Franz U. Noncommutative mathematics for quantum systems (Cambridge; Delhi, 2016). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаFranz U. Noncommutative mathematics for quantum systems / U.Franz, A.Skalski. - Cambridge: Cambridge University Press; Delhi: IISc Press, 2016. - xviii, 180 p.: ill. - (Cambridge - IISc series). - ISBN 978-1-107-14805-5
Шифр: (И/В31-F85) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
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 (fig.1+,+) ... 25
     1.4.5  Levy-Khintchine formulae on cones .................. 25
     1.4.6  The Lévy-Кhintchine formula on (fig.1d, +) ............. 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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