7336_OWS_lieb_titelei 19.5.2005 16:48 Uhr Seite 1 8398_Cuntz_OWS_TITELEI.indd 1 04.06.2007 09:36:47 Oberwolfach Seminars Volume 36 8398_Cuntz_OWS_TITELEI.indd 2 04.06.2007 09:36:47 Joachim Cuntz Ralf Meyer Jonathan M. Rosenberg Topological and Bivariant K-Theory Birkhäuser Basel · Boston · Berlin 8398_Cuntz_OWS_TITELEI.indd 3 04.06.2007 09:36:47 Joachim Cuntz Ralf Meyer Mathematisches Institut Mathematisches Institut Westfälische Wilhelms-Universität Münster Georg-August-Universität Göttingen Einsteinstraße 62 Bunsenstraße 3–5 48149 Münster 37073 Göttingen Germany Germany e-mail: [email protected] e-mail: [email protected] Jonathan M. Rosenberg Department of Mathematics University of Maryland College Park, MD 20742 USA e-mail: [email protected] 2000 Mathematical Subject Classification: primary 19-XX, secondary 46L80, 46L85, 58J20, 81T75 Library of Congress Control Number: 2007929010 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 978-3-7643-8398-5 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media (cid:51)(cid:85)(cid:76)(cid:81)(cid:87)(cid:72)(cid:71)(cid:3)(cid:82)(cid:81)(cid:3)(cid:68)(cid:70)(cid:76)(cid:71)(cid:16)(cid:73)(cid:85)(cid:72)(cid:72)(cid:3)(cid:83)(cid:68)(cid:83)(cid:72)(cid:85)(cid:3)(cid:83)(cid:85)(cid:82)(cid:71)(cid:88)(cid:70)(cid:72)(cid:71)(cid:3)(cid:73)(cid:85)(cid:82)(cid:80)(cid:3)(cid:70)(cid:75)(cid:79)(cid:82)(cid:85)(cid:76)(cid:81)(cid:72)(cid:16)(cid:73)(cid:85)(cid:72)(cid:72)(cid:3)(cid:83)(cid:88)(cid:79)(cid:83)(cid:17)(cid:3)(cid:55)(cid:38)(cid:41)(cid:3)(cid:146)(cid:3) Printed in Germany ISBN 978-3-7643-8398-5 e-ISBN 978-3-7643-8399-2 9 8 7 6 5 4 3 2 1 www.birkhauser.ch 8398_Cuntz_OWS_TITELEI.indd 4 04.06.2007 09:36:47 Contents Preface ix 1 The elementary algebra of K-theory 1 1.1 Projective modules, idempotents, and vector bundles . . . . . . . . 2 1.1.1 General properties . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Similarity of idempotents . . . . . . . . . . . . . . . . . . . 5 1.1.3 Relationship to vector bundles . . . . . . . . . . . . . . . . 5 1.2 Passage to K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Euler characteristics of finite projective complexes . . . . . 9 1.2.2 Definition of K for non-unital rings . . . . . . . . . . . . . 10 0 1.3 Exactness properties of K-theory . . . . . . . . . . . . . . . . . . . 12 1.3.1 Half-exactness of K . . . . . . . . . . . . . . . . . . . . . . 12 0 1.3.2 Invertible elements and the index map . . . . . . . . . . . . 13 1.3.3 Nilpotent extensions and local rings . . . . . . . . . . . . . 15 2 Functional calculus and topological K-theory 19 2.1 Bornologicalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 Spaces of continuous maps. . . . . . . . . . . . . . . . . . . 22 2.1.2 Bornologicaltensor products . . . . . . . . . . . . . . . . . 23 2.1.3 Local Banach algebras and functional calculus . . . . . . . 24 2.2 Homotopy invariance and exact sequences for local Banach algebras 27 2.2.1 Homotopy invariance of K-theory . . . . . . . . . . . . . . . 28 2.2.2 Higher K-theory . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.3 The Puppe exact sequence. . . . . . . . . . . . . . . . . . . 31 2.2.4 The Mayer–Vietoris sequence . . . . . . . . . . . . . . . . . 32 2.2.5 Projections and idempotents in C∗-algebras . . . . . . . . . 34 2.3 Invariance of K-theory for isoradialsubalgebras . . . . . . . . . . . 36 2.3.1 Isoradialhomomorphisms . . . . . . . . . . . . . . . . . . . 36 2.3.2 Nearly idempotent elements . . . . . . . . . . . . . . . . . . 38 2.3.3 The invariance results . . . . . . . . . . . . . . . . . . . . . 39 2.3.4 Continuity and stability . . . . . . . . . . . . . . . . . . . . 41 vi Contents 3 Homotopy invariance of stabilised algebraic K-theory 45 3.1 Ingredients in the proof . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.1 Split-exact functors and quasi-homomorphisms . . . . . . . 46 3.1.2 Inner automorphisms and stability . . . . . . . . . . . . . . 49 3.1.3 A convenient stabilisation . . . . . . . . . . . . . . . . . . . 51 3.1.4 Hölder continuity . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 The homotopy invariance result . . . . . . . . . . . . . . . . . . . . 54 3.2.1 A key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.3 Weak versus full stability . . . . . . . . . . . . . . . . . . . 60 4 Bott periodicity 63 4.1 Toeplitz algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 The proof of Bott periodicity . . . . . . . . . . . . . . . . . . . . . 65 4.3 Some K-theory computations . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 The Atiyah–Hirzebruch spectral sequence . . . . . . . . . . 72 5 The K-theory of crossed products 75 5.1 Crossed products for a single automorphism . . . . . . . . . . . . . 75 5.1.1 Crossed Toeplitz algebras . . . . . . . . . . . . . . . . . . . 77 5.2 The Pimsner–Voiculescu exact sequence . . . . . . . . . . . . . . . 79 5.2.1 Some consequences of the Pimsner–Voiculescu Theorem . . 83 5.3 A glimpse of the Baum–Connes conjecture . . . . . . . . . . . . . . 83 5.3.1 Toeplitz cones . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3.2 Proof of the decomposition theorem . . . . . . . . . . . . . 89 6 Towards bivariant K-theory: how to classify extensions 91 6.1 Some tricks with smooth homotopies . . . . . . . . . . . . . . . . . 91 6.2 Tensor algebras and classifying maps for extensions . . . . . . . . . 94 6.3 The suspension-stable homotopy category . . . . . . . . . . . . . . 99 6.3.1 Behaviour for infinite direct sums . . . . . . . . . . . . . . . 105 6.3.2 An alternative approach . . . . . . . . . . . . . . . . . . . . 106 6.4 Exact triangles in the suspension-stable homotopy category . . . . 108 6.5 Long exact sequences in triangulated categories . . . . . . . . . . . 113 6.6 Long exact sequences in the suspension-stable homotopy category. 116 6.7 The universal property of the suspension-stable homotopy category 119 7 Bivariant K-theory for bornological algebras 123 7.1 Some tricks with stabilisations . . . . . . . . . . . . . . . . . . . . 124 7.1.1 Comparing stabilisations. . . . . . . . . . . . . . . . . . . . 124 7.1.2 A general class of stabilisations . . . . . . . . . . . . . . . . 125 7.1.3 Smooth stabilisations everywhere . . . . . . . . . . . . . . . 128 7.2 Definition and basic properties . . . . . . . . . . . . . . . . . . . . 129 7.3 Bott periodicity and related results . . . . . . . . . . . . . . . . . . 132 Contents vii 7.4 K-theory versus bivariant K-theory . . . . . . . . . . . . . . . . . . 135 7.4.1 Comparisonwith other topological K-theories . . . . . . . . 137 7.5 The Weyl algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8 A survey of bivariant K-theories 141 8.1 K-Theory with coefficients . . . . . . . . . . . . . . . . . . . . . . . 143 8.2 Algebraic dual K-theory . . . . . . . . . . . . . . . . . . . . . . . . 146 8.3 Homotopy-theoreticKK-theory . . . . . . . . . . . . . . . . . . . . 148 8.4 Brown–Douglas–Fillmoreextension theory . . . . . . . . . . . . . . 149 8.5 Bivariant K-theories for C∗-algebras . . . . . . . . . . . . . . . . . 152 8.5.1 Adapting our machinery . . . . . . . . . . . . . . . . . . . . 152 8.5.2 Another variant related to E-theory . . . . . . . . . . . . . 156 8.5.3 Comparisonwith Kasparov’s definition. . . . . . . . . . . . 157 8.5.4 Some remarks on the Kasparovproduct . . . . . . . . . . . 164 8.6 Equivariant bivariant K-theories . . . . . . . . . . . . . . . . . . . 171 9 Algebras of continuous trace, twisted K-theory 173 9.1 Algebras of continuous trace . . . . . . . . . . . . . . . . . . . . . . 173 9.2 Twisted K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10 Crossed products by R and Connes’ Thom Isomorphism 185 10.1 Crossed products and Takai Duality . . . . . . . . . . . . . . . . . 185 10.2 Connes’ Thom Isomorphism Theorem . . . . . . . . . . . . . . . . 189 10.2.1 Connes’ originalproof . . . . . . . . . . . . . . . . . . . . . 189 10.2.2 Another proof. . . . . . . . . . . . . . . . . . . . . . . . . . 191 11 Applications to physics 195 11.1 K-theory in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 11.2 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 12 Some connections with index theory 203 12.1 Pseudo-differential operators . . . . . . . . . . . . . . . . . . . . . 204 12.1.1 Definition of pseudo-differential operators . . . . . . . . . . 204 12.1.2 Index problems from pseudo-differential operators . . . . . 207 12.1.3 The Dolbeault operator . . . . . . . . . . . . . . . . . . . . 208 12.2 The index theorem of Baum, Douglas, and Taylor. . . . . . . . . . 210 12.2.1 Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . 210 12.2.2 A formula for the boundary map . . . . . . . . . . . . . . . 212 12.2.3 Application to the Dolbeault operator . . . . . . . . . . . . 215 12.3 The index theorems of Kasparov and Atiyah–Singer . . . . . . . . 216 12.3.1 The Thom isomorphism and the Dolbeault operator . . . . 220 12.3.2 The Dolbeault element and the topological index map . . . 222 viii Contents 13 Localisation of triangulated categories 225 13.1 Examples of localisations . . . . . . . . . . . . . . . . . . . . . . . 229 13.1.1 The Universal Coefficient Theorem . . . . . . . . . . . . . . 230 13.1.2 The Baum–Connes assembly map via localisation . . . . . . 233 13.2 The Octahedral Axiom . . . . . . . . . . . . . . . . . . . . . . . . . 234 Bibliography 241 Notation and Symbols 249 Index 257 Preface The new field of noncommutative geometry (see [29,63]) applies ideas from geom- etryto mathematicalstructures determinedby noncommuting variables,andvice versa.Typically,acrucialpartoftheinformationisencodedinanoncommutative algebrawhoseelementsrepresentthesenoncommutingvariables.Suchalgebrasare naturallyassociated—forinstanceasalgebrasofdifferentialorpseudo-differential operators, algebras of intertwining operators for representations, Hecke algebras, algebras of observables in quantum mechanics — with many different geometric structuresarisingfromsubjectsrangingfrommathematicalphysicsanddifferential geometrytonumbertheory.Thefundamentaltoolsforthestudyoftopologicalin- variants attached to noncommutative structures are given by K-theory and cyclic homology. These generalised homology theories are naturally given as bivariant theories, that is, as functors of two variables. For instance, bivariant K-theory specialises both to ordinary topological K-theory and to its dual, K-homology. ThisbookgrewoutofanOberwolfachSeminarorganisedbythethreeauthors in May 2005.Our aim in this seminar was to introduce young mathematicians to the various forms of topological K-theory for (noncommutative) algebras without assuming too much background on the part of our audience. A second aim was to sketch some typical applications of these techniques, including bivariant ver- sionsoftheAtiyah–SingerIndexTheorem,twistedK-theory,someapplicationsto mathematical physics, and the Baum–Connes conjecture. An important part of our book is devoted to a complete and unified descrip- tion of a formalism that has been developed over the past 10 years in [36,37,39], and which allows us to construct topological K-theory and associated bivariant theorieswithgoodpropertiesformanydifferentcategoriesofalgebrasoverRorC suchasC∗-algebras,Banachalgebras,locallyconvexalgebras,Ind-orPro-Banach algebras. Since the construction has to be adapted to the different possible cat- egories, one first problem that we have to address is to fix the setting in which to present the construction. Here we have settled for the category of bornolog- ical algebras. This setting has been advocated in various contexts in [82,84,85]. It is particularly flexible and elegant and covers many interesting examples (for instanceitis especiallywellsuitedfor smoothgroupalgebras).Anotherargument for this choice is the fact that the construction of bivariant K-theory for locally convex algebras is already available in published form in [36,37,39]. So we can