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Springer Monographs in Mathematics Ilijas Farah Combinatorial Set Theory of C*-algebras Springer Monographs in Mathematics Editors-in-Chief IsabelleGallagher,Paris,France MinhyongKim,Oxford,UK SeriesEditors SheldonAxler,SanFrancisco,USA MarkBraverman,Princeton,USA MariaChudnovsky,Princeton,USA TadahisaFunaki,Tokyo,Japan SinanC.Güntürk,NewYork,USA ClaudeLeBris,MarnelaVallee,France PascalMassart,Orsay,France AlbertoA.Pinto,Porto,Portugal GabriellaPinzari,Napoli,Italy KenRibet,Berkeley,USA RenéSchilling,Dresden,Germany PanagiotisSouganidis,Chicago,USA EndreSüli,Oxford,UK ShmuelWeinberger,Chicago,USA BorisZilber,Oxford,UK Thisseriespublishesadvancedmonographsgivingwell-writtenpresentationsofthe “state-of-the-art”infieldsofmathematicalresearchthathaveacquiredthematurity neededforsuchatreatment.Theyaresufficientlyself-containedtobeaccessibleto morethanjusttheintimatespecialistsofthesubject,andsufficientlycomprehensive toremainvaluablereferencesformanyyears.Besidesthecurrentstateofknowledge initsfield,anSMMvolumeshouldideallydescribeitsrelevancetoandinteraction with neighbouring fields of mathematics, and give pointers to future directions of research. Moreinformationaboutthisseriesathttp://www.springer.com/series/3733 Ilijas Farah Combinatorial Set Theory of C*-algebras 123 IlijasFarah DepartmentofMathematicsandStatistics YorkUniversity Toronto,ON,Canada MatematicˇkiInstitutSANU Beograd,Serbia ISSN1439-7382 ISSN2196-9922 (electronic) SpringerMonographsinMathematics ISBN978-3-030-27091-9 ISBN978-3-030-27093-3 (eBook) https://doi.org/10.1007/978-3-030-27093-3 MathematicsSubjectClassification:03E75,03E65,03E05,46L05,46L30,46L40,03C20,03C98 ©SpringerNatureSwitzerlandAG2019 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,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Tothememoryofmymother,VeraGajic´ andmyuncle,RadovanGajic´. Chapter17isdedicatedtoColette. Anothersmart,generous,andwittysoul gonetoosoon. Reader, IHereputintothyHands,whathasbeenthediversionofsomeofmyidleandheavy Hours:Ifithasthegoodlucktoprovesoofanyofthine,andthouhastbuthalfsomuch Pleasureinreading,asIhadinwritingit,thouwiltaslittlethinkthyMoney,asIdomy Pains, ill bestowed. Mistake not this, for a Commendation of my Work; nor conclude, becauseIwaspleasedwiththedoingofit,thatthereforeIamfondlytakenwithitnow itisdone. JohnLocke,TheEpistletotheReader AnEssayConcerningHumanUnderstanding,1660 Well, that’s about it for tonight ladies and gentlemen, but remember if you’ve enjoyed watchingtheshowjusthalfasmuchaswe’veenjoyeddoingit,thenwe’veenjoyedittwice asmuchasyou.Ha,ha,ha. MontyPython’sFlyingCircus,Episode23,‘ScottoftheAntarctic’,1970 Preface This book is shorter than In Search of Lost Time, is easier to read than Principia Mathematica, and has more mathematical content than War and Peace.1,2 It ∗ provides an introduction to set-theoretic methods in the field of C -algebras, functionalanalysis,andgenerallargemetricalgebraicstructures.Themainobjects ∗ ofthestudyarethetwoclassesofC -algebras:(1)nonseparablebutusuallynuclear, ∗ and even approximately finite, C -algebras and (2) properties of massive quotient ∗ C -algebras such as coronas, ultraproducts, and relative commutants of separable subalgebrasofmassivealgebras. Whilewritingthisbook,Ihadinmindfourtypesofreaders: ∗ 1. GraduatestudentswhohadalreadytakenanintroductorycourseinC -algebras andwouldliketolearnset-theoreticmethods 2. Graduatestudentswhohadalreadytakenanintroductorycourseincombinatorial ∗ settheoryandwouldliketoapplytheirknowledgetoC -algebras 3. Graduate students who had taken a first course in functional analysis, and possibly a first course in mathematical logic (the latter can be replaced by ‘sufficient mathematical maturity’), and are interested in learning about set- ∗ theoreticmethodsinfunctionalanalysis,andC -algebrasinparticular 4. Maturemathematiciansinterestedinlearningaboutapplicationsofsettheoryto ∗ C -algebras This book can be used as a text for an advanced two-semester graduate course. Alternatively,onecanuseChapters1–8,Section9.2,andChapters10and11fora ∗ one-semestercourseonconstructionsofnonseparableC -algebras. 1If you thought there wasn’t much mathematical content in War and Peace then you haven’t made it asfar asthe second epilogue, where the following sentence canbe found: ‘Arriving at infinitesimals,mathematics,themostexactofsciences,abandonstheprocessofanalysisandenters onthenewprocessoftheintegrationofunknown,infinitelysmall,quantities’.Tolstoyproceeded tospeculateonapplicationsofcalculustohistory.Thiswaswritteninthe1860s,barely10years afterthebirthofRiemann’sintegralandfull80yearsbeforeAsimov’s‘Foundation’. 2...andsomeofthejokeswerenotstolenfromDouglasAdams. vii viii Preface AnotheralternativeistouseChapters1and2,andallofPartIII(exceptSection 12.5,whichreliesonChapter5)foraone-semestercourseonset-theoreticaspects oftheCalkinalgebra,othercoronas,andultraproducts. ∗ YetanotherpossibilityforaminicourseonrepresentationsofC -algebraswould betouseonlyChapters1–5.Thisoptioninvolvesnosettheory,butitcoversaspects ∗ oftherepresentationtheoryofC -algebrasnotcoveredelsewhere. Ifacourseisgiventostudentswithasolidbackgroundinsettheory,thenallof Chapter6andpartsofChapters7and8shouldbeomitted. In the dual situation, when the audience consists of students with a solid ∗ backgroundinC -algebras,Chapters1–3canbeomitted. Acknowledgements First of all, I should thank Paul Szeptycki and Ray Jayawardhana for kindly arranging a half-course teaching reduction in the fall 2016 semester that greatly helpedinthepreparationofthisbook.IwouldalsoliketothankBruceBlackadar, Sarah L. Browne, George A. Elliott, Saeed Ghasemi, Eusebio Gardella, Bradd Hart,Se–JinSamKim,AkitakaKishimoto,BorišaKuzeljevic´,PaulLarson,Mikkel Munkholm,NarutakaOzawa,N.ChristopherPhillips,AssafRinot,RalfSchindler, HannesThiel,AndreaVaccaro,AlessandroVignati,andBeatrizZamora–Avilesfor their critical remarks on the early drafts. I am indebted to Bruce Blackadar, Se– JinSamKim,andNarutakaOzawawhoprovidedasignificantmathematicalinput, includingsimpleproofsofLemma2.3.11,Example2.4.5,andLemma3.1.13(B.B.) and Lemma 1.10.7, Theorem 1.10.8, Lemma 3.2.10, and Theorem 3.2.9, as well as Lemma 3.4.3 and its proof (N.O.). Special thanks to Chris Schafhauser for the occasionalilluminatingremark.Iamindebtedtomytwowonderfuleditors:Eugene Ha,whomademestartthisproject(heisforgiven),andElizabethLoew,forexpertly andpatientlynavigatingmethroughoutthisendeavour.Whileweareattheeditors, manythankstoAssafRinotforsuggestingthatItryusingTexpad;itmadewriting the last few sections of this text feel even more drastically different than Richard Strauss’swriting‘AnAlpineSymphony’.Mostofthisbookhadbeenwrittenusing certain well-known LATEX editor that shall remain nameless.3 Last but not least, I owespecialthankstomydaughter,Gala,forherimpeccableandgenerouslinguistic support.4 Toronto,ON,Canada IlijasFarah July4,2019 3Thisoccasionallydidfeellike‘ajobthat,whenall’ssaidanddone,amusesmeevenlessthan chasingcockroaches’—whichishowStraussdescribedtheprocessofwritingthesaidpiece. 4I claim credit for the remaining mistakes, obscurities, and all missing or misplaced articles in particular. Contents ∗ PartI C -algebras ∗ 1 C -algebras,Abstract,andConcrete..................................... 3 ∗ 1.1 OperatorTheoryandC -algebras ................................... 3 ∗ 1.2 C -algebras........................................................... 6 ∗ 1.3 AbelianC -algebras.................................................. 11 ∗ 1.4 ElementsofC -algebras:ContinuousFunctionalCalculus........ 15 1.5 Projections............................................................ 18 ∗ 1.6 PositivityinC -algebras............................................. 21 1.7 PositiveLinearFunctionals.......................................... 25 1.8 ApproximateUnitsandStrictlyPositiveElements................. 30 1.9 Quasi-CentralApproximateUnits................................... 31 1.10 TheGNSConstruction............................................... 34 1.11 Exercises.............................................................. 38 ∗ 2 ExamplesandConstructionsofC -algebras............................ 47 2.1 PuttingtheBuildingBlocksTogether............................... 47 ∗ 2.2 Finite-DimensionalC -algebras,AFAlgebras,andUHFAlgebras 50 ∗ 2.3 UniversalC -algebrasDefinedbyBoundedRelations............. 54 2.4 TensorProducts,GroupAlgebras,andCrossedProducts.......... 59 2.5 QuotientsandLifts................................................... 66 ∗ 2.6 AutomorphismsofC -algebras...................................... 68 2.7 RealRankZero....................................................... 70 2.8 Exercises.............................................................. 72 ∗ 3 RepresentationsofC -algebras........................................... 79 3.1 TopologiesonB(H)andvonNeumannAlgebras................. 80 3.2 CompletelyPositiveMaps........................................... 88 3.3 AveragingandConditionalExpectation............................. 93 3.4 TransitivityTheoremsI:TheKadisonTransitivityTheorem ...... 96 3.5 Transitivity Theorems II: Direct Sums of Irreducible Representations....................................................... 100 ix