, µ). Note, however that a rel-atively independent joinings consists not only of invariant σ-subalgebras, butalso the ways in which we are including these subalgebras into the larger alge-bras. For instance, if α ∈ Aut(L∞(X,A, µ)) is a Γ-equivariant automorphism,then we obtain a new relatively independent joining by considering the alter-nate embedding α : L∞(X,A, µ) → L∞(X,A, µ) ⊂ L∞(X,B, µ). In general,the actions ΓyX ×A X and ΓyX ×A=α(A) X need not be isomorphic.

3.3.2 The basic construction

Relatively independent joinings over the trivial σ-subalgebra corresponds totaking a diagonal action on a product space. We have seen previously that auseful tool in analyzing the structure of product actions ΓyX × X was theidentification between L2(X×X) and the Hilbert-Schmidt operators on L2(X).This allowed us to use tools such as functional calculus. There is an analog ofthe Hilbert-Schmidt operators in the setting of relatively independent joiningswhich we will now describe.

Suppose (X,B, µ) is a probability space and A ⊂ B is a σ-subalgebra. Thebasic construction associated to the inclusion L∞(X,A, µ) ⊂ L∞(X,B, µ)is the algebra L∞(X,A, µ)′ ⊂ B(L2(X,B, µ)), of operators in B(L2(X,B, µ))which commute with L∞(X,A, µ). We will denote the basic construction by


〈L∞(X,B, µ), eA〉, where eA ∈ B(L2(X,B, µ)) is the orthogonal projection ontoL2(X,A, µ).

Note that we have

L∞(X,B, µ)eAL∞(X,B, µ) = spfeAg | f, g ∈ L∞(X,B, µ) ⊂ 〈L∞(X,B, µ), eA〉,

and that this is an algebra since if f ∈ L∞(X,B, µ) we have

eAfeA = EA(f)eA.

Note that we distinguish here the the projection eA, which is an operator onL2(X,A, µ), from the conditional expectation EA, which is an operator onL∞(X,A, µ).

Exercise 3.3.2. Suppose S ∈ 〈L∞(X,B, µ), eA〉, has polar decomposition S =V |S|. Show that V ∈ 〈L∞(X,B, µ), eA〉.Lemma 3.3.3. Suppose (X,B, µ) is a probability space and A ⊂ B is a σ-subalgebra. For each S ∈ 〈L∞(X,B, µ), eA〉 there exists a unique φ(S) ∈L∞(X,A, µ) such that

eASeA = φ(S)eA.

Moreover, the map S 7→ φ(S) is a unital, positivity preserving extension ofEA, which is L∞(X,A, µ)-bimodular, and continuous with respect to the weakoperator topology.

Proof. By Lemma 2.5.2 L∞(X,A, µ) is a maximal abelian subalgebra of B(L2(X,A, µ)).Thus, since eASeA restricted to L2(X,A, µ) commutes L∞(X,A, µ), there existsa unique element φ(S) ∈ L∞(X,A, µ) such that eASeAf = φ(S)f = φ(S)eAf ,for all f ∈ L2(X,A, µ). If f ∈ L2(X,A, µ)⊥ ⊂ L2(X,B, µ) then we haveeASeAf = 0 = φ(S)eA.

That x 7→ φ(x) is unital, positivity preserving, and weak operator topologycontinuous, follows from the fact that it is the composition of the map x 7→eAxeA and the ∗-homomorphism aeA 7→ a.

Exercise 3.3.4 (Generalized Cauchy-Schwartz inequality). Prove that for allx, y ∈ 〈L∞(X,B, µ), eA〉 we have

|φA(y∗x)|2 ≤ φA(y

∗y)φA(x∗x).

In the case where the σ-algebra A is trivial we have that eA is the rank 1projection on to the subspace C1 ⊂ L2(X,B, µ), and thus operators of the formfeAg were rank 1 projections. Rather than working with a Hilbert space basisξi ⊂ L2(X,B, µ) as before, we could have just as easily worked with the familyof partial isometries from C1 to Cξi. This motivates the following lemma.

Lemma 3.3.5. Suppose (X,B, µ) is a probability space and A ⊂ B is a σ-subalgebra. There exists a family of partial isometries vii∈I ⊂ 〈L∞(X,B, µ), eA〉such that

(a). viv∗i ≤ eA, for all i ∈ I;


(b). viv∗j = 0, for all i, j ∈ I, i 6= j;

(c). Σi∈Iv∗i eAvi = 1.

Proof. A simple argument with Zorn’s Lemma shows that there is a maximal(with respect to inclusion) family of partial isometries vii∈I satisfying condi-tions (a) and (b) above.

Let P = Σi∈Iv∗i eAvi, and consider S ∈ (1 − P ) · 〈L∞(X,B, µ), eA〉 · eA. By

considering the polar decomposition S = V |S|, we have that V ∗V = ProjRange(S) ≤eA, V V ∗ = ProjRange(S∗) ≤ (1−P ), and V ∈ 〈L∞(X,B, µ), eA〉. Thus by max-

imality of the family vii∈I we must have that V = 0, and hence S = 0.Thus 0 = (1−P )·〈L∞(X,B, µ), eA〉·eA·L2(X,B, µ) ⊃ (1−P )·L∞(X,B, µ)·

1. Since L∞(X,B, µ) · 1 is dense in L2(X,B, µ) this shows that P = 1.

A family of partial isometries which satisfy the conditions above will becalled an operator basis for 〈L∞(X,B, µ), eA〉. Note that since viv

∗i ≤ eA we

have that viv∗i = φA(viv

∗i ) ∈ L∞(X,A, µ) for each i ∈ I.

Definition 3.3.6. Suppose (X,B, µ) is a probability space and A ⊂ B is a σ-subalgebra. Let vii∈I be an operator basis for 〈L∞(X,B, µ), eA〉. An operatorS ∈ 〈L∞(X,B, µ), eA〉 is Hilbert-Schmidt class with Hilbert-Schmidt normif

‖S‖2HS = Σi∈I

∫φA(viS

∗Sv∗i ) dµ < ∞.

The quantity ‖S‖HS is the Hilbert-Schmidt norm of S.

This definition does not depend on the operator basis vii∈I , this can beseen from the following analogue of Perseval’s identity. If wjj∈J is anotheroperator basis, then we have

Σi∈I

∫φA(viS

∗Sv∗i ) dµ = Σi∈IΣj∈J

∫φA(viS

∗(w∗j eAwj)Sv

∗i ) dµ

= Σi∈IΣj∈J

∫φA(viS

∗w∗j )φA(wjSv

∗i ) dµ

= Σj∈JΣi∈I

∫φA(vjSvi)φA(viS

∗w∗j ) dµ

= Σj∈J

∫φA(wjS

∗Sw∗j ).

Exercise 3.3.7. Show that φA is a contraction from the Hilbert-Schmidt normto L2(X,A, µ).

Note that it follows from above that we may approximate S in the Hilbert-Schmidt norm with finite sums of the form Σi,jw

∗j (wjSv

∗i )vi. Also, if ε > 0, and

we consider gj, hi ∈ L∞(X,B, µ) such that ‖gj−w∗j (1)‖2, ‖viv∗i −hiv

∗i (1)‖2 < ε,

then we have

= ‖φA((wjSv∗i )− (wjgjφA(wjSv

∗i )eAhiv

∗i ))‖2


= ‖φA(wjSv∗i )− φA(wjgj)φA(wjSv

∗i )φA(hiv

∗i )‖2

≤ ‖φA(wjSv∗i )‖∞(‖wjw

∗j − φA(wjgj)‖2 + ‖viv∗i − φA(hiv

∗i )‖2)

≤ 2‖S‖ε.It also follows that

Σ(k,l)∈I×J,(k,l) 6=(i,j)‖φA(wlgj)‖22‖φA(hiv∗k)‖22 < 2ε2 + ε4,

and hence

‖w∗j (wjSv

∗i )vi − gjφA(wjSv

∗i )eAhi‖2HS ≤ 2(‖S‖+ 1)ε2 + ε4.

Since ε > 0 was arbitrary we may then use the triangle inequality to concludethat the ∗-algebra L∞(X,B, µ)eAL∞(X,B, µ) is dense in the Hilbert-Schmidtnorm.

If T ∈ 〈L∞(X,B, µ), eA〉 then T ∗T ≤ ‖T ‖2, and since φA is positivity pre-serving it follows that

‖TS‖2HS = Σi∈I

∫φA(viS

∗T ∗TSv∗i ) dµ ≤ ‖T ‖2‖S‖2HS.

Also, it follows from the argument above that the adjoint operator S 7→ S∗ isan anti-linear isometry, and hence we also have

‖ST ‖2HS ≤ ‖T ‖2‖S‖2HS.

In particular, we see that the Hilbert-Schmidt class in 〈L∞(X,B, µ), eA〉 is atwo sided ideal.

Exercise 3.3.8. Given a Hilbert-Schmidt class operator S ∈ 〈L∞(X,B, µ), eA〉show that ‖S‖HS = 0 if and only if S = 0.

The Hilbert-Schmidt norm has an associated inner product

〈S, T 〉HS = Σi∈I

∫φA(viT

∗Sv∗i ) dµ,

which is well defined by the generalized Cauchy-Schwartz inequality, and doesnot depend on the operator basis from the arguments above.

Thus, the class of Hilbert-Schmidt operators is an inner-product space. Thisspace is not complete in general1, we denote the Hilbert space completion byL2〈L∞(X,B, µ), eA〉.

Even though the class of Hilbert-Schmidt operators is not a complete spacein general we do have that it is complete when we restrict to the Hilbert-Schmidtoperators whose uniform norm is bounded by some fixed constant.

1Consider the case when A = B, then it is easy to see that the class of Hilbert-Schmidtoperators coincides with L∞(X,B, µ), and the inner-product structure is the usual inner-product on L2(X,B, µ).


Proposition 3.3.9. Suppose (X,B, µ) is a probability space and A ⊂ B is aσ-subalgebra. Suppose K > 0 and consider the convex set

BK = S ∈ 〈L∞(X,B, µ), eA〉 | ‖S‖ ≤ K.

Then BK is complete in the Hilbert-Schmidt norm.

Proof. Fix an operator basis vii∈I for 〈L∞(X,B, µ), eA〉. Since φA is a con-traction from the Hilbert-Schmidt norm to L2(X,A, µ), if Sn ∈ BK is Cauchyin the Hilbert-Schmidt norm, then for all i, j ∈ I, φ(vjSnv

∗i ) ∈ L∞(X,A, µ) is

Cauchy in L2(X,A, µ) and we also have that ‖φ(vjSnv∗i )‖∞ ≤ K. Hence, there

exists gi,j ∈ L∞(X,A, µ) such that ‖gi,j‖∞ ≤ K, and

limn→∞

‖v∗j (vjSnv∗i )vi − v∗j gi,jvi‖HS = lim

n→∞‖v∗jφ(vjSnv

∗i )eAvi − v∗j gi,jvi‖HS

≤ limn→∞

‖φ(vjSnv∗i )eA − gi,jeA‖HS = 0.

Since Range(v∗j ) are pairwise orthogonal subspaces we may then considerthe sum

S = Σi,j∈Iv∗j gi,jvi ∈ B(L2(X,B, µ)).

Then ‖S‖ ≤ K and it is easy to see that S ∈ 〈L∞(X,B, µ), eA〉. More over forany finite set I0 ⊂ I it follows from the triangle inequality that

limn→∞

‖Σi,j∈I0v∗j (vjSnv

∗i )vi − v∗j (vjSv

∗i )vi‖HS = 0.

Since Sn is Cauchy in the Hilbert-Schmidt norm this implies that S is Hilbert-Schmidt class and ‖Sn − S‖HS → 0.

If Γy(X,B, µ) is a measure preserving action of a countable group Γ, suchthat A is Γ-invariant, then we may consider the relatively independent self join-ing ΓyX ×A X defined above. We may then define a map Ξ : L∞(X,B, µ)⊗A

L∞(X,B, µ) → L∞(X,B, µ)eAL∞(X,B, µ) by linearly extending the formula

Ξ(b ⊗ a) = beAa,

for all a, b ∈ L∞(X,B, µ).If vii∈I is an operator basis for 〈L∞(X,B, µ), eA〉, we then have that for

all Σnk=1bk ⊗ ak ∈ L∞(X,B, µ)⊗A L∞(X,B, µ)

‖Σnk=1bk ⊗ ak‖2L2(X×AX) =

∫Σn

k,l=1EA(b∗kbl)ala

∗k dµ

=

∫Σn

k,l=1φA(b∗kbl)φA(ala

∗k) dµ

= Σi,j∈I

∫Σn

k,l=1φA(b∗kv

∗i eAvibl)φA(alv

∗j eAvja

∗k) dµ

= Σi,j∈I

∫Σn

k,l=1φA(b∗kv

∗i )φA(vibl)φA(alv

∗j )φA(vja

∗k) dµ


= Σi,j∈I

∫Σn

k,l=1φA(vibleAalv∗j )φA(vja

∗keAb

∗kv

∗i ) dµ

= ‖Ξ(Σnk=1bk ⊗ ak)‖2HS.

Hence Ξ is well defined and extends to a unitary operator (which we will alsodenote by Ξ) from L2(X ×A X) to L2〈L∞(X,B, µ), eA〉.

Moreover, this unitary implements an equivalence between the Koopmanrepresentation of ΓyX×AX and the representation of Γ on L2〈L∞(X,B, µ), eA〉,given by S 7→ σγSσγ−1 .

Suppose Γy(X,B, µ), and Γy(Y,A, ν) are two probability measure pre-serving actions and we have Γ-equivariant embeddings α : L∞(Z0, C, η) →L∞(X,B, µ) and β : L∞(Z0, C, η) → L∞(Y,A, ν). We may consider the class ofoperators S ∈ B(L2(X,B, µ), L2(Y,A, µ)) such that Sα(g) = β(g)S and |S∗S|1/2is in the Hilbert-Schmidt class of 〈L∞(X,B, µ), eα(C)〉. In this case we can con-

sider the norm given by ‖S‖HS = ‖|S∗S|1/2‖HS, and consider the completionunder this norm.

It then follows that this is a Hilbert space, and we may consider the mapΞ which linearly extends the formula Ξ(b ⊗ a) = beβ(C)=α(C)a (here we vieweβ(C)=α(C) as an operator from L2(X,B, µ) to L2(Y,A, ν)). Then just as aboveΞ extends to a unitary operator which implements an isomorphism between theKoopman representation ΓyY ×β(C)=α(C) X and the representation given byS 7→ σA

γ SσBγ−1 .

Exercise 3.3.10. Fill in the details to the previous paragraphs.

3.4 Ergodic, weak mixing, and compact exten-

sions

Definition 3.4.1. Let Γ be a countable group. An extension Γy(X,B, µ) of aprobability measure preserving action Γy(X,A, µ) is an ergodic extension ifthe σ-algebra I of Γ-invariant sets is contained in A.

Definition 3.4.2. Let Γ be a countable group. An extension Γy(X,B, µ)of a probability measure preserving action Γy(X,A, µ) is a weak mixing

extension if for any ε > 0 and any finite set F ⊂ L2(X,B, µ) such that EA(f) =0 for all f ∈ F , there exists γ ∈ Γ such that for all f, g ∈ F we have

‖EA(σγ(f)g)‖1 < ε.

Definition 3.4.3. Let Γ be a countable group and let Γy(X,B, µ) be an ex-tension of a probability measure preserving action Γy(X,A, µ). A functionf ∈ L2(X,B, µ) is almost periodic relative to L∞(X,A, µ) if for all ε > 0there exist g1, . . . , gn ∈ L∞(X,B, µ) such that for all γ ∈ Γ we have

σγ(f) ∈ε L2(X,A, µ)g1 + · · ·+ L2(X,A, µ)gn.

3.4. ERGODIC, WEAK MIXING, AND COMPACT EXTENSIONS 65

That is to say, for each γ ∈ Γ there exist kγ1 , . . . , kγn ∈ L2(X,A, µ) such that

‖σγ(f)− Σnj=1k

γj gj‖2 < ε.

The proof of the following proposition is a straight forward generalization ofthe proof of Proposition 2.2.22 and we leave the details to the reader.

Proposition 3.4.4. Let Γ be a countable group and let Γy(X,B, µ) be anextension of a probability measure preserving action Γy(X,A, µ). Let C ⊂ Bbe the σ-algebra generated by all functions which are almost periodic relative toL∞(X,A, µ). Then A ⊂ C, C is Γ-invariant, and f ∈ L2(X,B, µ) is almostperiodic relative to L∞(X,A, µ) if and only if f is C measurable.

Definition 3.4.5. Let Γ be a countable group. An extension Γy(X,B, µ) of aprobability measure preserving action Γy(X,A, µ) is a compact extension ifevery f ∈ L2(X,B, µ) is almost periodic relative to L∞(X,A, µ).

Note that Proposition 3.4.4 shows that there is a unique maximal compactextension of Γy(X,A, µ) which is a factor of Γy(X,B, µ).

We can now generalize the relationship between ergodic, weak mixing, andcompact actions that we have seen before.

Theorem 3.4.6. Let Γ be a countable group and let Γy(X,B, µ) be an ex-tension of a probability measure preserving action Γy(X,A, µ). The followingconditions are equivalent:

(1). The extension is weak mixing.

(2). There are no non-trivial functions f ∈ L2(X,B, µ) which are almostperiodic relative to L∞(X,A, µ).

(3). The relatively independent self joining ΓyX ×A X is an ergodic exten-sion of Γy(X,A, µ).

(4). For any ergodic extension Γy(Y, C, ν) of Γy(X,A, ν), the relativelyindependent self joining ΓyX ×A Y is an ergodic extension of Γy(X,A, ν).

(5). The relatively independent self joining ΓyX ×A X is a weak mixingextension of Γy(X,A, µ).

(6). For any weak mixing extension Γy(Y, C, ν) of Γy(X,A, ν), the rel-atively independent self joining ΓyX ×A Y is a weak mixing extension ofΓy(X,A, ν).

Proof. For (1) =⇒ (2) suppose that Γy(X,B, µ) is a weak mixing extensionand f ∈ L∞(X,B, µ) is almost periodic relative to L∞(X,A, µ), such thatEA(f) = 0. Consider ε > 0, then there exists g1, . . . , gn ∈ L∞(X,B, µ) suchthat for every γ ∈ Γ, there are kγ1 , . . . , k

γn ∈ L2(X,A, µ) such that

‖σγ(f)− Σnj=1k

γj gj‖2 < ε. (3.1)


Note that since supγ∈Γ ‖σγ(f)‖∞ = ‖f‖∞ we may assume that

K = sup1≤j≤n, γ∈Γ

‖kγj ‖∞ < ∞.

Also, by replacing gj with gj − EA(gj) we may assume that EA(gj) = 0 for1 ≤ j ≤ n.

Since Γy(X,B, µ) is a weak mixing extension there exists γ ∈ Γ such thatfor all 1 ≤ j ≤ n we have

‖EA(σγ(f)gj)‖1 < ε.

Hence, for all 1 ≤ j ≤ n we also have

|〈σγ(f), kγj gj〉| ≤ ‖EA(σγ(f)gjk

γj )‖1 = ‖EA(σγ(f)gj)k

γj ‖1 ≤ Kε. (3.2)

Combining (3.1) and (3.2) we then have

‖f‖22 ≤ ‖σγ(f)‖22 + ‖Σnj=1k

γj gj‖22

= ‖σγ(f)− Σnj=1k

γj gj‖22 − 2ℜ〈σγ(f),Σ

nj=1k

γj gj〉 ≤ ε2 + 2Knε.

Since ε > 0 was arbitrary this shows that f = 0.For (2) =⇒ (3) suppose that the relatively independent self joining ΓyX×A

X is not an ergodic extension. Hence, there exists a non-trivial Γ-invariantfunction f ∈ L2(X ×A X) such that EA(f) = 0. Using the identification fromSection ?? we then produce a non-trivial element S ∈ L2〈L∞(X,B, µ), eA〉 suchthat φA(S) = 0, and Ad(σγ)(S) = S for all γ ∈ Γ. Note that we may assume‖S‖HS = 1, and hence by considering an operator S0 ∈ 〈L∞(X,B, µ), eA〉 suchthat ‖S0‖HS = 1, and ‖S0 − S‖HS < 1/4, we have that ‖Ad(σγ)(S0)− S0‖HS <1/2, for all γ ∈ Γ.

If we consider then the closed (in the Hilbert-Schmidt norm) convex hullof Ad(σγ)(S0) | γ ∈ Γ, then the element T of minimal norm is Ad(σγ)-invariant, satisfies T 6∈ L∞(X,A, µ), and is bounded in the uniform norm byProposition 3.3.9. Consider the unit ball B1 = f ∈ L2(X,B, µ) | ‖f‖2 ≤ 1,then T (B1) ⊂ L2(X,B, µ) is Γ-invariant.

If ε > 0 is given and we consider an operator basis vii∈I for 〈L∞(X,B, µ), eA〉then we may find a finite set I0 ⊂ I such that

‖T − Σi,j∈I0v∗jφA(vjTv

∗i )vi‖HS < ε/2‖T ‖.

It is then not hard to see that if we consider fj ∈ L∞(X,B, µ) such that ‖fj −v∗j (1)‖2 < ε/2|I0|, then for all f ∈ T (B1) we have

f ∈ε Σj∈I0L2(X,A, µ)fj .

This shows that every function in T (B1) is almost periodic relative to L∞(X,A, µ).The same argument also shows the same for every function in T ∗(B1). Since

3.4. ERGODIC, WEAK MIXING, AND COMPACT EXTENSIONS 67

T 6= eATeA this shows that there exists a function which is not in L2(X,A, µ)which is almost periodic relative to L∞(X,A, µ).

For (3) =⇒ (1) if the action is not a weak mixing extension, then thereexists c > 0 and a finite set F ⊂ L2(X,B, µ) such that EA(f) = 0 for all f ∈ F ,and for all γ ∈ Γ we have

Σf,g∈F‖EA(σγ(f)g)‖22 ≥ Σf,g∈F‖EA(σγ(f)g)‖21 ≥ c.

If we consider the operator T = Σf∈FfeAf ∈ 〈L∞(X,B, µ), eA〉 then we havethat T has finite Hilbert-Schmidt norm, and for each γ ∈ Γ we have

〈Ad(σγ)(T),T〉HS

= Σf,g∈F〈σγ(f)eAσγ(f), geAg〉HS

= Σf,g∈F‖EA(σγ(f)g)‖22 ≥ c

Therefore, by Proposition 1.5.2 there exists a non-trivial Ad(σγ)-invariant vectorin L2〈L∞(X,B, µ), eA〉. From Section ?? it then follows that ΓyX×AX is notan ergodic extension.

The proof that (3) ⇔ (4) then follows from the remarks at the end of Sec-tion ??. If ΓyX ×A Y is not an ergodic extension then just as above weconstruct an operator T ∈ B(L2(Y, C, ν), L2(X,B, µ)) which commutes with theembeddings of A measurable functions, intertwines the Koopman representa-tions of Γ, and such that T 6= eATeA. by considering the operators (T ∗T )1/2

and (TT ∗)1/2, the result follows easily.The equivalence of (1)-(4) shows that if ΓyX ×A X is an ergodic extension

then Γy(X ×A X)×A (X ×A X) is also an ergodic extension. This then showsthat (1)-(5) are equivalent, and a similar argument works for (6) as well.

Corollary 3.4.7. Let Γ be a countably infinite amenable group with a Følnersequence Fn ⊂ Γ, and let Γy(X,B, µ) be an extension of a probability measurepreserving action Γy(X,A, µ). Then Γy(X,B, µ) is a weak mixing extensionif and only if for all f, g ∈ L2(X,B, µ) such that EA(f) = EA(g) = 0 we havethat

limn→∞

1

|Fn|Σγ∈F−1

n

∫|EA(σγ(f)g)|2 dµ = 0.

Proof. It is clear that this condition implies that Γy(X,B, µ) is a weak mixingextension and so we need only prove the converse.

Assume that Γy(X,B, µ) is a weak mixing extension then by Theorem 3.4.6the relatively independent self joining ΓyX ×A X is an ergodic extension, andhence if f, g ∈ L∞(X,B, µ) are such that EA(f) = EA(g) = 0, then we havethat ‖EI(f ⊗ g)‖2 ≤ ‖EA(f ⊗ g)‖2 = 0, where I is the σ-algebra of Γ-invariantsets for X ×A X .

By von Neumann’s Ergodic Theorem we then have

limn→∞

1

|Fn|Σγ∈F−1

n

∫|EA(σγ(f)g)|2 dµ


= limn→∞

1

|Fn|Σγ∈F−1

n〈(σγ ⊗A σγ)(f ⊗ f), g ⊗ g〉

= 〈EI(f ⊗ f), g ⊗ g〉 = 0.

We then obtain the general case when f, g ∈ L2(X,B, µ) by continuity.

Index

actioncompact, 39ergodic, 16, 36mixing, 39strongly ergodic, 37weak mixing, 37

adjoint Hilbert space, 5almost invariant vectors, 16almost periodic function, 40amenable group, 18asymptotically invariant sequence, 37

basic construction, 59Bernoulli shift, 32Birkoff’s Ergodic Theorem, 48

cocycle, 9composition of cocycles, 15conditional expectation, 46conjugacy, 32cyclic vector, 6

ergodic extension, 64

Følner net, 18factor, 40functions of positive type, 8fundamental domain, 11Furstenberg’s Correspondence Princi-

ple, 34

Gaussian action, 43generalized Bernoulli shift, 32generic measure, 55

Hilbert-Schmidt class, 7, 61

invariant measure, 31

invariant vectors, 16

joining, 59

Kac’s Theorem, 51Kakutani tower, 51Khintchine’s Theorem, 52Koopman representation, 35

lattice, 33

mean, 18

odometer action, 32

Poincare’s Recurrence Theorem, 50

quasi-invariant measure, 31

randomorphism, 34relatively independent joining, 59representation, 5

adjoint, 5amenable, 29compact, 24induced, 12, 13irreducible, 6left-regular, 5mixing, 22, 23quasi-regular, 5tensor product, 6trivial, 5weak mixing, 22, 26, 27

Rohlin’s Theorem, 51

spectral gap, 16, 37standard probability space, 41symmetric creation operator, 43

69

70 INDEX

uniform ordering, 33

Von Neumann’s Ergodic Theorem, 21,45

weak containment, 28weak mixing, 64weak topology, 39

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