Introduction

Quantum probability is a generalization of both classical probability theory and quantum mechanics that allows to describe the probabilistic aspects of quantum mechanics. This generalization is formulated in two steps. First, the theory is reformulated in terms of algebras of functions on probability spaces. Therefore, the notion of a probability space (Ω, ℱ, P) is replaced by the pair (L∞ (Ω), E(.) = ∫Ω. dP) consisting of the commutative von Neumann algebra of bounded random variables and the expectation functional. Then, the commutativity condition is dropped. In this way we arrive at the notion of a (von Neumann) algebraic probability space (N,Φ) consisting of a von Neumann algebra N and a normal (faithful tracial) state Φ. As we have seen this includes classical probability spaces in the form (L∞(Ω), E), it also includes quantum mechanical systems modeled by a Hilbert space H and a pure state ψ ∊ H (or a mixed state ρ ∊ S(H)), if we take N = B(H) and Ф the state defined by Ф(X) = 〈 ψ, X ψ 〉 (or Ф (X) = tr(ρX)) for X ∊ B(H). Note that in this course we shall relax the conditions on N and Φ and work with involutive algebras and positive normalized functionals, that is, *-algebraic probability spaces.

A striking feature of quantum probability (also called noncommutative probability) is the existence of several notions of independence. This is the starting point of this course, which intends to give an introduction to the theory of quantum stochastic processes with independent increments.

However, before we come to these processes, we will give a general introduction to quantum probability. In Section 1.2, we recall the basic definitions of quantum probability and discuss some fundamental differences between classical probability and quantum probability. In Section 1.3 we address the question ‘Why do we need Quantum Probability?’ We discuss the EPR experiment and the Kochen–Specker Theorem, which show that we cannot model quantum physics with classical probability spaces because the values of observable quantities do not exist unless we specify which measurement we will carry out and which quantities we will determine. In this sense quantum physics requires a more radical description of chance.

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.

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’.

Notational conventions

For a subset X of a Banach space the closed linear span of X will be denoted by (and the usual linear span by Lin X). Notation F ⊂⊂ Z will be used to signify that F is a finite subset of Z. Sometimes we will write for. All vector spaces (algebras, Hilbert spaces, and so on) will be considered over ℂ. As in the first part of this volume, scalar products are linear on the right (and not on the wrong!) side.

Noncommutative Mathematics – Gelfand–Naimark Theorem

Definition 2.1.1 A Banach algebra A with involution (that is, a complex algebra with involutive antilinear antimultiplicative map * : A → A, equipped with a submultiplicative norm making A a Banach space and such that the involution is an isometry) is called a C*-algebra if

‖ a*a‖ = ‖a2‖, a ∊ A.

For proofs of various general statements related to C*-algebras that will be used below we refer, for example, to the monograph [Mu].

Noncommutative mathematics might be considered as one of the major new trends of mathematics that arose in the twentieth century. Initially motivated by the formulation and development of quantum physics due to Heisenberg and von Neumann, the idea of ‘making a theory noncommutative’ has been since extended to many areas of pure and applied mathematics and turned out to be extremely fruitful.

In 1925, Heisenberg suggested replacing the observables of a quantum system, such as its position Q and momentum P, by noncommutative quantities, most commonly realised as operators on a Hilbert space. This idea allowed the correct prediction of the results of the experiments of quantum physics; for example, it led to a description of the observed spectra of atoms. It has also led to a cornucopia of new mathematics, in particular to the theory of von Neumann algebras and C *-algebras developed, respectively, by Murray and von Neumann, and by Gelfand and Naimark.

The general pattern of the noncommutative mathematics is the following: take a ‘classical’ mathematical theory, say topology, measure theory, differential geometry, or group theory, and reformulate it in terms of algebras. These algebras are algebras of functions on the spaces appearing in the ‘classical’ theory, that is, continuous functions (with values in ℂ) on a topological space, measurable functions on a measure space, smooth functions on a manifold, and so on. They inherit some additional structure, for example, an involution, a norm or topology, a coproduct, from the space on which they are defined. Finally, these algebras are always commutative. Axiomatizing the additional structure and dropping the commutativity condition one arrives at a noncommutative generalization of the original theory. In this way the theory of von Neumann algebras or C *-algebras can be viewed, respectively, as noncommutative measure theory and noncommutative topology. The theory of Hopf algebras can be viewed as noncommutative group theory (here a warning is in place: the term ‘noncommutative’ refers to the function algebra, not to the group multiplication, which can also be noncommutative in classical ‘commutative’ group theory).

The procedure of making a theory noncommutative is not canonical, the choice of the axioms of the new noncommutative theory is not unique and each time has to be adapted to the classical theory to which it is applied.

This monograph arose from the lectures delivered by the authors during the graduate school, being a part of the meeting ’Recent advances in Operator Theory and Operator Algebras’, which took place between 31 December 2012 and 12 January 2013 in the Indian Statistical Institute in Bangalore. We are very grateful to Rajarama Bhat, Tirthankar Bhattacharyya and Jaydeb Sarkar for organizing the meeting, inviting us to speak and providing excellent working conditions during our stay at ISI, and the audience of the school for their active participation in our courses. The authors are strong believers in the usefulness of organizing graduate schools for mathematical students – in fact we first met during an analogous event in Greifswald (Germany) in 2003, respectively, as a lecturer and a participating PhD student, and have collaborated ever since.

Our lectures during the Bangalore school treated notions of independence and quantum Lévy processes in quantum probability (by the first named author) and quantum processes understood as noncommutative incarnations of classical dynamical systems (by the second named author). Both these topics, born from necessity to incorporate quantum models into mathematical approaches to study physical systems, have now become very active, broad areas of modern mathematical research. The monograph consists of two chapters: ‘Independence and Lévy processes in quantum probability’, authored by Uwe Franz, and ‘Quantum dynamical systems from the point of view of noncommutative mathematics’, authored by Adam Skalski. These can be read independently, but we believe that there is an added value in placing them together; not in the least because they present alternative approaches to the noncommutative/quantum generalizations of classical concepts and results. The monograph is essentially self-contained, with several references to both sources of necessary background and to current research literature, and should thus form an appropriate entrance point for graduate students interested in the area.

We would like to express our gratitude to Rajarama Bhat and Gadadhar Misra for encouraging us to transform the lecture notes into the monograph form, and facilitating the contacts with the publisher, Cambridge University Press India, whose assistance during the editing process is also gratefully acknowledged.