Table Of ContentLecture Notes in Mathematics 2306
Ruy Exel
David R. Pitts
Characterizing
Groupoid
C*-algebras
of Non-Hausdorff
Etale Groupoids
Lecture Notes in Mathematics
Volume 2306
Editors-in-Chief
Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
SeriesEditors
KarinBaur,UniversityofLeeds,Leeds,UK
MichelBrion,UGA,Grenoble,France
AlessioFigalli,ETHZurich,Zurich,Switzerland
AnnetteHuber,AlbertLudwigUniversity,Freiburg,Germany
DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA
IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK
AngelaKunoth,UniversityofCologne,Cologne,Germany
ArianeMézard,IMJ-PRG,Paris,France
MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg
MarkPolicott,MathematicsInstitute,UniversityofWarwick,Coventry,UK
SylviaSerfaty,NYUCourant,NewYork,NY,USA
LászlóSzékelyhidi , Institute of Mathematics, LeipzigUniversity, Leipzig,
Germany
GabrieleVezzosi,UniFI,Florence,Italy
AnnaWienhard,RuprechtKarlUniversity,Heidelberg,Germany
This series reports on new developments in all areas of mathematics and their
applications-quickly,informallyandatahighlevel.Mathematicaltextsanalysing
newdevelopmentsinmodellingandnumericalsimulationarewelcome.Thetypeof
materialconsideredforpublicationincludes:
1. Researchmonographs
2. Lecturesonanewfieldorpresentationsofanewangleinaclassicalfield
3. Summerschoolsandintensivecoursesontopicsofcurrentresearch.
Textswhichareoutofprintbutstillindemandmayalsobeconsiderediftheyfall
withinthesecategories.Thetimelinessofamanuscriptissometimesmoreimportant
thanitsform,whichmaybepreliminaryortentative.
Titles from this series are indexed by Scopus, Web of Science, Mathematical
Reviews,andzbMATH.
Ruy Exel (cid:129) David R. Pitts
Characterizing Groupoid
C*-algebras of
Non-Hausdorff Étale
Groupoids
RuyExel DavidR.Pitts
DepartamentodeMatemática DepartmentofMathematics
UniversidadeFederaldeSantaCatarina UniversityofNebraska
Florianópolis,SC,Brazil Lincoln,NE,USA
This work was supported by CNPq to Ruy Exel, Simons Foundation (http://dx.doi.org/10.13039/
100000893),#316952(forDavidR.Pitts).
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-031-05512-6 ISBN978-3-031-05513-3 (eBook)
https://doi.org/10.1007/978-3-031-05513-3
MathematicsSubjectClassification:46L05,46L45,22A22,46L55
©SpringerNatureSwitzerlandAG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse
ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Abstract
Given a not-necessarily Hausdorff, topologically free, twisted étale groupoid
(G,L), we consider its essential groupoid C*-algebra, denoted C∗ (G,L),
ess
obtained by completing C (G,L) with the smallest among all C*-seminorms
c
coincidingwiththeuniformnormonC (G(0)).TheinclusionofC*-algebras
c
(cid:2) (cid:3)
C (G(0)),C∗ (G,L)
0 ess
is then proven to satisfy a list of properties characterizing it as what we call a
weak Cartan inclusion. We then prove that every weak Cartan inclusion (cid:3)A,B(cid:4),
with B separable, is modeled by a topologically free, twisted étale groupoid, as
above.In oursecondmain result,we givea necessary andsufficientconditionfor
aninclusionofC*-algebras(cid:3)A,B(cid:4)tobemodeledbyatwistedétalegroupoidbased
on the notion of canonicalstates. A simplicity criterion for C∗ (G,L) is proven
ess
andmanyexamplesareprovided.
v
Contents
1 Introduction .................................................................. 1
2 Inclusions ..................................................................... 7
2.1 LocalModules......................................................... 7
2.2 RegularIdealsandtheLocalizingProjection ........................ 14
2.3 RegularInclusions..................................................... 20
2.4 InvariantIdeals......................................................... 24
2.5 ExtendedMultiplicationforNormalizers ............................ 25
2.6 RegularityofMaximalIdealsinRegularInclusions................. 28
2.7 Extension of Pure States, Relative Free Points
andSmoothNormalizers.............................................. 33
2.8 FreePoints............................................................. 37
2.9 FourierCoefficients.................................................... 39
2.10 OpaqueandGrayIdeals............................................... 46
2.11 TopologicallyFreeInclusions......................................... 52
2.12 Pseudo-Expectations................................................... 63
3 Groupoids..................................................................... 67
3.1 ÉtaleGroupoids........................................................ 67
3.2 TwistsandLineBundles .............................................. 70
3.3 TheC*-AlgebraofaTwistedGroupoid.............................. 72
3.4 TopologicallyFreeGroupoids ........................................ 75
3.5 TheEssentialGroupoidC*-Algebra.................................. 84
3.6 Kwasniewski and Meyer’sVersion of the Essential
GroupoidC*-Algebra ................................................. 89
3.7 TheRelativeWeylGroupoid.......................................... 95
3.8 FellBundlesOverInverseSemigroups............................... 102
3.9 TopologicalFreenessoftheWeylGroupoid
andtheMainTheorem ................................................ 111
3.10 Semi-Masas............................................................ 115
3.11 CanonicalStates ....................................................... 119
vii
viii Contents
4 ExamplesandOpenQuestions............................................. 127
4.1 Example:Non-SmoothNormalizers.................................. 127
4.2 Example:PeriodicFunctionsontheInterval......................... 131
4.3 Example:TheGrayIdealofTwistedGroupoid
C*-Algebras............................................................ 134
4.4 SomeOpenQuestions................................................. 139
5 Appendix:IsotropyProjection ............................................. 143
References......................................................................... 151
SymbolIndex..................................................................... 153
ConceptIndex .................................................................... 155
Chapter 1
Introduction
In their landmark paper [14], Feldman and Moore gave methods for describing
certainvonNeumannalgebrasusingtwistedmeasuredequivalencerelationsarising
from a Cartan MASA. Their result may be thought of as a deep and far-reaching
generalizationof the processof describingall linear mapson a finite dimensional
Hilbertspaceusingafixedorthonormalbasis.Nearlyadecadelater,Kumjian[21]
defined the notion of a diagonal A in a C*-algebra B, and provided a theory
of twisted topological equivalence relations which allowed him to classify the
inclusion A ⊆ B in terms of such twisted topological equivalence relations.
However, because Kumjian’s results require that each pure state of A uniquely
extendto a purestate on B, there are naturalsettings for which his results do not
apply. In a 2007 paper, Renault [32] introduced a notion of Cartan MASA in a
C*-algebra,andgave anelegantclassification theoremforC*-algebrascontaining
a Cartan MASA completely analagous to the Feldman-Moore results for von
Neumann algebras. Renault’s result replaces the measured equivalence relations
found in the von Neumann algebra setting with an appropriate class of twisted
locallycompactgroupoidswhichareHausdorffandétale.
This is a satisfying state of affairs. However, there still are desirable settings
where one has an inclusion of C*-algebras A ⊆ B (which we write as (cid:3)A,B(cid:4))
whichsatisfiesthesameregularityconditionasaCartansubalgebra,butwhichmay
notsatisfyotherconditionsfoundinthedefinitionofaCartanMASA.Forexample,
the crossed product constructions found in [28, Section 6.1] give a large class of
examples where A is a regular MASA in B, but which do not have a conditional
expectation E : B → A. In such settings, it is possible to use the constructions
ofKumjianandRenaulttoproduceatwistedgroupoid(G,L),buttheunderlying
groupoidG isnecessarilynon-Hausdorff(see[29,Theorem4.4]).Itisnotpossible
todescribesuchinclusions(cid:3)A,B(cid:4)usingthemethodsofRenaultandKumjian.
Thepresentworkhasseveralmaingoals.Thefirstistogiveagroupoidmodelfor
inclusionsof C*-algebrassatisfying certain propertiesinspired by,althoughmuch
weakerthan,thepropertiescharacterizingCartaninclusions,asdefinedbyRenault
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1
R.Exel,D.R.Pitts,CharacterizingGroupoidC*-algebrasofNon-Hausdorff
ÉtaleGroupoids,LectureNotesinMathematics2306,
https://doi.org/10.1007/978-3-031-05513-3_1
2 1 Introduction
in[32].Thesecondmaingoalisintimatelyrelatedtothefirst:wedefinetheessential
groupoidC*-algebraassociatedto anotnecessarilyHausdorff,topologicallyfree,
étale groupoid G, equipped with a Fell line bundle L. Taken together, these two
results provide a classification for weak Cartan inclusions in the spirit of Renault
andKumjianusingtwisted,butpossiblynotHausdorff,étalegroupoids.Ourresults
applyinconsiderablegenerality,andincludetheregularMASAinclusions(cid:3)A,B(cid:4)
mentionedinthepreviousparagraph,buttheyalsoapplytomanyregularinclusions
whereAisanabeliansubalgebraofBwhichisnotmaximalabelian.Letusbemore
specific.
LetGbeatopologicallyfree,étalegroupoid,equippedwithaFelllinebundleL.
WhilewedonotassumeG isHausdorff,wedoassumeitsunitspaceisHausdorff.
Well-knownconstructionsassociatetwonormsonthe∗-algebraC (G,L)formed
c
bythecontinuouslocalcross-sectionsofL:themaximumandthereducednorms,
respectively(see(3.3.3)fortheprecisedefinition).Theircompletionsyieldthefull
andthereducedtwistedgroupoidC*-algebrasofG,usuallydenotedbyC∗(G,L)
andC∗ (G,L).
red
When G is topologically free, that is, when the units of G with trivial isotropy
form a dense set, we show in Theorem 3.5.6 that there is a very canonical third
choice, namely the smallest (cid:4)amon(cid:5)g all C*-seminorms on Cc(G,L) coinciding
with the uniform norm on C G(0) . The completion of C (G,L) under this C*-
c c
seminormistheessentialgroupoidC*-algebrareferredtoabove,whichwedenote
by
C∗ (G,L).
ess
Inspiredbyanearlierversionofthepresentwork,KwasniewskiandMeyer[23]
introduced a different notion of essential C*-algebra which applies to groupoids
whicharenotnecessarilytopologicallyfree.Theirconstructionisdescribedindetail
inSect.3.6.
In case G is Hausdorff, C∗ (G,L) turns out to be isomorphic to C∗ (G,L),
ess red
butotherwisethisisnotnecessarilyso.Infactweshowin(2.10.11.iii)that,inthe
generalcase,thereexistsalargestideal
(cid:2) (cid:2)C∗ (G,L),
red red
(cid:4) (cid:5)
havinga trivialintersectionwithC G(0) , whichwecallthegrayideal,andsuch
0
that
C∗ (G,L)(cid:7)C∗ (G,L)/(cid:2) .
ess red red
ThepairofC*-algebras
(cid:2) (cid:4) (cid:5) (cid:3)
C G(0) ,C∗ (G,L) (1.1)
0 ess
