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Lecture 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). 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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