Abel Symposia 12 Toke M. Carlsen Nadia S. Larsen Sergey Neshveyev Christian Skau E ditors Operator Algebras and Applications The Abel Symposium 2015 ABEL SYMPOSIA Edited by the Norwegian Mathematical Society Moreinformationaboutthisseriesathttp://www.springer.com/series/7462 5. 1 0 2 ms um osiTo pw me Syndr elA Abby en he tk atta so ntot cipaPh arti P Toke M. Carlsen • Nadia S. Larsen (cid:129) Sergey Neshveyev (cid:129) Christian Skau Editors Operator Algebras and Applications The Abel Symposium 2015 123 Editors TokeM.Carlsen NadiaS.Larsen DepartmentofScienceandTechnology DepartmentofMathematics UniversityoftheFaroeIslands UniversityofOslo Tórshavn,FaroeIslands Oslo,Norway SergeyNeshveyev ChristianSkau DepartmentofMathematics DepartmentofMathematicalSciences UniversityofOslo NorwegianUniversityofScience Oslo,Norway andTechnology Trondheim,Norway ISSN2193-2808 ISSN2197-8549 (electronic) AbelSymposia ISBN978-3-319-39284-4 ISBN978-3-319-39286-8 (eBook) DOI10.1007/978-3-319-39286-8 LibraryofCongressControlNumber:2016945020 MathematicsSubjectClassification(2010):46Lxx,37Bxx,19Kxx ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Foreword TheNorwegiangovernmentestablishedtheAbelPrizeinmathematicsin2002,and the first prize was awarded in 2003. In addition to honoring the great Norwegian mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstanding scientificworkinthefieldofmathematics,theprizeshallcontributetowardraising the status of mathematics in society and stimulate the interest for science among schoolchildrenandstudents.Inkeepingwiththisobjective,theNielsHenrikAbel Board has decided to finance annual Abel Symposia. The topic of the symposia maybeselectedbroadlyintheareaofpureandappliedmathematics.Thesymposia should be at the highest international level and serve to build bridges between thenationalandinternationalresearchcommunities.TheNorwegianMathematical Societyisresponsiblefortheevents.Ithasalsobeendecidedthatthecontributions fromthese symposia shouldbe presentedin a series of proceedings,and Springer Verlag has enthusiastically agreed to publish the series. The Niels Henrik Abel Boardisconfidentthattheserieswillbeavaluablecontributiontothemathematical literature. ChairoftheNielsHenrikAbelBoard HelgeHolden v Preface Måletforvårvitenskaperpådenenesideåoppnånye resultater,ogpådenannensideåsammenfatteogbelyse tidligereresultatersettfraethøyereståsted. SophusLie1 TheAbelSymposium2015focusedonoperatoralgebrasandthewideramifications the field has spawned.Operator algebrasform a branchof mathematicsthat dates back to the work of John von Neumann in the 1930s. Operator algebras were proposedasaframeworkforquantummechanics,withtheobservablesreplacedby self-adjointoperatorsonHilbertspacesandclassicalalgebrasoffunctionsreplaced by algebrasof operators.Spectacularbreakthroughsby the Fields medalistsAlain Connes and Vaughan Jones marked the beginning of an impressive development, inthecourseofwhichoperatoralgebrasestablishedimportanttieswithotherareas ofmathematics,suchasgeometry,K-theory,numbertheory,quantumfieldtheory, dynamicalsystems,andergodictheory. The first Abel Symposium, held in 2004, also focused on operator algebras. It is interesting to see the developmentand the remarkable advances that have been made in this field in the years since, which strikingly illustrate the vitality of the field. The Abel Symposium 2015 took place on the ship Finnmarken, part of the Coastal Express line (the Norwegian Hurtigruten), which offered a spectacular venue.TheshipleftBergenonAugust7andarrivedatitsfinaldestination,Harstad intheLofotenIslands,onAugust11.Thescenerytheparticipantssawontheway northwasmarvelous;forexample,theshipsailedintoboththeGeirangerfjordand Trollfjord. There were altogether 26 talks given at the symposium. In keeping with the organizers’ goals, there was no single main theme for the symposium, but rather a variety of themes, all highlighting the richness of the subject. It is perhaps appropriate to draw attention to one of the themes of the talks, which is the classificationprogramfornuclearC(cid:2)-algebras.Infact,atrulymajorbreakthrough 1“The goal of our science is on the one hand to obtain new results, and on the other hand to summarizeandilluminateearlierresultsasseenfromahighervantagepoint.”SophusLie vii viii Preface inthisareaoccurredjustafewweeksbeforetheAbelSymposium2015—amazing timing! Some of the protagonists in this effort—one that has stretched over more than25yearsandhasinvolvedmanyresearchers—gavetalksonthisverytopicat thesymposium.ThesurveyarticlebyWilhelmWinterinthisproceedingsvolume offersa panoramicviewofthedevelopmentsin theclassificationprogramleading uptothebreakthroughmentionedabove. Alain Connes and Vaughan Jones were also among the participants, and they gavetalks on topicsranging,respectively,from gravity and the standardmodelin physics to subfactors, knot theory, and the Thompson group, thus illustrating the broadramificationsofoperatoralgebrasinmodernmathematics. OlaBratteliandUffeHaagerup,twomaincontributorstothetheoryofoperator algebras,tragicallypassedawayinthemonthsbeforethesymposium.Theirlegacy was commemorated and honored in a talk by Erling Størmer. One of the articles inthisvolumeisbyUffeHaagerup,anditspublicationwasmadepossiblewiththe helpofthreeofHaagerup’scolleaguesfromtheUniversityofCopenhagen,towhom hehadprivatelycommunicatedtheresultsshortlybeforehisuntimelypassing. Thearticlesinthisvolumeareorganizedalphabeticallyratherthanthematically. Someareresearcharticlesthatpresentnewresults,othersaresurveysthatcoverthe development of a specific line of research, and yet others offer a combination of survey and research. These contributionsoffer a multifaceted portrait of beautiful mathematics that both newcomers to the field of operator algebras and seasoned researchersalikewillappreciate. Tórshavn,FaroeIslands TokeM.Carlsen Oslo,Norway NadiaS.Larsen Oslo,Norway SergeyNeshveyev Trondheim,Norway ChristianSkau April2016 Contents C(cid:2)-TensorCategoriesandSubfactorsforTotallyDisconnected Groups............................................................................. 1 YukiAranoandStefaanVaes DecomposableApproximationsRevisited..................................... 45 NathanialP.Brown,JoséR.Carrión,andStuartWhite ExoticCrossedProducts......................................................... 61 AlcidesBuss,SiegfriedEchterhoff,andRufusWillett OnHongandSzyman´ski’sDescription ofthePrimitive-IdealSpaceofaGraphAlgebra............................. 109 TokeM.CarlsenandAidanSims CommutatorInequalitiesviaSchurProducts ................................ 127 ErikChristensen C(cid:2)-AlgebrasAssociatedwithAlgebraicActions.............................. 145 JoachimCuntz ANewLookatC(cid:2)-SimplicityandtheUniqueTraceProperty ofaGroup......................................................................... 161 UffeHaagerup EquilibriumStatesonGraphAlgebras........................................ 171 AstridanHuefandIainRaeburn SemigroupC(cid:2)-Algebras......................................................... 185 XinLi TopologicalFullGroupsofÉtaleGroupoids.................................. 197 HirokiMatui TowardsaClassificationofCompactQuantumGroupsofLieType ...... 225 SergeyNeshveyevandMakotoYamashita ix