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This work has arisen from lecture courses given by the authors on important topics within functional analysis. The authors, who are all leading re- searchers, give introductions to their subjects at a level ideal for beginning graduatestudents,aswellasothersinterestedinthesubject.Thecollectionhas beencarefullyeditedtoformacoherentandaccessibleintroductiontocurrent researchtopics. The first part of the book, by Professor Dales, introduces the general theory ofBanachalgebras,whichservesasabackgroundtotheremainingmaterial. DrWillisthenstudiesacentrallyimportantBanachalgebra,thegroupalgebraof alocallycompactgroup.TheremainingchaptersaredevotedtoBanachalgebras ofoperatorsonBanachspaces:ProfessorEschmeiergivesallthebackground fortheexcitingtopicofinvariantsubspacesofoperators,anddiscussessome key open problems; Dr Laursen and Professor Aiena discuss local spectral theoryforoperators,leadingintoFredholmtheory. LONDONMATHEMATICALSOCIETYSTUDENTTEXTS Managingeditor:ProfessorW.Bruce,DepartmentofMathematics UniversityofLiverpool,UnitedKingdom 3 Localfields,J.W.S.CASSELS 4 Anintroductiontotwistertheory,secondedition,S.A.HUGGET&K.P.TOD 5 Introductiontogeneralrelativity,L.P.HUGHSTON&K.P.TOD 7 Thetheoryofevolutionanddynamicalsystems,J.HOFBAUER&K.SIGMUND 8 SummingandnuclearnormsinBanachspacetheory,G.J.O.JAMESON 9 AutomorphismsofsurfacesafterNielsonandThurston,A.CASSON&S.BLEILER 11 Spacetimeandsingularities,G.NABER 12 Undergraduatealgebraicgeometry,M.REID 13 AnIntroductiontoHankeloperators,J.R.PARTINGTON 15 Presentationsofgroups,secondedition,D.L.JOHNSON 17 Aspectsofquantumfieldtheoryincurvedspacetime,S.A.FULLING 18 Braidsandcoverings:Selectedtopics,V.LUNDSGAARDHANSEN 19 Stepsincommutativealgebra,R.Y.SHARP 20 Communicationtheory,C.M.GOLDIE&R.G.E.PINCH 21 RepresentationsoffinitegroupsofLietype,P.DIGNE&J.MICHEL 22 Designs,graphs,codes,andtheirlinks,P.J.CAMERON&J.H.VANLINT 23 Complexalgebraiccurves,F.KIRWAN 24 Lecturesonellipticcurves,J.W.S.CASSELS 25 Hyperbolicgeometry,B.IVERSEN 26 AnintroductiontothetheoryofL-functionsandEisensteinseries,H,HIDA 27 Hilbertspace:Compactoperatorsandthetracetheorem,J.R.RETHERFORD 28 Potentialtheoryinthecomplexlane,T.RANSFORD 29 Undergraduatecummutativealgebra,M.REID 31 TheLaplacianonaRlemannianmanifold,S.ROSENBERG 32 LecturesonLiegroupsandLiealgebras,R.CARTER,G.SEGAL&J.MACDONALD 33 AprimerofalgebraicD-modules,S.C.COUTINHO 34 Complexalgebraicsurfaces,A.BEAUVILLE 35 Youngtableaux,W.FULTON 37 Amathematicalintroductiontowavelets,P.WOJTASZCZYK 38 Harmonicmaps,loopgroupsandintegrablesystems,M.GUEST 39 Settheoryfortheworkingmathematician,K.CIESIELSKI 40 Ergodictheoryanddynamicalsystems,M.POLLICOTT&M.YURI 41 Thealgorithmicresoutionofdiophantineequations,N.P.SMART 42 Equlibriumstatesinergodictheory,G.KELLER 43 Fourieranalysisonfinitegroupsandapplications,A.TERRAS 44 Classicalinvarianttheory,P.J.OLVER 45 Purmutationgroups,P.J.CAMERON 46 Rlemannsurfaces:Aprimer,A.BEARDON 47 Introductorylecturesonringsandmodules,J.BEACHY 48 Settheory,A.HAJNAL&P.HAMBURGER 49 AnintroductiontoK-theoryforC∗-algebras,M.RØRDAM,F.LARSEN& N.JLAUSTSEN 50 Abriefguidetoalgebraicnumbertheory,H.P.F.SWINNERTON-DYER 51 Stepsincommutativealgebra,R.Y.SHARP 52 FiniteMarkovchainsandalgorithmicapplications,O.HAGGSTROM 53 Theprimenumbertheorem,G.J.O.JAMESON 54 Topicsingraphautomorphismsandreconstruction,J.LAURI&R.SCAPELLATO 55 Elementarynumbertheory,grouptheoryandRamanujangraphs,P.SARNAK, G.DAVIDOFF&A.VALETTE 56 Logic,inductionandsets,T.FORSTER INTRODUCTION TO BANACH ALGEBRAS, OPERATORS, AND HARMONIC ANALYSIS H. GARTH DALES UniversityofLeeds,UK PIETROAIENA Universita`degliStudi,Palermo,Italy JO¨RGESCHMEIER Universita¨tdesSaarlandes,Saarbru¨cken KJELDLAURSEN UniversityofCopenhagen,Denmark GEORGEA.WILLIS UniversityofNewcastle,NewSouthWales,Australia    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521828932 © Cambridge University Press 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback - ---- paperback - --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface pagevii PartI Banachalgebras H.GarthDales 1 1 Definitionsandexamples 3 2 Idealsandthespectrum 12 3 Gelfandtheory 20 4 Thefunctionalcalculus 30 5 Automaticcontinuityofhomomorphisms 38 6 Modulesandderivatives 48 7 Cohomology 58 PartII Harmonicanalysisandamenability GeorgeA.Willis 73 8 Locallycompactgroups 75 9 Groupalgebrasandrepresentations 86 10 Convolutionoperators 98 11 Amenablegroups 109 12 Harmonicanalysisandautomaticcontinuity 121 PartIII Invariantsubspaces Jo¨rgEschmeier 135 13 Compactoperators 137 14 Unitarydilationsandthe H∞-functionalclass 143 15 Hyperinvariantsubspaces 154 16 Invariantsubspacesforcontractions 160 17 Invariantsubspacesforsubnormaloperators 166 18 Invariantsubspacesforsubdecomposableoperators 171 19 Reflexivityofoperatoralgebras 178 20 Invariantsubspacesforcommutingcontractions 186 AppendixtoPartIII 193 PartIV Localspectraltheory KjeldBaggerLaursen 199 21 Basicnotionsfromoperatortheory 201 22 Classesofdecomposableoperators 212 v vi Contents 23 Dualitytheory 226 24 Preservationofspectraandindex 230 25 MultipliersoncommutativeBanachalgebras 241 AppendixtoPartIV 254 PartV Single-valuedextensionpropertyandFredholm theory PietroAiena 265 26 Semi-regularoperators 267 27 Thesingle-valuedextensionproperty 285 28 SVEPforsemi-Fredholmoperators 298 Indexofsymbols 319 Subjectindex 321 Preface Thisvolumeisbasedonacollectionoflecturesintendedforgraduatestudents and others with a basic knowledge of functional analysis. It surveys several areas of current research interest, and is designed to be suitable preparatory readingforthoseembarkingongraduatework.Thevolumeconsistsoffiveparts, whicharebasedonseparatesetsoflectures,eachbydifferentauthors.Eachpart providesanoverviewofthesubjectthatwillalsobeusefultomathematicians workinginrelatedareas.Thechapterswereoriginallypresentedaslecturesat instructional conferences for graduate students, and we have maintained the stylesoftheselectures. Thesetsoflecturesareanintroductiontotheirsubjects,intendedtoconvey theflavourofcertaintopics,andtogivesomebasicdefinitionsandmotivating examples:theyarecertainlynotcomprehensiveaccounts.Referencesaregiven tosourcesintheliteraturewheremoredetailscanbefound. ThechaptersinPartIarebyH.G.Dales.Theseareanintroductiontothe general theory of Banach algebras, and a description of the most important examples:B(E),thealgebraofallboundedlinearoperatorsonaBanachspace E; L1(G), the group algebra of a locally compact group G, taken with the convolutionproduct;commutativeBanachalgebras,includingBanachalgebras offunctionsoncompactsetsinCandradicalBanachalgebras.Chapters3–6 coverGelfandtheoryforcommutativeBanachalgebras,theanalyticfunctional calculus,and,inachapteron‘automaticcontinuity’,thelovelyresultsthatshow theintimateconnectionbetweenthealgebraicandtopologicalstructuresofa Banachalgebra.Chapters6and7areanintroductiontothecohomologytheory ofBanachalgebras,atpresentaveryactiveareaofresearch;weconcentrateon thebasicstructure,thatofderivationsintomodules. The chapters in Part II, by G. A. Willis, develop the theory of one of the examplesdiscussedbyDales:thesearethegroupalgebras L1(G).Chapters8 and9giveadescriptionoflocallycompactgroupsGandtheirstructuretheory, vii viii Preface andthendescribethealgebras L1(G),andtherelatedmeasurealgebra M(G), as a Banach algebra. The Gelfand theory for general commutative Banach algebras,asdescribedbyDales,becomesFouriertransformtheoryinthespecial case of the algebras L1(G). In Chapter 10, Willis discusses compact groups, abelian groups, and free groups, and then, in Chapter 11, moves to a very importantclass,thatofamenablegroups:manycharacterizationsofamenability ariseindiverseareasofmathematics.WillisthenexpandsanotionfromPartI bydiscussingtheautomaticcontinuityoflinearmapsfromgroupalgebras. Parts III–V of this book develop the theory of another example mentioned by Dales: this is the algebra B(E) for a Banach space E. However, they also concentrateonthepropertiesofsingleoperatorsofvarioustypeswithinB(E). Aseminalquestioninfunctionalanalysisisthe‘invariantsubspaceproblem’. Let E be a Banach space, and let T ∈B(E). A closed subspace F of E is invariant for T if Ty ∈ F (y ∈ F); F is trivial if F ={0} or F = E. Does such an operator T always have a non-trivial invariant subspace? A positive answertothisquestioninthecasewhereE isfinite-dimensional(ofdimension atleast2)isthefirststepinthestructuretheoryofmatrices.Thequestionfor Banach spaces has been the spur for a huge amount of research in operator theorysincethequestionwasfirstraisedinthe1930s.Thequestionisstillopen in the case where E is a Hilbert space – this is one of the great problems of oursubject–butcounter-examplesareknownwhen E isanarbitraryBanach space. Nevertheless, there are many positive results for operators T ∈B(E) whichbelongtoaspecialclass. ThechaptersinPartIII,byJ.Eschmeier,discussinparticularoneveryimpor- tanttechniqueforestablishingpositiveresults:itdescendsfromoriginalwork ofScottBrownin1978.Oneclassofoperatorsconsideredisthatofsubdecom- posableoperators.PartIIIconcludeswithremarksabouttheextensions,mainly duetotheauthor,ofthepositiveresultston-tuplesofcommutingoperators. AsexplainedbyDales,everyelementaofaBanachalgebrahasaspectrum, called σ(a); this is a non-empty, compact subset of the complex plane C. In particular,eachoperatorT ∈B(E)hassuchaspectrum,σ(T).Inthecasewhere E isfinite-dimensional,σ(T)isjustthesetofeigenvaluesofT.Thenotionof thespectrumforageneraloperatorT isattheheartoftheremainingchapters, byK.B.LaursenandP.Aiena. Laursendiscussesthespectraltheoryofoperatorsinseveraldifferentclasses; theseincludeinparticularthedecomposableoperators,whichwerealsointro- ducedbyEschmeier.WeunderstandthenatureofanoperatorT bylookingatthe decompositionofσ(T)intosubsetswithspecialpropertiesandalsoatspecial closedsubspacesof E onwhichT acts‘inaniceway’.Inparticular, Laursen Preface ix discussessuper-decomposableandgeneralizedscalaroperators.InChapter 25, Laursen relates his description to notions introduced by Dales by discussing whenmultiplicationoperatorsoncommutativeBanachalgebrashavethevar- ious properties that he has introduced. A valuable appendix to Part IV sketches the background theory, involving distributions, to the ‘functional model’ of Albrecht and Eschmeier that is the natural setting for many of the dualityresultsthathavebeenobtained. The final chapters, those of P. Aiena, are closely related to those of Laursen. The basic examples of the decomposable operators of Eschmeier’s andLaursen’schaptersarecompactoperatorsonaBanachspaceandnormal operatorsonaHilbertspace.Itisnaturaltostudythedecomposableoperators whichhavesimilarpropertiestothoseoftheseimportantspecificexamples:we areledtotheclassof‘Fredholmoperators’andrelatedclasses,amaintopicof Aiena’slectures. In these chapters, we see again, from a different perspective, some of the keyideas – decompositionofthespectrum,invariantsubspace,single-valued extension property, actions of analytic functions, divisible subspaces – that havefeaturedinearlierchapters.ThefinalchapterbyAienasummarizesrecent workoftheauthorandothers. Thelecturesonwhichthisbookisbasedweregivenattwoconferences.The firstwasheldinMussomeli,Sicily,from22to29September1999.Wearevery gratefultoDrGianluigiOliveri,whoorganizedthisconference,andtotheAs- sociazoneCulturaleArchimedeofSicilywhosponsoredit.Thelecturesgiven at this conference were those of Dales, Eschmeier, Laursen, and Aiena. The secondconferencewasheldattheSadarPatelUniversity,VallabhVidyanagar, Gujarat, India, from 8 to 15 January 2002. We are very grateful to Professor SubhashbhaiBhattandDrHareshDedaniafororganizingthisconference,and totheIndianBoardforHigherMathematicsandtotheLondonMathematical Society,whosupportedtheconferencefinancially.ThelecturesgiveninGujarat werethoseofDales,Willis,andLaursen. As we said, the original lectures were intended for graduate students and otherswithabasicknowledgeoffunctionalanalysisandwithabackgroundin complexanalysisandalgebratypicalofafirstdegreeinmathematics.Inboth casesthestudentswereenthusiasticandhelpful;theirsuggestionsledtomany improvementsintheexposition,andwearegratefultothemforthis. Infact,theactuallecturesasgivendidnotincludeallthatiswrittendownhere: modest additions have been made subsequently. There is more in a ‘lecture’ thancaneasilybeabsorbedinonehour.However,wehavemaintainedthefairly informalstyleofthelecturetheatre.Atvariouspoints,thereaderisinvitedto