Operator Algebras and Applications: The Abel Symposium 2015

345 Pages·2016·2.61 MB·English

by  Toke M. Carlsen

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
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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
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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
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