Table Of Content

Lectures on von Neumann Algebras ThefirsteditionofthisbookwasfirstpublishedinRomanianin1975,andthentranslated intoEnglishin1979.Theauthorshaverevisedthesecondeditionwiththeadditionofseveral topics. This valuable text discusses the fundamental concepts of von Neumann algebras including bounded and unbounded linear operators in Hilbert spaces, the von Neumann bicommutanttheorem,theKaplanskydensitytheorem,reducedandinducedvonNeumann algebras, the geometry of projections and the classification of von Neumann algebras, the typerelationwiththecommutant,adetailedstudyofthepredualofavonNeumannalgebra, the finite von Neumann algebras, the von Neumann BT-theorem and the Tomita theory of standard forms of von Neumann Algebras. Each chapter of the book has a section of exercisesandasectionofcomments. Therevisedtextcoversnewmaterialincludingfactorsassociatedtodiscretegroups,the first two examples of non-isomorphic type II factors, the original Murray–von Neumann 1 proofoftheexistenceofthetraceonfactorsoftypeII ,theDixmieraveragingtheoremin 1 caseoffactors,theproofoftheBT-theoremofvonNeumann,adetailedtheoryofstandard forms for abelian, finite and semi-finite von Neumann algebras (these being the classical cases),aswellasTomita’stheoremforvonNeumannalgebrashavingacyclicandseparating vector,asapreludetothegeneraltheoryofstandardforms.AlsotheJonesindexisbriefly describedinasectionofcomments. The book is written in an easy to understand manner, with complete proofs, the only prerequisitesbeingsomeelementaryunderstandingoffunctionalanalysis. S¸erbanValentinStrătilăisaprofessorintheDepartmentofMathematicsattheUniversity ofBucharestandseniorresearcherattheInstituteofMathematics‘SimionStoilow’ofthe RomanianAcademy.Hisareasofresearchincludefunctionalanalysis,harmonicanalysis, operatoralgebrasandrepresentationtheory. László Zsidó, former researcher at the Institute of Mathematics ‘Simion Stoilow’ of the RomanianAcademy,isaprofessorintheDepartmentofMathematicsattheUniversityof Rome ‘Tor Vergata’. His areas of research include functional analysis, harmonic analysis, operatoralgebras,generalizedfunctionsandergodictheory. Both authors received the ‘Simion Stoilow’ Prize of the Romanian Academy in 1975 for thisbook. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/EDEB93BAEFB2CFF1F4806E1EBC28E161 CAMBRIDGE–IISc SERIES Cambridge–IISc Series aims to publish the best research and scholarly work on different areas of scienceandtechnologywithemphasisoncutting-edgeresearch. The books will be aimed at a wide audience including students, researchers, academicians and professionalsandwillbepublishedunderthreecategories:researchmonographs,centenarylectures andlecturenotes. Theeditorialboardhasbeenconstitutedwithexpertsfromarangeofdisciplinesindiversefieldsof engineering,scienceandtechnologyfromtheIndianInstituteofScience,Bangalore. IIScPressEditorialBoard: AmareshChakrabarti,ProfessorandChair,CentreforProductDesignandManufacturing DiptimanSen,Professor,CentreforHighEnergyPhysics PrabalKumarMaiti,Professor,DepartmentofPhysics S.P.Arun,AssociateProfessor,CentreforNeuroscience Titlesinprintinthisseries: • ContinuumMechanics:FoundationsandApplicationsofMechanicsbyC.S.Jog • FluidMechanics:FoundationsandApplicationsofMechanicsbyC.S.Jog • NoncommutativeMathematicsforQuantumSystemsbyUweFranzandAdamSkalski • Mechanics,WavesandThermodynamicsbySudhirRanjanJain • FiniteElements:TheoryandAlgorithmsbySashikumaarGanesanandLutzTobiska • Ordinary Differential Equations: Principles and Applications by A. K. Nandakumaran, P.S. DattiandRajuK.George • BiomaterialsScienceandTissueEngineering:PrinciplesandMethodsbyBikramjitBasu • Knowledge Driven Development: Bridging Waterfall and Agile Methodologies by Manoj KumarLal Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/EDEB93BAEFB2CFF1F4806E1EBC28E161 Cambridge IISc Series Lectures on von Neumann Algebras Second Edition S¸erban Valentin Strătilă László Zsidó Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/EDEB93BAEFB2CFF1F4806E1EBC28E161 “9781108496841FM_pi-xii” — 2019/3/6 — 16:45 — page iv — #4 UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,vic3207,Australia 314to321,3rdFloor,PlotNo.3,SplendorForum,JasolaDistrictCentre,NewDelhi110025,India 79AnsonRoad,#06-04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108496841 ©CambridgeUniversityPress2019 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. FirstRomanianeditionpublishedbyEdituraAcademiei,1975 FirstEnglishtranslationpublishedbyEdituraAcademieiandAbacusPress,1979 SecondeditionpublishedbyCambridgeUniversityPress,2019 PrintedinIndia AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData Names:Strat̆ila,̆ S¸erban,1943-author.|Zsidó,László,author. Title:LecturesonvonNeumannalgebras/S¸erbanValentinStrat̆ila,̆ LászlóZsidó. Othertitles:LectiidealgebrevonNeumann.English Description:2ndedition.|Cambridge;NewYork,NY:CambridgeUniversity Press,2019.|Series:Cambridge–IIScseries|Originallypublishedin Romanianin1975.FirstEnglishtranslation,1979.|Includes bibliographicalreferencesandindexes. Identifiers:LCCN2019001647|ISBN9781108496841(hardback:alk.paper) Subjects:LCSH:VonNeumannalgebras.|Hilbertspace. Classification:LCCQA326.S76132019|DDC512/.556–dc3 LCrecordavailableathttps://lccn.loc.gov/2019001647 ISBN978-1-108-49684-1Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/EDEB93BAEFB2CFF1F4806E1EBC28E161 To the memory of Sanda Strătilă (1944–2011) Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:40, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/27662EA14515A0D4B3C1F36DBEFD315C Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:40, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/27662EA14515A0D4B3C1F36DBEFD315C CONTENTS Prefaceto2ndedition ix Prefaceto1stedition xi Introduction 1 1 Topologiesonspacesofoperators 4 2 BoundedlinearoperatorsinHilbertspaces 15 3 vonNeumannalgebras 60 4 ThegeometryofprojectionsandtheclassificationofvonNeumannalgebras 87 5 Linearformsonalgebrasofoperators 115 6 RelationsbetweenavonNeumannalgebraanditscommutant 145 7 FinitevonNeumannalgebras 162 8 Spatialisomorphismsandrelationsbetweentopologies 197 9 UnboundedlinearoperatorsinHilbertspaces 215 10 ThetheoryofstandardvonNeumannalgebras 276 Appendix 399 Bibliography 403 Subjectindex 422 Notationindex 426 vii Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/43BF5B838A9A2CD95B158381FAEDB1AE Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/43BF5B838A9A2CD95B158381FAEDB1AE PREFACE TO 2nd EDITION This book is an elementary self-contained exposition, with complete proofs, of the fundamentalsofthetheoryofvonNeumannalgebras.Althoughthistheoryhasconsiderably evolvedinthelast40yearssincethefirsteditionofthisbookwasprinted,thefundamentals of the theory remain the same. Besides many achievements in the operator algebras themselves, new connected powerful mathematical theories have appeared, such as the NoncommutativeGeometryofAlainConnesandtheFreeProbabilityTheoryofDan-Virgil Voiculescu,andthismakesitevenmorenecessarytohaveanelementarypresentationofthe fundamentalsofthetheory.Thefirsteditionwasintensivelyusedinthemainmathematical centresintheworldandwasverysoonoutofprint,andwereceivedmanydemandsforthe bookthatwecouldnotfulfil. Inthisneweditionwehaveadded,amongotherthings,somematerialconcerningfactors associatedtoICCgroups,theBT-TheoremofJ.vonNeumann,theMurray–vonNeumann proofoftheexistenceofthetraceontypeII factors,theAveragingTheoremofJ.Dixmier 1 incaseoffactors,andapreliminarypresentationofTomita’sTheoremfirstintheclassical caseofacyclictracevectorandtheninanintroductiontoChapterX,inthecaseofacyclic andseparatingvector.Inthecommentssections,wehavequotedsomenewresults,among whicharetheJonesindexofsubfactorsandtheConnesclassificationofhyperfinitefactors. The extensive bibliography of the first edition has been considerably reduced, mainly to quotedbooksorarticles. ThemainconsequencesofTomita’stheory,presentedinChapter10,arethesubjectofa direct continuation of the present book, namely Modular Theory in Operator Algebras by S¸.Strat̆ ilă(1981). We are grateful to V. Sunder (Chennai Institute of Mathematical Sciences, India) and Gadadhar Misra (Indian Institute of Science, Bangalore, India) for proposing a revised edition of this book, especially to Gadadhar Misra for the help in the LaTeX conversion ofthebook,andtotheCambridgeUniversityPressforpublishingthissecondeditionofthe book. ix Downloaded from https://www.cambridge.org/core. University of Sussex Library, on 29 Apr 2019 at 02:49:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108654975.001