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Notes on the Brown–Douglas–Fillmore Theorem
The Brown–Douglas–Fillmore (BDF) theorem provides a remarkable solution to a
long-standing open question in operator theory. The solution is obtained by first finding an
equivalent formulation of the problem involving extensions of C∗-algebras. This does not
make the original problem any simpler. None the less, Brown, Douglas, and Fillmore used
newtoolsfromalgebraictopologyandhomologicalalgebratosolvetheproblemofextensions
of commutative C∗-algebras. This new approach also inspired the study of extensions of
non-commutativeC∗-algebras. ThespurtofactivityaroundtheBDFtheoremintheseventies
wastheprecursortotheimportantfieldsofnon-commutativegeometryandK-theory.
The book presents the original proof of BDF in its entirety. Some parts of this proof, which
were only briefly indicated in the BDF paper “Unitary equivalence modulo the compact
operatorsandextensionsofC*-algebras”publishedintheSpringerLectureNotes345,appear
here in complete detail. These notes have attempted to preserve the crucial connection of the
BDF theorem with concrete problems in operator theory. This connection, for instance, is
manifestinthestudyoftheArevson–Douglasconjecture.
Sameer Chavan is Professor in the Department of Mathematics and Statistics at the Indian
InstituteofTechnology,Kanpur. Hispublicationsinsingle-andmulti-variableoperatortheory
useavarietyoftoolsfromgraphtheoryandclassicalanalysis. HeisanAssociateEditorofthe
IndianJournalofPureandAppliedMathematicssince2020.
Gadadhar Misra is Professor in the Department of Mathematics at the Indian Institute of
Science, Bangalore. His research in operator theory is mainly focused on operators in the
Cowen–Douglasclassandthestudyofhomogeneousoperators. HehasbeentheEditorofthe
ProceedingsofMathematicalSciencesduringtheperiod2005–2012andoftheIndianJournal
ofPureandAppliedMathematicsduringtheperiod2015–2020.
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Cambridge–IISc Series
Notes on the
Brown–Douglas–Fillmore Theorem
Sameer Chavan
Gadadhar Misra
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Names: Chavan,Sameer,author. |Misra,Gadadhar,author.
Title: NotesontheBrown-Douglas-Fillmoretheorem/SameerChavan,
GadadharMisra.
Description: Cambridge;NewYork,NY:CambridgeUniversityPress,2021. |
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index.
Identifiers: LCCN2021013072(print)|LCCN2021013073(ebook)|ISBN
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Subjects: LCSH:Operatortheory. |C*-algebras. |Topologicalalgebras. |
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TothememoryofR.G.Douglaswithgreatadmiration,
muchaffectionanddeeprespect.
ThesecondnamedauthorisgratefultoR.G.Douglasforthemathematical
traininghehasreceivedfromhim,firstasadoctoralstudentandthenwhile
workingtogetheronseveraljointresearchprojects.
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Contents
Preface ix
Overview 3
1 SpectralTheoryforHilbertSpaceOperators 7
1.1 PartialIsometriesandPolarDecomposition 7
1.2 CompactandFredholmOperators 9
1.3 FredholmIndexandAbstractIndex 19
1.4 SchattenClasses 34
1.5 IsometriesandvonNeumann–WoldDecomposition 37
1.6 ToeplitzOperatorswithContinuousSymbols 43
1.7 NotesandExercises 47
2 Ext(X)asaSemigroupwithIdentity 57
2.1 EssentiallyNormalOperators 57
2.2 Weyl–vonNeumann–Berg–SikoniaTheorem 60
2.3 ExtensionsandEssentiallyUnitaryOperators 66
2.4 AbsorptionLemma 70
2.5 WeaklyandStronglyEquivalentExtensions 73
2.6 ExistenceandUniquenessofTrivialClass 76
2.7 IdentityElementforExt(X) 82
2.8 NotesandExercises 86
3 SplittingandtheMayer–VietorisSequence 91
3.1 Splitting 91
3.2 DisjointSumofExtensions 94
3.3 FirstSplittingLemma 97
3.4 Surjectivityof(i ) 100
A,X ∗
3.5 Ext(A)→Ext(X)→Ext(X/A)isExact 108
3.6 Mayer–VietorisSequence 112
3.7 NotesandExercises 114
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viii Contents
4 DeterminationofExt(X)asaGroupforPlanarSets 117
4.1 SecondSplittingLemma 117
4.2 ProjectiveLimitsandIteratedSplittingLemma 124
4.3 Ext(X)isaGroup 130
4.4 γ isInjective 133
X
4.5 BDFTheoremandItsConsequences 140
4.6 NotesandExercises 144
5 ApplicationstoOperatorTheory 147
5.1 BergmanOperatorsandSurjectivityofγ 147
X
5.2 HyponormalOperatorsandm-Isometries 152
5.3 EssentiallyNormalCircularOperators 157
5.4 EssentiallyHomogeneousOperators 159
5.5 EssentiallyReductiveQuotientandSubmodules 164
5.6 NotesandExercises 175
Epilogue 179
• OtherProofs 179
◦ PropertiesofExt(X) 180
◦ Theshortexactsequence 180
◦ Arveson’sproofof“Ext(X)isagroup” 181
• RelatedDevelopments 182
◦ Ext(A,B) 182
◦ Homotopyinvariance 183
◦ K-theory 185
• OpenProblems 187
◦ Essentiallynormaltuples 187
◦ Essentiallyhomogeneoustuples 189
◦ Arveson–Douglasconjecture 190
AppendixA PointSetTopology 193
AppendixB LinearAnalysis 201
AppendixC TheSpectralTheorem 217
References 233
SubjectIndex 241
IndexofSymbols 245
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