2.2
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
1.6
Amenability
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
subse
1.7
Mixing properties
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
that
| ( ), | < ,
for all F .
The representati
1.5
Invariant vectors
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 cont
1.4
Induced representations
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 linearl
1.2
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
1.3
Cocycles
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
1.7.1
Compact representations
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
1.7.2
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
2.5
Ergodic theorems
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
2.6
Recurrence theorems
Theorem 2.6.1 (Poincars Recurrence Theorem [Poi90]). Let Z T (X, B , )
e
be a measure preserving action on a probability space (X, B , ). If A X is
measurable, such that (A) >
2.4
Gaussian actions
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 acti
2.3
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.
1.8
Weak containment
Denition 1.8.1. Let be a group, and : U (H), : U (K) two
unitary representations. The representation is weakly contained in the
representation (written
) if for each H, F nite, a
2.1
Examples
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 ha
1.1
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. o