Noncommutative probability:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht u.a.
Kluwer
1994
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| Series: | Mathematics and its applications
305 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 353 S. |
| ISBN: | 0792331338 |
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MARC
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Record in the Search Index
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| adam_text | Table of Contents
PREFACE Xiii
CHAPTER 1. Central limit theorem on L(H) 1
1. 1.Introduction 1
1. 2.Generalities about Hilbert spaces and linear operators on them 2
1. 3.The polar decomposition in L(H) 3
1. 4.Linear functionals on L(H) 5
1. 5.The predual of L(H) 6
1. 6.Direct sums and tensor products of Hilbert spaces 6
1. 7.The conjugate of a Hilbert space 7
1. 8.Finite rank operators 7
1. 9.Structure on the dual space of L(H) 8
1.10.Order relation in L(H)* 9
1.11.The L, and L_ spaces of operators on H 11
1.12.First characterisation of elements in 1 (H) 12
1.13.Spectral characterisation of the operators in L_(H) 12
1.14.Spectral characterisation of the operators in L 13
1.15.Structure of the nonnegative f s in L(H) 14
1.16.Definition of Tr 14
1.17.Weak convergence of functionals 15
1.18.L(H), is the dual of L(H)f 15
1.19.Other characterisation of elements in L(H)A 16
1.20.Generalisation of the convergence theorem 16
1.21.The canonical Weyl system (representation of commutation 17
relations)
1.22.Lie group representation view 17
1.23.Extending the Weyl system in 1.21 18
1.24.Irreducibility 19
1.25.Characteristic functions 21
1.26.Convergence of characteristic functions 21
1.27.Square integrability of characteristic functions 22
1.28.Example 22
1.29.Tensor product of functionals 23
1.30.Constructions of Weyl systems 23
1.31.The idea for convolution 24
vi Table of Contents
1.32.Representations of L(H) 24
1.33.General form of unidimensional Weyl systems 26
1.34.The analogue of convolution 27
1.35.The Stone theorem 27
1.36.Central limit theorem for Weyl systems 28
1.37.Infinitesimal operator of a group of unitaries and second 28
moments
1.38.Spectral families attached to a Weyl system 30
1.39.Criterion for the existence of the second derivatives of the 31
characteristic function
1.40.The uncertainity relation 32
1.41.Analogues of gaussian distributions 33
1.42.Centering 34
1.43.The moment method 34
1.44.The noncommutative analogue of the moment method theorem 35
1.45.The algebra generated by the operators in the canonical 36
commutation relations
1.46.Infinitesimal operator in tensor product 37
1.47.The algebraic part of the central limit theorem 38
1.48.Symmetric Fock space defined using exponential vectors 40
1.49.Weyl operators 42
1.50.Noncommutative analogues of brownian motion 42
1.51.Orthonormal basis in the symmetric Fock space 43
1.52.Symmetric tensor product construction of the Fock space 45
1.53.Preliminaries about the infinitesimal operators 47
1.54.Infinitesimal operators for groups of Weyl operators 48
1.55.Example 50
1.56.Bibliographical sources 52
CHAPTER 2. Probability theory on von Neumann algebras 53
2. 1.Generalities 53
2. 2.C algebras and von Neumann algebras 53
2. 3 . Isomorphisms .special facts 54
2. 4.Isomorphisms of von Neumann algebras 57
2. 5.Tensor product of two von Neumann algebras 59
2. 6.Characterising positive operators in the tensor product 60
2. 7.Infinite tensor products of Hilbert spaces and of von Neumann 62
algebras
2. 8.Eliminating the vectors from the definition of the infinite 62
Table of Contents vii
tensor product of von Neumann algebras
2. 9.A probability topic 65
2.10.A second topic from probability theory 66
2.11.A part of Kaplansky s theorem 67
2.12.Universality property for the tensor product 68
2.13.Small projections 70
2.14.Trace states 71
2 .15 . r finite and finite von Neumann algebras 71
2.16.Equivalence of projectors 73
2.17.Closed operators 74
2.18.Closed operators affiliated to a von Neumann algebra 75
2.19.Operators affiliated to a finite von Neumann algebra 76
2.20.Distribution of a self adjoint operator 77
2.21.Convergence 78
2. 22. Properties and pathologies 79
2.23.Kolmogorov s criterion of path continuity 81
2.24.L2 and GNS 82
2.25.L1 and the predual 84
2.26.Other remarks 85
2.27.Convergence in the general case 86
2.28.The commutation theorem 89
2.29.Stochastic processes 91
2.30.Bibliographical sources 93
CHAPTER 3. Free independence 95
3. 1.Introduction 95
3. 2.Free product of algebras with 1 95
3. 3.Free families of subalgebras with respect to a linear 95
functional
3. 4.Construction of the free product; first step 96
3. 5.Free product of linear spaces 97
3. 6.Construction of the free product of algebras; second step 97
3. 7.Third step:the functional 98
3. 8.Universality property 99
3. 9.Associativity and desassociativity of free independence 99
3.10.Definition of the free convolution B 200
3.11.Properties of the convolution ffl 201
3.12.Free convolution as an operation with sequences 202
3.13.Cumulants 102
viii Table of Contents
3.14.Convolution of functionals on the free algebra generated by an 104
arbitrary set
3.15.Homothety 105
3.16.Central limit theorem for equal factors 105
3.17.Tensor Pock space 106
3.18.Operators on the tensor Pock space 106
3.19.Example of free convolution 108
3.20.Construction of elements with given cumulants 109
3.21.Other examples of free families 110
3.22.Processes with independent increments 111
3.23.Free * algebras 112
3.24.Example 113
3.25.Free product of C algebras 113
3.26.Central limit theorem for unequal components 115
3.27.The noncommutative gaussian law 116
3.28.Free products of von Neumann algebras and trace states 117
3.29.Free convolution of arbitrary probabilities on R US
3.30.Remarks concerning multiplicative convolution 119
3.31.Another expression for the relation between moments and 120
cumulants
3.32. Characteristic functions for free convolution 121
3.33.Bibliographical sources 123
CHAPTER 4. The Clifford algebra 124
4. 1.Construction of the antisymmetric (alternated) tensor product 124
4. 2.Properties of the alternated tensor product 125
4. 3.Alternated product on HH s 125
4. 4.Definition and structure of the antisymmetric Fock space 126
4. 5.Inclusion between alternate Fock spaces 126
4. 6.Alternate Fock space of a direct sum 127
4. 7.Remark concerning order of terms 127
4. 8.Left multiplication operators 128
4. 9.Properties of the left multiplication operators 128
4.10.Example 129
4.11.Generalisation in the case of a symmetric Fock space. 131
4.12.Generalisation in the case of an antisymmetric Fock space 132
4.13.The Clifford C* algebra 134
4.14.Free construction leading to the Clifford algebra 135
Table of Contents ix
4.15.Uniqueness of the C norm on CQ(H) 136
4.16.Representations of C(H) 136
4.17.The anti Fock representation 137
4.18.Operations with representations: particular case 138
4.19.Example 139
4.20.The * automorphism uH 140
4.21.Even States 141
4.22.Commutativity 143
4.23.Clifford convolution 143
4.24.Cumulants 144
4.25.Notation 144
4.26.Towards the formula for the cumulants 145
4.27.General case 146
4.28.Central limit theorem 147
4.29.Quasifree states 147
4.30.Operator defined by a state 147
4.31.Construction of quasifree states in the general case 149
4.32.Gauge invariance 150
4.33.Another way of introducing the Clifford C algebra 151
4.34.Structure of the Clifford algebra 153
4.3 5. Expressing f . /_ 154
(1/2)1/2
4.36.The von Neumann algebra generated by f ,_ (C(H)) 155
(1/2)1/2
4.37.Quasifree states in the new setup 156
4.38 .Quasifree stochastic processes 157
4.39.Bibliographical sources 158
CHAPTER 5. Stochastic integrals 160
5. l.Filtrations and adapted integrands 260
5. 2.A variant of the stochastic integral 161
5. 3.A variant of the It6 Clifford stochastic integral 163
5. 4.Generalisation 164
5. 5.Description of the measure appearing in the Wick integral 167
5. 6.Properties of the It6 Clifford integral in 5.3 168
5, 7.Representing B products as stochastic integrals 169
5. S.Representability by stochastic integrals 170
5. 9.Calculating stochastic integrals 171
5.10.General formulas expressing the stochastic integral in 5.3 272
x Table of Contents
5.11.Relation between the antisymmetric and the symmetric Fock 174
spaces
5.12.Direct proof of the result in 5.11 175
5.13.The operator stochastic integral; introduction 176
5 .14 . Example : ITTgC(H) for dimH= oo 176
5.15.Non Fock representations 177
5.16.Operator stochastic integral 177
5.17.Elements related to a direct sum decomposition 180
5.18.A formula involving v~ in F (H)= L2{v~) 182
5.19.Formulas for the operator stochastic integral 283
5.20.Statement of the existence theorem for the operator stochastic 285
integral
5.21.Other type of stochastic integral 187
5.22.The alternate sum of an operator 190
5.23.Constructing families of operators as in 5.21 191
5.24.Operators on tensor products 192
5.25.Hilbert integrals 194
5.25.Decomposition of F (H) 195
5.27.Another type of stochastic integral 198
5.28.Existence theorem 200
5.29.Formulas non involving the decomposition in direct sum of H 201
5.30.A unicity result 202
5.31.The first step towards an Ito formula 203
5.32.Partially continuous integrands 222
5.33.Proof 212
5.34.Ito formula for a product 215
5.35.Once more about the relations between the symmetric and 228
antisymmetric Fock spaces
5.36.Symmetric stochastic integrals 219
5.37.Exponential vectors in defining symmetric stochastic integrals 222
5.38.Formulas noninvolving the decomposition H= ® H. 224
iel x
5.39.Unicity result 225
5.40.ItS type formulas in the case of exponential vectors 226
5.41.Usual Stieltjes integrals 228
5.42.Example 229
5.43.Inequalities for norms of the stochastic integrals 230
5.44. Inequalities for ( ( x 231
5.45.Bibliographical sources 232
Table of Contents x1
CHAPTER 6 . Conditional mean values 234
6. 1.First definition 234
6. 2.Tomiyama s result 234
6. 3.Another related result 23S
6. 4.Normal conditional mean values 237
6. 5.The support of a conditional mean value 23s
6. 6.Conditional mean values and GNS representations 239
6. 7.Sequences of von Neumann subalgebras 242
6. 8.The martingale convergence theorem 244
6. 9.Ascending case 24s
6.10.Comments 24S
6.11.Tensor product of conditional mean values 246
6.12.Examples 247
6.13.Connection with the operator stochastic integral 24s
6.14.Example related to quasifree representations 249
6.15.The modular Tomita theory 253
6.16.The operator S 254
6.17.The modular operator A and the involution J 254
6.18.Some of the main results 255
6.19.Proof of AltMA~Xt= M 256
6.20.Constructing elements in Mv 25S
6.21.Proof of JMJ= M 260
6.22.The modular automorphisms 262
6.23.Coming back to conditional mean values 262
6. 24. Properties of E^M 263
6.25.Conditions for the existence of a conditional mean value 253
6.26.A more precise result 265
6.27.Stinespring s theorem 268
6.28.Noncommutative Markov processes 269
6.29.Markov property 270
6.30.Another concept of Markov triple 273
6.31.Construction of Markov families 274
6.32.A problem leading to the standard form of a von Neumann 275
algebra
6.33.More about the Tomita theory 276
6.34.The relative A operators 279
6.35.Examples 281
xii Table of Contents
6.36.Eliminating the dependence on the semistandard form 282
S.37.Arbitrary o—finite von Neumann algebras with a r finite 283
commutant
6.38.Properties of super and substandardness 285
6.39.(Df:Dg ) in the general case 286
6.40.Direct definition of (Df:Dg ) 287
6.41.Martingale convergence type results 288
6.42.Bibliographical sources 291
CHAPTER 7. Jordan algebras 293
7. 1.Introduction 293
7. 2.Homogenisation 293
7. 3.Jordan algebras generated by one element 294
7. 4.Idempotents and Peirce decomposition 294
7. 5.Free Jordan algebra generated by two elements 296
7. 6.The operators L . 297
7. 7.Linear independent elements in the free Jordan algebra 298
generated by two elements
7. 8.Description of the free Jordan algebra with 1 generated by two 300
elements
7. 9.Proof of the fact that the monomials in 7.7 have as linear 300
hull an algebra
7.10.The Cohn Shirshov theorem 302
7.11.Construction of a Jordan algebra 302
7.12.Exceptional Jordan algebras 305
7.13.Example:the Cayley algebra 307
7.14.Order relation in the exceptional Jordan algebra 309
7.15.Jordan von Neumann algebras 311
7.16.Spin factors 312
7.17.Traces on Jordan algebras 313
7.18.Bibliographical sources 314
REFERENCES 3I7
INDEX 349
|
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| id | DE-604.BV009895907 |
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| indexdate | 2024-07-09T17:42:46Z |
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| isbn | 0792331338 |
| language | English |
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| series2 | Mathematics and its applications |
| spelling | Cuculescu, Ioan Verfasser aut Noncommutative probability by I. Cuculescu and A. G. Oprea Dordrecht u.a. Kluwer 1994 XIV, 353 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 305 Mathematische Physik Mathematical physics Probabilities Von Neumann algebras Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Nichtkommutative Algebra (DE-588)4304013-5 gnd rswk-swf Nichtkommutative Wahrscheinlichkeit (DE-588)4362758-4 gnd rswk-swf Nichtkommutative Algebra (DE-588)4304013-5 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Nichtkommutative Wahrscheinlichkeit (DE-588)4362758-4 s DE-188 Oprea, Ana Gabriela Verfasser (DE-588)1013259432 aut Mathematics and its applications 305 (DE-604)BV008163334 305 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006553077&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cuculescu, Ioan Oprea, Ana Gabriela Noncommutative probability Mathematics and its applications Mathematische Physik Mathematical physics Probabilities Von Neumann algebras Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Nichtkommutative Algebra (DE-588)4304013-5 gnd Nichtkommutative Wahrscheinlichkeit (DE-588)4362758-4 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4304013-5 (DE-588)4362758-4 |
| title | Noncommutative probability |
| title_auth | Noncommutative probability |
| title_exact_search | Noncommutative probability |
| title_full | Noncommutative probability by I. Cuculescu and A. G. Oprea |
| title_fullStr | Noncommutative probability by I. Cuculescu and A. G. Oprea |
| title_full_unstemmed | Noncommutative probability by I. Cuculescu and A. G. Oprea |
| title_short | Noncommutative probability |
| title_sort | noncommutative probability |
| topic | Mathematische Physik Mathematical physics Probabilities Von Neumann algebras Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Nichtkommutative Algebra (DE-588)4304013-5 gnd Nichtkommutative Wahrscheinlichkeit (DE-588)4362758-4 gnd |
| topic_facet | Mathematische Physik Mathematical physics Probabilities Von Neumann algebras Wahrscheinlichkeitstheorie Nichtkommutative Algebra Nichtkommutative Wahrscheinlichkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006553077&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT cuculescuioan noncommutativeprobability AT opreaanagabriela noncommutativeprobability |