Table Of ContentThis work has arisen from lecture courses given by the authors on impor-
tant 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
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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
