CHAPTER 0
General framework and notational conventions
0.1. Groups
Throughout this section and the whole book G is a countable discrete group. Its
identity element will always be denoted by e.
0.2. Probability spaces
A measurable space is a pair (X, S) where X is a set and S is a σ-algebra of subsets
of X. Members of S are called measurable sets. Given measurable spaces (X, S) and
(Y, R), a map ϕ : X → Y is measurable if ϕ−1 (B) ∈ S for all B ∈ R. We say that the
measurable spaces (X, S) and (Y, R) are isomorphic if there is a bijection ϕ : X → Y
such that ϕ and ϕ−1 are both measurable, and such a map is called an isomorphism (of
measurable spaces). If (X, S) is a measurable space and A is a measurable subset of X,
then A itself is a measurable space when paired with the σ-algebra of all intersections of
members of S with A.
A measure on a measurable space (X, S) is a map µ : S → [0, ∞] such that
(i) µ(∅) = 0, and
S
P∞
∞
(ii) µ( ∞
A
)
=
n
n=1
n=1 µ(An ) for every countable collection {An }n=1 of pairwise disjoint sets in S (countable additivity).
Usually we speak of a measure on X, with S not being explicitly mentioned. The measure
µ is finite if µ(X) is finite, and is called a probability measure if µ(X) = 1. It is atomless
if µ({x}) = 0 for every x ∈ X. A measurable set A is null if µ(A) = 0 and conull if
µ(X\A) = 0.
By a measure space we mean a measurable space (X, S) along with a measure µ
on (X, S). A measurable subset of a measure space is itself a measure space under the
restriction of the measure. We will usually write a measure space as a pair (X, µ) and talk
about measurable sets without explicitly naming the σ-algebra. When µ is a probability
measure we refer to (X, µ) as a probability space.
If (X, µ) and (Y, ν) are measure spaces then a map ϕ : X → Y is a measure isomorphism if it is an isomorphism of measurable spaces which satisfies ν(A) = µ(ϕ−1 (A))
for all measurable A ⊆ Y . A measure-preserving transformation of a probability space
(X, µ) is an automorphism T : X → X of measurable spaces which satisfies µ(A) =
27
28
0. GENERAL FRAMEWORK AND NOTATIONAL CONVENTIONS
µ(T −1 A) for all measurable A ⊆ Y (in other words, a measure isomorphism from X to
itself). Usually this terminology is used without the assumption of invertibility, so that
T is merely measurable and measure-preserving, but in the spirit of our main interest in
group actions we make invertibility part of the definition, although some facts we discuss
about measure-preserving transformations remain valid in the noninvertible case.
0.3. Measure algebras
In ergodic theory we are mainly interested in phenomena which are most naturally
expressed at the level of the measure algebra and its automorphisms. Weak mixing, compactness, and entropy all fall into this category, while the pointwise ergodic theorem could
be regarded as an exception, although it too admits a measure algebra formulation. It is
rarely imperative however to work with the measure algebra (although analysis in spaces
like L2 and L∞ , which are effectively completions of the linearization of the measure
algebra, is often vital), and we will adhere to the language of points and sets, as is customary. This is also important when trying to leverage any extra topological or geometric
structure that the dynamics might have, for example via the Borel structure which actually
grounds our definition of probability-measure-preserving action in Definition 0.2. Nevertheless, the measure algebra picture is helpful for motivating and understanding, among
other things, the notion of standardness for probability spaces, as discussed further below.
A Boolean algebra is a set containing distinguished elements 0 and 1 with operations
∨, ∧, and ¬ which satisfy the same axioms as union, intersection, and complementation
do for subsets of a fixed set, with 0 and 1 playing the roles of the empty set and the whole
set, respectively. A measure algebra is a pair (M, µ) where M is a Boolean algebra and
µ is a map M → [0, ∞] such that
(i) µ(A) = 0 if and only if A = 0, and
W
P∞
∞
(ii) µ( ∞
n=1 An ) =
n=1 µ(An ) for every countable collection {An }n=1 of elements in M which are pairwise disjoint in the sense that Ai ∧ Aj = 0 for i 6= j
(countable additivity).
Let (X, µ) be a probability space and S its σ-algebra. The collection of sets A ∈ S
satisfying µ(A) = 0 (the null sets) forms a σ-ideal N in S, meaning that it is closed
under taking countable unions and taking subsets within S. We define an equivalence
relation on S by declaring that A ∼ B if A∆B ∈ N. Then the quotient S/N by this
relation is a measure algebra when equipped with the map defined on equivalence classes
by [A] 7→ µ(A). We call this the measure algebra of µ.
We often speak of two measurable sets being equal modulo a null set, by which we
mean that their equivalence classes in the measure algebra are equal.
0.4. STANDARD PROBABILITY SPACES
29
0.4. Standard probability spaces
We will typically restrict our scope to probability spaces (X, µ) which are standard in
the following sense, which is natural not only because it enables us to leverage tools from
descriptive set theory at the level of the space and translates as separability at the level of
the measure algebra (which is equivalent to the separability of L2 (X, µ)), but also because
of the way it connects these two levels in the presence of dynamics through Theorem 0.5.
This standardness is analogous to metrizability for a compact space Y (which is equivalent to the separability of C(Y )) and similarly allows us to avoid certain pathologies and
additional technicalities, especially in discussions where factors, extensions, or equivalence relations are involved, or where spaces of actions come into play as in Section ??.
The basic theory of Polish spaces and standard Borel spaces which underlies all of this is
covered in Appendix A.
The Borel σ-algebra of a topological space X is the σ-algebra B generated by the
open subsets of X, and the members of B are called Borel sets (we may also refer to them
as measurable sets if we are viewing (X, B) abstractly as a measurable space). A Borel
probability measure on X is a probability measure on the Borel σ-algebra of X.
A Borel space is a measurable space (X, B) such that there is a topology on X for
which B is the Borel σ-algebra. If this topology can be chosen to be Polish (i.e., so
that X is separable and admits a compatible metric under which it is complete) then
the Borel space is said to be standard. For economy we usually just write X without
naming the σ-algebra. If we wish to stress the special nature of (X, B) among measurable
spaces then we will refer to the members of B as Borel sets instead of using the generic
qualifier “measurable”. This will be the case when discussing equivalence relations (as in
Section 3.7) where descriptive set theory and Polishness play an especially decisive role.
The term “Borel space” is often used to mean the same thing as a measurable space,
especially in older literature, but the definition above is more consistent with the expressions “measurable space” and “standard Borel space”, both of which are nowadays
customary. In any case this terminological issue will not concern us as we will only be
working with standard Borel spaces.
There is little variety among standard Borel spaces. Indeed as soon as such a space is
uncountable it is must be isomorphic (as a measurable space) to the unit interval with its
usual Borel σ-algebra, as follows from Corollary A.5 and Theorem A.17.
D EFINITION 0.1. A standard probability space is a probability space (X, µ) such that,
denoting its σ-algebra by B, the pair (X, B) is a standard Borel space.
For some purposes (although it will not be an issue for us) one might wish to work
with measure spaces which are complete in the sense that the σ-algebra is closed under
taking subsets of null sets. A measure space (X, µ) with σ-algebra S can be turned into a
30
0. GENERAL FRAMEWORK AND NOTATIONAL CONVENTIONS
complete measure space by first noticing that the σ-algebra generated by S and the subsets
of null sets in S consists of sets of the form A ∪ N where A ∈ S and N is a subset of
a null set in S, and then extending µ to this larger σ-algebra by defining the measure
of A ∪ N to be µ(A). In general a standard Borel space with a probability measure is
not complete, such as the unit interval with Lebesgue measure on its Borel σ-algebra.
The definition of standardness is often taken to include completeness and is in addition
frequently expressed in a more concrete or even purely measure-theoretic way, but these
formulations are invariably equivalent to the completion of a space as in Definition 0.1.
The abstract Borel framework of Definition 0.1 has several advantages, as Appendix A
illustrates.
0.5. Group actions
By an action of the group G on a set X we mean a map α : G × X → X such that,
writing the first argument as a subscript, αs (αt (x)) = αst (x) and αe (x) = x for all x ∈ X
and s, t ∈ G. Most of the time we will write the action as G y X and not give it a
name, with the image of a pair (s, x) written as sx. If we need to give some name α to an
action, such as when needing to distinguish two or more actions, we will use the notation
α
G y X. For sets A ⊆ X and K ⊆ G and an s ∈ G we write
sA = {sx : x ∈ A},
Kx = {sx : s ∈ K},
KA = {sx : x ∈ A and s ∈ K}.
The G-orbit of a point x ∈ X is the set Gx.
Our basic objects of study will be continuous actions on compact Hausdorff spaces
and probability-measure-preserving actions, the latter of which we define as follows.
D EFINITION 0.2. By a p.m.p. (probability-measure-preserving) action of G we mean
an action of G on a standard probability space (X, µ) by measure-preserving transformations. In this case we will combine together the notation and simply write G y (X, µ).
In the absence of an action we do not assume a probability space to be standard unless
otherwise stated, but in order to avoid extra technicalities and because it covers all of our
examples of interest we include standardness as part of the above definition of a p.m.p.
action, whether or not it is actually necessary for a given result.
Given an action G y X on a set, we say that a set A ⊆ X is G-invariant if GA = A,
which is equivalent to GA ⊆ A. When the action is probability-measure-preserving and
A is a measurable set, then we interpret G-invariance we mean GA = A modulo a null
set, i.e., µ(sA∆A) = 0 for all s ∈ G.
The natural notion of isomorphism for group actions on ordinary sets is conjugacy.
Two such actions G y X and G y Y are conjugate if there is a bijection ϕ : X → Y
such that ϕ(sx) = sϕ(x) for all x ∈ X and s ∈ G. Such a map ϕ is called a conjugacy. In
0.6. MEASURE CONJUGACY VS. MEASURE ALGEBRA CONJUGACY
31
the case of continuous actions or p.m.p. actions, the terms “conjugate” and “conjugacy”
will be meant in a more restricted sense which is tailored to the structural context, as made
precise in the following definitions.
D EFINITION 0.3. Two continuous actions G y X and G y Y of the same group
on compact Hausdorff spaces are said to be (topologically) conjugate if there is a homeomorphism ϕ : X → Y such that ϕ(sx) = sϕ(x) for all s ∈ G and x ∈ X.
D EFINITION 0.4. Two p.m.p. actions G y (X, µ) and G y (Y, ν) of the same group
are said to be (measure) conjugate if there are conull sets X ′ ⊆ X and Y ′ ⊆ Y with
GX ′ ⊆ X ′ and GY ′ ⊆ Y ′ and an isomorphism ϕ : X ′ → Y ′ of measurable spaces such
that ϕ(sx) = sϕ(x) for all s ∈ G and x ∈ X ′ and ν(ϕ(A)) = µ(A) for all measurable
sets A ⊆ X ′ .
In both of the above definitions, the map ϕ is called a conjugacy.
In the case G = Z we can alternatively describe an action as a single transformation
T : X → X, which corresponds to the generator 1 in Z and generates an action n 7→ T n
through iteration. The T -orbit of a point x ∈ X is defined to be the set {T n x : n ∈ Z},
i.e., the orbit of x under the Z-action that T generates. For a set A ⊆ X we write T A for
the set {T x : x ∈ A}. We say that A is T -invariant if T A ⊆ A. This does not mean that
A is invariant for the associated Z-action, for we might have T −1 A 6⊆ A. However, if T is
a measure-preserving transformation of a probability space (X, µ) and A is a measurable
subset of X, then by measure-preservingness we see that the following are equivalent,
with invariance now being interpreted modulo a null set (i.e., µ(T A∆A) = 0 in the case
of T ): (i) A is T -invariant, (ii) A is both T -invariant and T −1 -invariant, (ii) A is invariant
for the Z-action generated by T .
0.6. Measure conjugacy vs. measure algebra conjugacy
Standardness for a probability space has the consequence that every measure-preserving
action on the measure algebra essentially arises from a p.m.p. action on the space. To
make this statement precise, which we do in Theorem 0.5, we first need to introduce
some terminology.
By an action on a measure algebra (M, µ) we mean a map G × M → M, written
(s, A) 7→ sA, which satisfies s(tA) = (st)A, eA = A, and µ(sA) = µ(A) for all s, t ∈ G
and A ∈ M. A conjugacy between actions of G on measure algebras (M, µ) and (N, ν)
is a measure algebra isomorphism Φ : M → N (i.e., a Boolean algebra isomorphism
satisfying ν ◦ Φ = µ) such that Φ(sA) = sΦ(A) for all A ∈ M and s ∈ G. A p.m.p.
action G y (X, µ) induces an action on its corresponding measure algebra by the formula
s[A] = [sA] for measurable sets A ⊆ X and s ∈ G. A conjugacy between two p.m.p.
32
0. GENERAL FRAMEWORK AND NOTATIONAL CONVENTIONS
actions G y (X, µ) and G y (Y, ν) induces a conjugacy between the induced actions
on the measure algebras: if X ′ ⊆ X and Y ′ ⊆ Y are G-invariant measurable sets with
µ(X ′ ) = µ(Y ′ ) = 1 and ϕ : X ′ → Y ′ is a measure isomorphism satisfying ϕ(sx) =
sϕ(x) for all s ∈ G and x ∈ X and ν(ϕ(A)) = µ(A) for all measurable A ⊆ X ′ , then
setting Φ([A]) = [ϕ(A)] for all measurable A ⊆ X ′ defines a measure algebra conjugacy
Φ. The converse is also true because of the assumption of standardness in the definition
of p.m.p. action (and would be false in general without it):
T HEOREM 0.5. Let G y (X, µ) and G y (Y, ν) be p.m.p. actions of the same group
and let Φ be a conjugacy between the induced actions on the measure algebras. Then
there is a conjugacy between the actions which induces Φ.
P ROOF. It is enough to show that, given an isomorphism from the measure algebra
MX of X to the measure algebra MY of Y , we can find a measure isomorphism h : X →
Y which induces it. Clearly such an h is unique modulo sets of measure zero.
On every measure algebra we have the canonical metric d(A, B) = µ(A∆B), where
symmetric difference is interpreted via representatives. Since (X, µ) is standard, using
Propositions A.18 and A.19 we see that MX is separable with respect to this metric, and so
we can find a sequence {Pn } of finite partitions in MX such that the algebra they generate
is dense in MX . Using the sequence {Pn } we will construct a standard probability space
(W, µ′ ) and an identification of MX with the measure algebra MW of (W, µ′ ) such that
there is a Borel measure-preserving map f : X → W which induces the identity map
Q
MW → MX . Define W to be ∞
with the product topology, under which it is
n=1 Pn Q
compact and metrizable. To a cylinder set ∞
W (i.e., a product set where An =
n=1 An in
T∞ S
Pn for all but finitely many n) we assign the value µ( n=1 An ). This defines a measure
on the algebra of all finite disjoint unions of cylinder sets. By Kolmogorov’s extension
theorem this measure extends uniquely to a Borel probability measure µ′ on W .
Q
T∞ S
By associating to a cylinder set ∞
An in MX , we obtain a
n=1 An the class of
n=1
measure-preserving algebra homomorphism from the algebra of all finite disjoint unions
of cylinder sets in W to MX . Since the collection of finite disjoint unions of cylinder
sets is dense in MW and the algebra generated by the partitions Pn is dense in MX , this
homomorphism extends to an isometry from MW onto MX . As the algebraic operations
(union, intersection, and complementation) are all continuous, they are preserved by this
isometry. Define a map f : X → W by declaring f (x) to be the sequence whose nth
term is the member of the partition Pn which contains x. Since the preimage under f
of each cylinder set in W is Borel, the map f is Borel. As f preserves the measure on
finite disjoint unions of cylinder sets, it also preserves the measure on the σ-algebra they
generate, since the σ-algebra generated by an algebra of subsets is the smallest collection
of subsets containing the algebra which is closed under taking increasing unions and
0.7. FUNCTIONAL ANALYSIS
33
decreasing intersections indexed by N (Theorem 10.1(ii) of [82]). Thus f induces an
isometric embedding MW ֒→ MX . Clearly f induces the identity map on the algebra of
all finite disjoint unions of cylinder sets, and therefore induces the identity map MW →
MX .
Next, by Theorem A.17 we may assume that X is a compact metric space. So we may
assume that the maximum diameter of the members of Pn converges to 0 as n → ∞, since
we can always refine a given Pn to produce a partition whose members all have diameter
as small as we like.
The map f constructed above is injective. A standard result (Corollary 15.2 in [82])
then says that f (X) is a Borel subset of W and f is a Borel isomorphism from X onto
f (X). Thus f is a measure isomorphism modulo a set of measure zero.
Finally, we do the same for Y , starting from a sequence Pn′ of partitions of Y . By
taking the join of Pn and Pn′ , we may assume that Pn is mapped to Pn′ under the given
measure algebra isomorphism MX → MY . Then from Pn′ we also obtain (W, µ′ ). Now
we can take the composition X → W → Y .
0.7. Functional analysis
We will make frequent use of basic facts in integration theory and related aspects of
functional analysis, for which we refer the reader to [43]. One of the most important of
these is the dominated convergence theorem.
T HEOREM 0.6 (Dominated convergence theorem). Let (X, µ) be a probability space,
g : X → R an integrable function, and {fn } a sequence of real-valued measurable
functions on X such that |fn | ≤ g a.e.
R for all n ≥ 1 and
R fn → f pointwise a.e for some
function f . Then f is integrable and f dµ = limn→∞ fn dµ.
We will frequently encounter the spaces L1 , L2 , and L∞ , whose definitions we recall.
Let (X, µ) be a probability
space and 1 ≤ p < ∞. For an integrable function f : X → C
R
we define kf kp = ( |f |p dµ)1/p . The set of all f for which kf kp < ∞ is a linear
space whose quotient under the relation of equality on a set of full measure is a Banach
space which we denote by Lp (X), or Lp (X, µ) if we need to emphasize the measure.
We similarly write L∞ (X) or L∞ (X, µ) for the Banach space of equivalence classes of
essentially bounded functions f : X → C with the essential supremum norm. Following
custom we do not notationally distinguish between a function and its equivalence class
in Lp (X), and often refer to and manipulate elements of Lp (X) as if they were genuine
functions. We will thus frequently speak of equality and pointwise operations holding
a.e. (almost everywhere), meaning on a set of full measure. Equipped with pointwise a.e.
multiplication and complex conjugation as the involution, L∞ (X) is also a von Neumann
34
0. GENERAL FRAMEWORK AND NOTATIONAL CONVENTIONS
algebra (see Appendix E). We write 1X or simply 1 for the function taking constant value
one, which belongs to Lp (X) for every 1 ≤ p ≤ ∞.
0.8. Hilbert spaces, operators, and unitary representations
Let H be a Hilbert space, which we will always assume to be over the complex numbers unless otherwise indicated. We write the inner product as h·, ·i, or h·, ·iH in case of
possible confusion. We write B(H) for the algebra of all bounded linear operators on
H, which possesses an involution a 7→ a∗ given by taking the adjoint. As discussed in
Appendix E, this is a basic example of a von Neumann algebra. We write idH or id for
the identity operator on H, or 1 if we wish to emphasize its algebraic role as the unit in
the algebra B(H). Operators will be denoted by either lower or upper case letters, usually depending on whether or not the operator is being manipulated as an element of a
specified subalgebra of B(H).
An operator U ∈ B(H) is unitary if U ∗ U = U U ∗ = id. A unitary representation of
G on a Hilbert space H is a map G → B(H) which is a homomorphism into the group of
unitary operators on H. We write the codomain as B(H) instead of the unitary group since
it is typically the algebraic interaction with other elements of B(H) which is of interest.
For sets K ⊆ G and Ω ⊆ H we write π(K)Ω to mean the set {π(s)ξ : s ∈ K and ξ ∈ Ω},
and also write π(K)ξ and π(s)Ω with similar meanings in the case of a single vector ξ or
group element s. A set Ω ⊆ H is G-invariant if π(G)Ω = Ω, or equivalently π(G)Ω ⊆ Ω.
An operator T ∈ B(H) is compact if the image of the closed unit ball under T has
compact closure in H. In this case the spectrum σ(T ) (i.e., the set of all λ ∈ C such that
λ1−T fails to be invertible) is a countable set whose nonzero elements are all isolated and
are eigenvalues for T . If T is in addition normal, i.e., T ∗ T = T T ∗ , then the Hilbert space
H decomposes into a direct sum of eigenspaces for T . The compact operators form a
closed ideal in B(H), and when H is infinite-dimensional and separable this is the unique
nontrivial closed ideal in B(H). See Sections 1.4, 2.4, and 4.1 of [105].
Let B be an orthonormal basis√for H. An operator T ∈ B(H) is of trace class if
P
T ∗ T (as defined using the functional calculus for the
ξ∈B h|T |ξ, ξi < ∞ where |T | =
∗
positive operator T T ), and the trace of such an operator is defined by
X
Tr(T ) =
hT ξ, ξi.
ξ∈B
Using orthonormal expansions it is not hard to show that this sum is independent of the
choice of B. The trace class operators form an ideal in B(H) which is a dense subset
of the ideal of compact operators, and for all trace class operators T , bounded linear
operators S, and unitary operators U on H we have Tr(ST ) = Tr(T S) and Tr(U T U ∗ ) =
0.8. HILBERT SPACES, OPERATORS, AND UNITARY REPRESENTATIONS
35
Tr(T ). On the space of trace class operators we have the norm kT k1 = Tr(|T |). See
Section 2.4 of [105] for details.
The direct sum π⊕ρ of two unitary representations π : G → B(H) and ρ : G → B(K)
of the same group is the representation on H ⊕ K defined by
((π ⊕ ρ)(s))(ξ, ζ) = (π(s)ξ, ρ(s)ζ)
for s ∈ G and ξ, ζ ∈ H. The trivial representation of G is the unitary representation
on the one-dimensional space C in which every vector is fixed by every element of G.
Two unitary representations π : G → B(H) and ρ : G → B(K) of the same group
are (unitarily) equivalent if there is a unitary isomorphism U : H → K (i.e., a bijective
inner-product-preserving linear map) such that ρ(s) = U π(s)U −1 for all s ∈ G. A
subrepresentation of a unitary representation π : G → B(H) is a representation ρ : G →
B(K) obtained by restricting π to a closed G-invariant subspace K ⊆ H. When ρ is
unitarily equivalent to a subrepresentation of ρ then we say that π contains ρ. Given that
π(G) is closed under taking adjoints it is clear that the orthogonal complement K⊥ of K
is also G-invariant, so that π can be expressed as ρ ⊕ ρ′ where ρ′ is the restriction of π to
K⊥ .
To construct the Hilbert space tensor product H ⊗ K of two Hilbert spaces we first
define an inner product on the algebraic tensor product by setting
hξ1 ⊗ ζ1 , ξ2 ⊗ ζ2 i = hξ1 , ξ2 ihζ1 , ζ2 i
on elementary tensors and extending (one can use orthonormal bases to show that this is
well defined and positive definite), and then complete in the norm kξk := hξ, ξi1/2 . Note
that if B and C are orthonormal bases for H and K, respectively, then the linear map ϕ :
H ⊗ K → ℓ2 (B × C) given on elementary tensors by ϕ(ξ1 ⊗ ζ1 )(ξ2 , ζ2 ) = hξ1 , ξ2 ihζ1 , ζ2 i
is isometric with dense image and hence is a unitary isomorphism. If U and V are unitary
operators on H and K, respectively, then there is a unitary operator U ⊗ V on H ⊗ K
determined on elementary tensors by (U ⊗ V )(ξ ⊗ ζ) = U ξ ⊗ V ζ. The tensor product
π ⊗ ρ of unitary representations π : G → B(H) and ρ : G → B(K) is the unitary
representation of G given by s 7→ π(s) ⊗ ρ(s).
Let H be a Hilbert space. Its conjugate H is the Hilbert space which is the same
as H as an additive group but with the scalar multiplication (c, ξ) 7→ c̄ξ for c ∈ C and
inner product hξ, ζiH = hζ, ξiH . If U is a unitary operator on H, then the operator U
on H which formally coincides with U is also unitary. Given a unitary representation
π : G → B(H), its conjugate π̄ : G → B(H) is the unitary representation defined by
s 7→ π(s).
Let H and K be Hilbert spaces, and let B and C be orthonormal bases for H and K,
respectively. Write HS(H, K) (or simply HS(H) if K = H) for the set of all bounded
36
0. GENERAL FRAMEWORK AND NOTATIONAL CONVENTIONS
linear operators T : H → K such that
XX
(4)
|hT ξ, ζi|2 < ∞.
ξ∈B ζ∈C
A simple exercise using orthonormal expansions shows that the above double sum, whether
finite or infinite, is independent of the choice of B and C, and that it can be also exP
P
pressed as both ξ∈B kT ξk2 and ζ∈C kT ∗ ζk2 . We refer to the elements of HS(H, K)
as Hilbert-Schmidt operators. The set HS(H, K) forms a linear subspace of the linear
P
space of bounded linear operators H → K and kT k2 = ( ξ∈B kT ξk2 )1/2 defines a norm
on HS(H, K), as can be seen by applying the triangle inequality (i.e., Minkowski’s inequality) in ℓ2 (B × C). In fact we can endow HS(H, K) with the structure of a Hilbert
space by identifying it with ℓ2 (B × C) via the linear map ϕ : HS(H, K) → ℓ2 (B × C)
given by
ϕ(T )(ξ, ζ) = hT ξ, ζi.
This map is evidently isometric, and it is surjective (and hence a unitary isomorphism)
because for every x ∈ ℓ2 (B × C) we can set
X XX
T
cξ ξ =
cξ x(ξ, ζ)ζ
ξ∈B
ζ∈C ξ∈B
for all square-summable coefficients cξ , which defines a bounded linear operator T : H →
K by virtue of the application
2
X
X X
XX
X cξ x(ξ, ζ) ≤
|cξ |2
|x(ξ, ζ)|2 = kxk
cξ ξ ζ∈C
ξ∈B
ζ∈C
ξ∈B
ξ∈B
ξ∈B
of the Cauchy-Schwarz inequality, and T clearly satisfies (4) and maps to x under ϕ. The
inner product on HS(H, K) arising from this identification with ℓ2 (B × C) is then given
by
XX
hS, T i =
hSξ, ζihζ, T ξi.
ξ∈B ζ∈C
By passing through ℓ2 (B × C), we can now give a natural alternative description of
the tensor product H ⊗ K of two Hilbert spaces as the space HS(K, H) of all HilbertSchmidt operators from K to H. The unitary isomorphism ϕ : H ⊗ K → HS(K, H) that
sets up this identification sends the elementary tensor ξ ⊗ ζ to the rank-one operator η 7→
hη, ζiK ξ. Given unitary representations π : G → B(H) and ρ : G → B(K), the unitary
isomorphism ϕ conjugates the tensor product π ⊗ ρ to the conjugation representation
on HS(K, H), in which a group element s is represented as the unitary operator T 7→
π(s)T ρ̄(s)−1 .
0.9. THE KOOPMAN REPRESENTATION
37
It is readily seen that the Hilbert-Schmidt operators in B(H) form an ideal which is a
dense subset of the ideal of compact operators.
For more details on tensor products and Hilbert-Schmidt operators see Section 2.6 of
[76].
0.9. The Koopman representation
Given a p.m.p. action G y (X, µ), its Koopman representation is the unitary representation κ : G → B(L2 (X)) defined by κ(s)f (x) = f (s−1 x) for all s ∈ G, f ∈ L2 (X),
and a.e. x ∈ X. Note that the one-dimensional subspace C1X is always G-invariant and
the restriction of the Koopman representation to it is the trivial representation. We will
thus frequently focus our attention on the restriction of the Koopman representation to
2
2
the
R orthogonal complement L (X) ⊖ C1, which consists of all f ∈ L (X) such that
f dµ = 0.
X