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| Introduction | |
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| Basic concepts | |
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| Non-commutative probability spaces and distributions | |
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| Non-commutative probability spaces | |
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| *-distributions (case of normal elements) | |
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| *-distributions (general case) | |
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| Exercises | |
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| A case study of non-normal distribution | |
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| Description of the example | |
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| Dyck paths | |
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| The distribution of a + a[superscript *] | |
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| Using the Cauchy transform | |
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| Exercises | |
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| C[superscript *]-probability spaces | |
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| Functional calculus in a C[superscript *]-algebra | |
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| C[superscript *]-probability spaces | |
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| *-distribution, norm and spectrum for a normal element | |
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| Exercises | |
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| Non-commutative joint distributions | |
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| Joint distributions | |
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| Joint *-distributions | |
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| Joint *-distributions and isomorphism | |
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| Exercises | |
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| Definition and basic properties of free independence | |
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| The classical situation: tensor independence | |
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| Definition of free independence | |
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| The example of a free product of groups | |
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| Free independence and joint moments | |
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| Some basic properties of free independence | |
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| Are there other universal product constructions? | |
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| Exercises | |
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| Free product of *-probability spaces | |
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| Free product of unital algebras | |
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| Free product of non-commutative probability spaces | |
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| Free product of *-probability spaces | |
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| Exercises | |
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| Free product of C[superscript *]-probability spaces | |
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| The GNS representation | |
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| Free product of C[superscript *]-probability spaces | |
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| Example: semicircular systems and the full Fock space | |
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| Exercises | |
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| Cumulants | |
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| Motivation: free central limit theorem | |
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| Convergence in distribution | |
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| General central limit theorem | |
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| Classical central limit theorem | |
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| Free central limit theorem | |
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| The multi-dimensional case | |
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| Conclusion and outlook | |
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| Exercises | |
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| Basic combinatorics I: non-crossing partitions | |
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| Non-crossing partitions of an ordered set | |
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| The lattice structure of NC(n) | |
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| The factorization of intervals in NC | |
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| Exercises | |
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| Basic combinatorics II: Mobius inversion | |
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| Convolution in the framework of a poset | |
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| Mobius inversion in a lattice | |
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| The Mobius function of NC | |
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| Multiplicative functions on NC | |
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| Functional equation for convolution with [Mu subscript n] | |
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| Exercises | |
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| Free cumulants: definition and basic properties | |
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| Multiplicative functionals on NC | |
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| Definition of free cumulants | |
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| Products as arguments | |
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| Free independence and free cumulants | |
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| Cumulants of random variables | |
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| Example: semicircular and circular elements | |
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| Even elements | |
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| Classical cumulants | |
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| Exercises | |
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| Sums of free random variables | |
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| Free convolution | |
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| Analytic calculation of free convolution | |
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| Proof of the free central limit theorem via R-transform | |
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| Free Poisson distribution | |
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| Compound free Poisson distribution | |
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| Exercises | |
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| More about limit theorems and infinitely divisible distributions | |
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| Limit theorem for triangular arrays | |
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| Cumulants of operators on Fock space | |
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| Infinitely divisible distributions | |
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| Conditionally positive definite sequences | |
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| Characterization of infinitely divisible distributions | |
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| Exercises | |
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| Products of free random variables | |
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| Multiplicative free convolution | |
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| Combinatorial description of free multiplication | |
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| Compression by a free projection | |
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| Convolution semigroups ([Mu superscript Characters not reproduciblet])[subscript tge]1 | |
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| Compression by a free family of matrix units | |
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| Exercises | |
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| R-diagonal elements | |
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| Motivation: cumulants of Haar unitary elements | |
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| Definition of R-diagonal elements | |
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| Special realizations of tracial R-diagonal elements | |
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| Product of two free even elements | |
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| The free anti-commutator of even elements | |
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| Powers of R-diagonal elements | |
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| Exercises | |
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| Transforms and models | |
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| The R-transform | |
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| The multi-variable R-transform | |
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| The functional equation for the R-transform | |
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| More about the one-dimensional case | |
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| Exercises | |
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| The operation of boxed convolution | |
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| The definition of boxed convolution, and its motivation | |
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| Basic properties of boxed convolution | |
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| Radial series | |
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| The Mobius series and its use | |
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| Exercises | |
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| More on the one-dimensional boxed convolution | |
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| Relation to multiplicative functions on NC | |
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| The S-transform | |
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| Exercises | |
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| The free commutator | |
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| Free commutators of even elements | |
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| Free commutators in the general case | |
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| The cancelation phenomenon | |
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| Exercises | |
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| R-cyclic matrices | |
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| Definition and examples of R-cyclic matrices | |
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| The convolution formula for an R-cyclic matrix | |
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| R-cyclic families of matrices | |
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| Applications of the convolution formula | |
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| Exercises | |
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| The full Fock space model for the R-transform | |
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| Description of the Fock space model | |
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| An application: revisiting free compressions | |
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| Exercises | |
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| Gaussian random matrices | |
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| Moments of Gaussian random variables | |
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| Random matrices in general | |
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| Selfadjoint Gaussian random matrices and genus expansion | |
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| Asymptotic free independence for several independent Gaussian random matrices | |
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| Asymptotic free independence between Gaussian random matrices and constant matrices | |
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| Unitary random matrices | |
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| Haar unitary random matrices | |
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| The length function on permutations | |
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| Asymptotic freeness for Haar unitary random matrices | |
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| Asymptotic freeness between randomly rotated constant matrices | |
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| Embedding of non-crossing partitions into permutations | |
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| Exercises | |
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| Notes and comments | |
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| References | |
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| Index | |