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**Introduction**

First we give a brief overview of classical dynamics (partly drawn from [6] and [10]). A fundamental question in statistical mechanics concerns the existence of certain types of time averages. The problem may be formulated as follows: The state of a physical system at a certain time is described by specifying a point in a » phase space » X. When a mechanical system is subject to a principle of scientiﬁc determinism, e.g. when it is assumed to follow the classical Hamiltonian equations, it is known that an initial state x will, after t seconds have elapsed, have passed into a unique new state y. Since y is uniquely determined by x and t, a function T : X → X is deﬁned by the equation y = Tt(x). The ﬂow Tt in this case has the property that Tt(Ts(x)) = Tt+s(x) for all points x in phase space and for all times s and t.

**AMENABLE GROUPS AND FØLNER SEQUENCES **

Følner sequence satisfying certain conditions. The role of a Følner sequence is to replace the sequence of sets {1, …, n} appearing in the averages in the expressions above.One of the technical tools we use in this case is a so-called Van der Corput lemma which we discuss in Chapter 3.2. This type of lemma and related inequalities, inspired by the classical Van der Corput diﬀerence theorem and Van der Corput inequality, have been used by Bergelson et al [1], [3], Fursten berg [19], Niculescu, Str¨oh, and Zsid´o [28], and others, to study polynomial ergodic theorems, nonconventional ergodic averages, and noncommutative recurrence, for example.

**The GNS construction**

The Gelfand-Naimark-Segal (GNS) construction provides us with a powerful tool for the study of ergodic theory in non-commutative dynamical systems, as it enables us to approach some problems through Hilbert space theory. In the discussion below L(X) refers to the algebra of all linear operators X → X while L(X) refers to all bounded linear operators.

**Characterizations of weak mixing**

The characterizations of weak mixing given below will set the stage for our study of weak mixing of all orders in Section 3.3. In this section G need not be abelian and the properties of Følner sequences are only needed in Corollary 3.1.2. The material in this Section is fairly standard, except that we work with the notion “M-weak mixing” (and “M-ergodicity”), which is important in Section 3.3. Also note that throughout this Chapter, the operator ι and the Hilbert space H are those obtained from the GNS construction as discussed in Section 1.4.

**1 Background**

1.1 Introduction

1.2 Amenable groups and Følner sequences

1.3 Density limits

1.4 The GNS construction

1.5 The Mean Ergodic Theorem

**2 Dynamical systems**

2.1 Deﬁnitions

2.2 Examples

2.2.1 A noncommutative compact system

2.2.2 A counterexample

2.2.3 An asymptotic abelian and weakly mixing system

**3 Weakly mixing systems**

3.1 Characterizations of weak mixing

3.2 A Van der Corput lemma

3.3 Weak mixing of all orders

**4 The Szemer´edi property**

4.1 Compact systems

4.2 Compact factors and ergodic systems

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The Szemer´edi property in noncommutative dynamical systems