Preface ........................................................ xi
Conference photo .............................................. xiv
Introduction ................................................... xv
1 Independence and Lévy Processes in Quantum Probability ..... 1
1.1 Introduction ............................................... 1
1.2 What is Quantum Probability? ............................... 4
1.2.1 Distinguishing features of classical and quantum
probability ........................................ 11
1.2.2 Dictionary 'Classical → Quantum' ................... 13
1.3 Why do we Need Quantum Probability? ....................... 15
1.3.1 Mermin's version of the EPR experiment ............. 15
1.3.2 Gleason's theorem .................................. 20
1.3.3 The Kochen-Specker theorem ......................... 21
1.4 Infinite Divisibility in Classical Probability ............ 23
1.4.1 Stochastic independence ............................ 23
1.4.2 Convolution ........................................ 23
1.4.3 Infinite divisibility continuous convolution
semigroups, and Lévy processes ..................... 23
1.4.4 The De Finetti-Lévy-Кhintchine formula on ( +,+) ... 25
1.4.5 Levy-Khintchine formulae on cones .................. 25
1.4.6 The Lévy-Кhintchine formula on (d, +) ............. 26
1.4.7 The Markov semigroup of a Lévy process ............. 27
1.4.8 Hunt's formula ..................................... 27
1.5 Lévy Processes on Involutive Bialgebras ................... 29
1.5.1 Definition of Lévy processes on involutive
bialgebras ......................................... 29
1.5.2 The generating functional of a Lévy process ........ 34
1.5.3 The Schurmann triple of a Lévy process ............. 36
1.5.4 Examples ........................................... 41
1.6 Lévy Processes on Compact Quantum Groups and their
Markov Semigroups ......................................... 42
1.6.1 Compact quantum groups ............................. 43
1.6.2 Translation invariant Markov semigroups ............ 48
1.7 Independences and Convolutions in Noncommutative
Probability ............................................... 55
1.7.1 Nevanlinna theory and Cauchy-Stieltjes
transforms ......................................... 55
1.7.2 Free convolutions .................................. 57
1.7.3 A useful Lemma ..................................... 60
1.7.4 Monotone convolutions .............................. 60
1.7.5 Boolean convolutions ............................... 73
1.8 The Five Universal Independences .......................... 83
1.8.1 Algebraic probability spaces ....................... 87
1.8.2 Classical stochastic independence and the product
of probability spaces .............................. 88
1.8.3 Products of algebraic probability spaces ........... 90
1.8.4 Classification of the universal independences ...... 94
1.9 Lévy Processes on Dual Groups ............................. 98
1.9.1 Dual groups ........................................ 98
1.9.2 Definition of Lévy processes on dual groups ....... 101
1.9.3 Time reversal ..................................... 103
1.10 Open Problems ............................................ 105
References ............................................... 107
2 Quantum Dynamical Systems from the Point of View of
Noncommutative Mathematics ............................... 121
2.1 Noncommutative Mathematics and Quantum/Noncommutative
dynamical systems ........................................ 121
2.1.1 Noncommutative Mathematics - Gelfand-Naimark
Theorem ........................................... 122
2.1.2 Quantum topological dynamical systems and some
properties of transformations of C*-algebras ...... 123
2.1.3 Cuntz algebras .................................... 126
2.1.4 Graph C*-algebras ................................. 128
2.1.5 C*-algebras associated with discrete groups ....... 130
2.2 Noncommutative Topological Entropy of Voiculescu ......... 131
2.2.1 Endomorphisms of Cuntz algebras and a quantum
shift ............................................. 132
2.2.2 The shift transformation on a graph С*-algebra .... 134
2.2.3 Classical topological entropy as defined by
Rufus Bowen and extensions to topological
pressure .......................................... 134
2.2.4 Finite-dimensional approximations in the theory
of C*-algebras .................................... 136
2.2.5 Noncommutative topological entropy (Voiculescu
topological entropy) .............................. 137
2.3 Voiculescu Entropy of the Shift and other Examples ....... 141
2.3.1 Estimating the Voiculescu entropy of the
permutative endomorphisms of Cuntz algebras ....... 141
2.3.2 The Voiculescu entropy of the noncommutative
shift and related questions ....................... 145
2.3.3 Generalizations to graph C*-algebras and to
noncommutative topological pressure ............... 146
2.3.4 Automorphism whose Voiculescu entropy is
genuinely noncommutative .......................... 147
2.3.5 Automorphism that leaves no non-trivial abelian
subalgebras invariant ............................. 149
2.4 Crossed Products and the Entropy ......................... 152
2.4.1 Crossed products .................................. 152
2.4.2 The Voiculescu entropy computations for the maps
extended to crossed products ...................... 154
2.5 Quantum 'Measurable' Dynamical Systems and Classical
Ergodic Theorems ......................................... 157
2.5.1 Measurable dynamical systems and individual
ergodic theorem ................................... 157
2.5.2 GNS construction and the passage from
topological to measurable noncommutative
dynamical systems ................................. 158
2.6 Noncommutative Ergodic Theorem of Lance; Classical and
Quantum Multi Recurrence ................................. 163
2.6.1 Mean ergodic theorem(s) in von Neumann algebras ... 163
2.6.2 Almost uniform convergence in von Neumann
algebras .......................................... 166
2.6.3 Lance's noncommutative individual ergodic
theorem and some comments on its proof ............ 167
2.6.4 Classical multirecurrence ......................... 168
2.6.5 Noncommutative extensions and counter examples
due to Austin, Eisner and Tao ..................... 169
2.6.6 Final remarks ..................................... 172
References ............................................... 173
Index ......................................................... 177
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