2

Quantum Dynamical Systems

from the Point of View of

Noncommutative Mathematics

A starting point of ‘noncommutative mathematics’ is based on the
observation that many features of classical spaces can be phrased
in terms of the algebras of functions defined on these spaces. In
these lectures we will present a few examples of how in presence of
dynamics this idea leads to the study of ‘quantum’ transformations.
Our focus will be on two building blocks of modern abstract theory
of dynamical systems: entropy and ergodic theorems.

2.1 Noncommutative Mathematics and Quantum/

Noncommutative Dynamical Systems

The notion of quantum processes, or quantum dynamical systems,
can be understood in several different ways, as can be viewed for
example by looking at some of the several hundred articles
classified in the MathSciNet database under the heading MSC
46L53 (corresponding to ‘Noncommutative dynamical systems’).
In these lectures we will present a purely mathematically
motivated approach to certain ‘quantum’ or ‘noncommutative’
versions of fundamental concepts and problems studied in
classical topological and measurable dynamics. The origins of the
need to study questions of that type can be naturally found in
quantum mechanics: readers interested in physical interpretations
of the topics studied below can consult, for example, the book
[AF], or the lecture of Uwe Franz in the first part of this volume.
Here our approach will be based on what is nowadays often called
the philosophy of ‘Noncommutative Mathematics’.