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ABEL SYMPOSIA Edited by the Norwegian Mathematical Society First row from left: George Elliott, Dorte Olesen, Masamichi Takesaki, Joachim Cuntz, Matilde Marcolli, Alain Connes, Erling Størmer, Ola Bratteli. Second row from left: Trond Digernes, Ken Dykema, Sergey Neshveyev, Nadia Larsen, Thierry Giordano, Heidi Dahl, Takeshi Katsura, Mikael Rørdam, Søren Eilers, Erik Alfsen, David Evans, Magnus Landstad, Christian Skau, John Rognes. Last row from left: Anders Hansen, Akitaka Kishimoto, Yosimichi Ueda, Sindre Duedahl, Dimitri Shlyakhtenko, Kjetil Røysland, Stanisław Lech Woronowicz, Georges Skandalis, Toke Meier Carlsen, Eberhard Kirchberg, Erik Bedos, Ragnar Winter, Ryszard Nest. Alain Connes with predecessors Niels Henrik Abel and Evariste Galois. Photo credits: Samfoto/Svein Erik Dahl . Ola Bratteli Sergey Neshveyev Christian Skau Editors Operator Algebras The Abel Symposium 2004 Proceedings of the First Abel Symposium, Oslo, September 3- 5, 2004 ABC Editors Ola Bratteli Sergey Neshveyev Department of Mathematics Department of Mathematics University of Oslo University of Oslo PB 1053 - Blindern PB 1053 - Blindern 0316 Oslo 0316Oslo Norway Norway e-mail:[email protected] e-mail:[email protected] Christian Skau Department of Mathematical Sciences Norwegian University of Science and Technology 7491 Trondheim Norway e-mail:[email protected] LibraryofCongressControlNumber:2006926217 MathematicsSubjectClassification(2000):37A20, 37A60, 46L35, 46L40, 46L54, 46L55, 46L60, 46L80, 46L85, 46L87 ISBN-10 3-540-34196-XSpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-34196-3SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductionon microfilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationorpartsthereofis permittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion, andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliableforprosecutionunderthe GermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 Printedin the Netherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsandSPi usingaSpringerLATEXmacro package Coverdesign: E. Kirchner, Heidelberg, Germany Printedonacid-freepaper SPIN:11429630 Va46/3100/SPi 543210 Preface to the Series TheNielsHenrikAbelMemorialFundwasestablishedbytheNorwegiangov- ernmentonJanuary1.2002.ThemainobjectiveistohonorthegreatNorwe- gian mathematician Niels Henrik Abel by awarding an international prize for outstanding scientific work in the field of mathematics. The prize shall con- tributetowardsraisingthestatusofmathematicsinsocietyandstimulatethe interest for science among school children and students. In keeping with this objective the board of the Abel fund has decided to finance one or two Abel Symposia each year. The topic may be selected broadly in the area of pure and applied mathematics. The Symposia should be at the highest interna- tionallevel,andservetobuildbridgesbetweenthenationalandinternational researchcommunities.TheNorwegianMathematicalSocietyisresponsiblefor the events. It has also been decided that the contributions from these Sym- posia should be presented in a series of proceedings, and Springer Verlag has enthusiastically agreed to publish the series. The board of the Niels Henrik AbelMemorialFundisconfidentthattheserieswillbeavaluablecontribution to the mathematical literature. Ragnar Winther Chairman of the board of the Niels Henrik Abel Memorial Fund Preface The theme of this symposium was operator algebras in a wide sense. In the last 40 years operator algebras has developed from a rather special disci- pline within functional analysis to become a central field in mathematics often described as “non-commutative geometry” (see for example the book “Non-Commutative Geometry” by the Fields medalist Alain Connes). It has branched out in several sub-disciplines and made contact with other subjects like for example mathematical physics, algebraic topology, geometry, dynam- ical systems, knot theory, ergodic theory, wavelets, representations of groups and quantum groups. Norway has a relatively strong group of researchers in the subject, which contributed to the award of the first symposium in the series of Abel Symposia to this group. The contributions to this volume give a state-of-the-art account of some of these sub-disciplines and the variety of topicsreflecttosomeextenthowthesubjecthasbranchedout.Wearehappy that some of the top researchers in the field were willing to contribute. The basic field of operator algebras is classified within mathematics as part of functional analysis. Functional analysis treats analysis on infinite dimensional spaces by using topological concepts. A linear map between two such spaces is called an operator. Examples are differential and integral operators. An important feature is that the composition of two operators is a non-commutative operation. It is often convenient not just to consider a sin- gleoperator,butawholeclassofoperatorswhichformanalgebraandsatisfy sometechnicalconditions.ThebasictheoryofOperatoralgebrasencompasses C∗-algebras and von Neumann algebras. The study of C∗-algebras could be called non-commutative topology and the study of von Neumann algebras non-commutative measure theory, since this study reduces to the study of topology and measure theory in the case that the algebras are abelian. The symposium took place in Oslo, September 3–5, 2004, and was orga- nized by • Ola Bratteli, University of Oslo • Alain Connes, Collège de France, Paris VIII Preface to the Series • Joachim Cuntz, Westfälische Wilhelms-Universität Münster • Sergei Neshveyev, University of Oslo • Christian Skau, Norwegian University of Science and Technology, Trondheim • Erling Størmer, University of Oslo ThesymposiumwasdedicatedtothememoryofGertKjærgaardPedersen, the pater familias of the operator algebraists in Denmark, who was invited to give a talk, but died March 15, 2004. One of his last contributions to mathematics is published in these proceedings. The following senior researchers from abroad participated and all gave talks: • Alain Connes, Paris • Matilde Marcolli, Bonn • Joachim Cuntz, Münster • Ryszard Nest, Copenhagen • Ken Dykema, Texas A&M • Dorte Olesen, Copenhagen • Søren Eilers, Copenhagen • Mikael Rørdam, Odense • George Elliott, Toronto • Dimitri Shlyakhtenko, UCLA • David Evans, Cardiff • Georges Skandalis, Paris • Thierry Giordano, Ottawa • Masamichi Takesaki, UCLA • Takeshi Katsura, Sapporo • Yoshimichi Ueda, Kyushu • Eberhard Kirchberg, Berlin • Stanisław Lech Woronowicz, • Akitaka Kishimoto, Sapporo Warsaw Senior researchers and postdocs from Oslo and Trondheim who participated: • Erik Alfsen • Magnus Landstad • Erik Bedos • Nadia Larsen • Ola Bratteli • Sergey Neshveyev • Toke Meier Carlsen • Christian Skau • Trond Digernes • Erling Størmer Doctoral students from Oslo and Trondheim who participated: • Sindre Duedahl • Kjetil Røysland • Heidi Dahl More information about the symposium may be found at this web page: http://abelsymposium.no/2004 Oslo and Trondheim 27 March 2006 Ola Bratteli Sergei Neshveyev Christian Skau Contents Interpolation by Projections in C∗-Algebras Lawrence G. Brown, Gert K. Pedersen.............................. 1 KMS States and Complex Multiplication (Part II) Alain Connes, Matilde Marcolli, Niranjan Ramachandran ............. 15 An Algebraic Description of Boundary Maps Used in Index Theory Joachim Cuntz .................................................. 61 On Rørdam’s Classification of Certain C∗-Algebras with One Non-Trivial Ideal Søren Eilers, Gunnar Restorff ..................................... 87 Perturbation of Hausdorff Moment Sequences, and an Application to the Theory of C∗-Algebras of Real Rank Zero George A. Elliott, Mikael Rørdam .................................. 97 Twisted K-Theory and Modular Invariants: I Quantum Doubles of Finite Groups David E Evans ..................................................117 The Orbit Structure of Cantor Minimal Z2-Systems Thierry Giordano, Ian F. Putnam, Christian F. Skau.................145 Outer Actions of a Group on a Factor Yoshikazu Katayama, Masamichi Takesaki ..........................161 Non-Separable AF-Algebras Takeshi Katsura .................................................165 X Contents Central Sequences in C∗-Algebras and Strongly Purely Infinite Algebras Eberhard Kirchberg...............................................175 Lifting of an Asymptotically Inner Flow for a Separable C∗-Algebra Akitaka Kishimoto ...............................................233 Remarks on Free Entropy Dimension Dimitri Shlyakhtenko .............................................249 Notes on Treeability and Costs for Discrete Groupoids in Operator Algebra Framework Yoshimichi Ueda.................................................259 ∗ Interpolation by Projections in C -Algebras Lawrence G. Brown and Gert K. Pedersen∗ Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, [email protected] Dedicated to the memory of Gert. K. Pedersen Note from author L.G.B.: This paper was begun in 2002 and was mainly completed in that year. There were some possible small changes still under discussion. In this version I have made only very minor changes that I’m sure Gert would have approved of. Summary. If x is a self-adjoint element in a unital C∗-algebra A, and if p and δ q denote the spectral projections of x corresponding to the intervals ]δ,∞[ and δ ]−∞,−δ[,weshowthatthereisaprojectionpinAsuchthatp ≤p≤1−q ,pro- δ δ vided that δ >dist{x,A−1}. This result extends to unbounded operators affiliated sa with a C∗−algebra, and has applications to certain other distance functions. 1 Introduction 1.1 LetxbeanoperatoronaHilbertspaceHwithpolardecompositionx=v|x|, and for each δ ≥ 0 let e and f denote the spectral projections of |x| and δ δ |x∗|, respectively, corresponding to the interval ]δ,∞[. Practically the first observationtobemadeinsingleoperatortheoryisthate andf areMurray– δ δ von Neumann equivalent; in fact, ve v∗ =f . The second observation is that δ δ 1 − e and 1 − f need not be equivalent if H is infinite dimensional; in δ δ fact, (1−e )H = kerx and (1−f )H = kerx∗, and these spaces may have 0 0 widely different dimensions. If, however, 1−e =w∗w and 1−f =ww∗ for δ δ some partial isometry w, then u=w+ve is a unitary conjugating e to f . δ δ δ Equivalently phrased, the operator xe can now be written xe =u|xe | with δ δ δ a unitary u. ∗Supported in part by SNF, Denmark.