Basic Noncommutative Geometry

257 Pages·2013·1.577 MB·English

by  Masoud Khalkhali

EMS Series of Lectures in Mathematics
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Martin J. Mohlenkamp andMaría Cristina Pereyra,Wavelets, Their Friends, and What They
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Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles
Masoud Khalkhali, Basic Noncommutative Geometry
Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods
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Hans Triebel,Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration
Koen Thas,A Course on ElationQuadrangles
Masoud Khalkhali
Basic
Noncommutative
Geometry
Second edition
Author:
Masoud Khalkhali
Department of Mathematics
The University of Western Ontario
London, Ontario N6A 5B7
Canada
E-mail: [email protected]
First edition published in 2009 by the European Mathematical Society Publishing House
2010 Mathematics Subject Classification: 58-02; 58B34
Key words: Noncommutative space, noncommutative quotient, groupoid, cyclic cohomology, Connes–Chern
character, index formula
ISBN 978-3-03719-128-6
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
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ForAzadehandSaman

Preface to the second edition
Thefirsteditionofthisbookwaspublishedaboutfouryearsago. Mygoalwastowrite
anintroductorygraduateleveltextbookmostlyontopologicalaspectsofNoncommu-
tativeGeometrytofillacertaingapintheliterature. Whenmypublishertoldmethat
the book is out of print and there is still good demand for it and we should perhaps
thinkaboutasecondedition,Ifeltvindicated!
Notmuchhaschangedinthissecondedition. Ihavejustaddedtwonewsections,
anddeletednone. Oneisaverybrief,andIamafraidawfullyterse,introductiontoavery
recent development in the subject concerning the Gauss–Bonnet theorem and scalar
curvature for curved noncommutative tori, and the second is a brief introduction to
Hopfcycliccohomology.Apropertreatmentofcurvatureinnoncommutativegeometry
requirestoolsbeyondthescopeofthisbookandcanonlybeadequatelytreatedwith
muchextrapreparatorymaterial. Thebibliographyisextendedandsomenewexamples
areoffered.
Theprogressinnoncommutativegeometryinthelastthirtyfiveyearscanberoughly
dividedintothreephasesinchronologicalorder: topological,spectral,andarithmeti-
cal. Astudentofthesubjectiswelladvisedtofollowthehistoricaldevelopmentand
acquaint herself with all three aspects before focusing on a particular research topic.
As said before, this book mostly covers topological issues: noncommutative spaces,
cycliccohomologyanditsrelationwithK-theoryandK-homology,noncommutative
indextheory,andnoncommutativequotients. Muchhashappenedinthefieldbetween
thesetwoeditionswhichunfortunatelycannotbedealtwithherewithoutsubstantially
increasing the number of pages. For an introduction to the circle of ideas relating
numbertheoryandalgebraicgeometrytononcommutativegeometryIrefertoConnes
and Consani [43] and their last paper [44] and references therein. For relations with
spectral geometry, high energy physics, and number theory, the reader is referred to
Connes–Marcolli’smonograph[55].
Howtousethisbook: Noncommutativegeometrydrawsonmanyideasandtechniques
fromdifferentareasofmathematicswhichareusuallynotallmasteredbyagraduate
student. Giventhis,Iwouldliketosayafewwordsaboutusingthisbookasatextbook
foragraduatecourse. Myexperienceisthatitisnotpossibletocoverallthematerial
inonetermandonereallyneedsafullyearcourseforthat. Onecanteachaoneterm
course based on Chapter 3, cyclic cohomology. Here the instructor needs to fill in
backgroundmaterialonhomologicalalgebrathatIhaveassumedandisnotdiscussed
inthetext: resolutions,derivedfunctors,longexactsequences,andspectralsequences.
Ihavealsoassumedbasicknowledgeofalgebraictopologybasedondifferentialforms
and de Rham cohomology. Alternatively a one term course can be taught based on
Chapter4,Connes–Cherncharacter. Thischapterismoreanalysisbasedandassumes
viii Prefacetothesecondedition
basicnotionsoffunctionalanalysisandoperatortheory. Theinstructorcanaddextra
materialonK-theory,K-homology,spectraltriples,pseudodifferentialoperators,and
theindextheorem. IthinkbasicideasofChapter1ondualitytheorems,andChapter2
onnoncommutativequotientsshouldbeincorporatedinanyintroductorycourseonthe
subject.
ItisagreatpleasuretothankManfredKarbeandIreneZimmermannattheEMS
PublishingHousefortheircare,support,andadvice,indifferentstagesoftheproduc-
tionofthisbook. Iwouldalsoliketothankallthosewhowroteinwithencouraging
wordsofsupport,orcriticalcommentsandsuggestions.
MasoudKhalkhali London,Ontario,September2013
Contents
Prefacetothesecondedition vii
Introduction xi
1 Examplesofalgebra-geometrycorrespondences 1
1.1 LocallycompactspacesandcommutativeC(cid:2)-algebras . . . . . . . . 1
1.1.1 Thespectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 TheGelfandtransform . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Noncommutativespaces . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Noncommutativespacesfromgroups . . . . . . . . . . . . . . 9
1.1.5 Noncommutativetori . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 Vectorbundles,finiteprojectivemodules,andidempotents . . . . . . 18
1.3 Affinevarietiesandfinitelygeneratedcommutativereducedalgebras . 24
1.4 Affineschemesandcommutativerings . . . . . . . . . . . . . . . . . 26
1.5 CompactRiemannsurfacesandalgebraicfunctionfields . . . . . . . 28
1.6 SetsandBooleanalgebras . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 FromgroupstoHopfalgebrasandquantumgroups . . . . . . . . . . 30
1.7.1 Symmetryinnoncommutativegeometry . . . . . . . . . . . . 39
2 Noncommutativequotients 47
2.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Groupoidalgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3 Moritaequivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 MoritaequivalenceforC(cid:2)-algebras . . . . . . . . . . . . . . . . . . 73
2.5 Noncommutativequotients . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Sourcesofnoncommutativespaces . . . . . . . . . . . . . . . . . . . 86
3 Cycliccohomology 87
3.1 Hochschildcohomology . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2 Hochschildcohomologyasaderivedfunctor. . . . . . . . . . . . . . 95
3.3 Deformationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.4 Topologicalalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.5 Examples: Hochschild(co)homology . . . . . . . . . . . . . . . . . 115
3.6 Cycliccohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.7 Connes’longexactsequence . . . . . . . . . . . . . . . . . . . . . . 136
3.8 Connes’spectralsequence . . . . . . . . . . . . . . . . . . . . . . . 140
3.9 Cyclicmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.10 Examples: cycliccohomology . . . . . . . . . . . . . . . . . . . . . 148
x Contents
3.11 Hopfcycliccohomology . . . . . . . . . . . . . . . . . . . . . . . . 154
4 Connes–Cherncharacter 163
4.1 Connes–CherncharacterinK-theory . . . . . . . . . . . . . . . . . . 163
4.1.1 BasicK-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.1.2 Pairingwithcycliccohomology . . . . . . . . . . . . . . . . . 166
4.1.3 NoncommutativeChern–Weiltheory . . . . . . . . . . . . . . 172
4.1.4 TheGauss–Bonnettheoremandscalarcurvaturein
noncommutativegeometry . . . . . . . . . . . . . . . . . . . . 176
4.2 Connes–CherncharacterinK-homology . . . . . . . . . . . . . . . . 177
4.3 Algebrasstableunderholomorphicfunctionalcalculus . . . . . . . . 194
4.4 Afinalword: basicnoncommutativegeometryinanutshell . . . . . . 199
Appendices 201
A Gelfand–Naimarktheorems 201
A.1 Gelfand’stheoryofcommutativeBanachalgebras . . . . . . . . . . . 201
A.2 StatesandtheGNSconstruction . . . . . . . . . . . . . . . . . . . . 205
B Compactoperators,Fredholmoperators,andabstractindextheory 212
C Projectivemodules 219
D Equivalenceofcategories 221
Bibliography 223
Index 235