Combinatorics of Free Probability Theory（英.PDF

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1、CombinatoricsoffreeprobabilitytheoryRolandSpeicherQueen’sUniversity,Kingston,OntarioAbstract.Thefollowingmanuscriptwaswrittenformylec-turesatthespecialsemester’Freeprobabilitytheoryandoperatorspaces’,IHP,Paris,1999.AcomplementaryseriesoflectureswasgivenbyA.Nica.Itisplannedtocom

2、binebothmanuscriptsintoabookA.NicaandR.Speicher:LecturesontheCombinatoricsofFreeProbability,whichwillbepublishedbyCambridgeUniversityPressContentsPart1.Basicconcepts5Chapter1.Basicconceptsofnon-commutativeprobabilitytheory71.1.Non-commutativeprobabilityspacesanddistributions71.

3、2.Haarunitariesandsemicircularelements10Chapter2.Freerandomvariables172.1.Definitionandbasicpropertiesoffreeness172.2.Thegroupalgebraofthefreeproductofgroups212.3.ThefullFockspace232.4.Constructionoffreeproducts27Part2.Cumulants35Chapter3.Freecumulants373.1.Motivation:Freecentra

4、llimittheorem373.2.Non-crossingpartitions463.3.PosetsandM¨obiusinversion503.4.Freecumulants52Chapter4.Fundamentalpropertiesoffreecumulants594.1.Cumulantswithproductsasentries594.2.Freenessandvanishingofmixedcumulants66Chapter5.Sumsandproductsoffreevariables755.1.Additivefreecon

5、volution755.2.Descriptionoftheproductoffreevariables845.3.Compressionbyafreeprojection875.4.Compressionbyafreefamilyofmatrixunits91Chapter6.R-diagonalelements956.1.DefinitionandbasicpropertiesofR-diagonalelements956.2.Theanti-commutatoroffreevariables1036.3.PowersofR-diagonalele

6、ments105Chapter7.FreeFisherinformation1097.1.Definitionandbasicproperties10934CONTENTS7.2.Minimizationproblems112Part3.Appendix123Chapter8.Infinitelydivisibledistributions1258.1.Generalfreelimittheorem1258.2.Freelyinfinitelydivisibledistributions127Part1BasicconceptsCHAPTER1Basicc

7、onceptsofnon-commutativeprobabilitytheory1.1.Non-commutativeprobabilityspacesanddistributionsDefinition1.1.1.1)Anon-commutativeprobabilityspace(A,ϕ)consistsofaunitalalgebraAandalinearfunctionalϕ:A→C;ϕ(1)=1.2)IfinadditionAisa∗-algebraandϕfulfillsϕ(a∗)=ϕ(a)foralla∈A,thenwecall(A,ϕ

8、)a∗-probabilityspace.3)IfAisaunitalC∗-algebraandϕastat