The Koopman representation
Denition 2.2.1. Let (X, B , ) be an action on a measure space which
preserves the measure class . The Koopman representation of associated
to this action is the unitary representation : U (L2 (X, ) given by
( ( )f )(x) = f (
Much of the material of this section has been taken from Section 2.6 in the book
of Brown and Ozawa [BO08].
Denition 1.6.1. Let be a group, a Flner net is a net of non-empty nite
subsets Fi such that |Fi Fi |/|Fi | 0, for all .
Note that w
Denition 1.7.1. Let be a group, a unitary representation : U (H) is
weak mixing if for each nite set F H, and > 0 there exists such
| ( ), | < ,
for all F .
The representation is (strong) mixing if | = , and for each nite set
Denition 1.5.1. Let be a group, a unitary representation : U (H)
contains invariant vectors if there exists a non-zero vector H such that
( ) = for all . The representation contains almost invariant vectors
if for each F , and > 0,
Given an action X of a group on a set X , a cocycle : X into
a group , and a unitary representation : U (K), we obtain an induced
representation Ind : U ( 2 X K) by linearly extending the formula
Ind ( )(x ) = x ( (, x) ).
Functions of positive type
Denition 1.2.1. Given a group , a function of positive type on is a
map : C such that for all u C we have
, (1 ) 0.
One way in which function of positive types appear is when we have a unitary
representation : U (K), togethe
Denition 1.3.1. Suppose X is an action of a group on a set X , and
is a group. A cocycle for the action into is a map : X such that
(1 2 , x) = (1 , 2 x)(2 , x),
for all 1 , 2 , and x X . Two cocycles , : X are cohomologous
if there is a map
If H is a Hilbert space the strong operator topology on B (H) is such that
limi Ti = T if and only if limi (Ti T ) = 0, for all H. The weak operator
topology on B (H) is such that limi Ti = T if and only if limi (Ti T ), = 0,
Weak mixing for amenable groups
Proposition 1.7.14. Let be an amenable group with a Flner net Fi ,
and let : U (H) be a unitary representation. Then is weak mixing if
and only if for each , H we have
1 | ( ), | 0.
|Fi | Fi
Proof. If is not weak
Using the Koopman representation, von Neumanns Ergodic Theorem for Hilbert
spaces can be rephrased for actions.
Theorem 2.5.1 (Von Neumanns Ergodic Theorem [vN32]). Let (X, ) be
a measure preserving action of a countable amenable grou
Theorem 2.6.1 (Poincars Recurrence Theorem [Poi90]). Let Z T (X, B , )
be a measure preserving action on a probability space (X, B , ). If A X is
measurable, such that (A) > 0 then for almost every point x A, the orbit Zx
Let : O(H) be an orthogonal representation of a countable group . The
aim of this section, which is taken from [PS09], is to describe the construction
of a measure-preserving action of on a non-atomic standard probability space
A remark about measure spaces
So far we have been considering actions of a countable group on a general
probability space (X, B , ). For most aspects of ergodic theory this is the proper
setting. However, occasionally this level of generality can be p
Denition 1.8.1. Let be a group, and : U (H), : U (K) two
unitary representations. The representation is weakly contained in the
) if for each H, F nite, and > 0,
there exists 1 , . . . , n K such that
| ( ), n
Denition 2.1.1. Let (X, B , ) be a -nite measure space, and let be a
countable group, an action X such that 1 (B ) = B , for all is measure preserving if for each measurable set A B we have ( 1 A) = (A).
Alternately, we will say that is an in
Denitions and constructions
Denition 1.1.1. Let be a group, a unitary representation (resp. an
orthogonal representation) of is a homomorphism : U (H) (resp.
: O(H) from into the unitary (resp. orthogonal) group of a complex
(resp. real) Hilbert spac