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Basic Noncommutative Geometry Second Edition Author 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. 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. © 2013 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed on acid-free paper produced from chlorine-free pulp. TCF °° Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany 9 8 7 6 5 4 3 2 1 For Azadeh and Saman Preface to the second edition The first edition of this book was published about four years ago. My goal was to write an introductory graduate level textbook mostly on topological aspects of Noncommu- tative Geometry to fill a certain gap in the literature. When my publisher told me that the book is out of print and there is still good demand for it and we should perhaps think about a second edition, I felt vindicated! Not much has changed in this second edition. I have just added two new sections, and deleted none. One is a very brief, and I am afraid awfully terse, introduction to a very 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 Hopf cyclic cohomology. A proper treatment of curvature in noncommutative geometry requires tools beyond the scope of this book and can only be adequately treated with much extra preparatory material. The bibliography is extended and some new examples are offered. The progress in noncommutative geometry in the last thirty five years can be roughly divided into three phases in chronological order: topological, spectral, and arithmeti- cal. A student of the subject is well advised to follow the historical development and acquaint herself with all three aspects before focusing on a particular research topic. As said before, this book mostly covers topological issues: noncommutative spaces, cyclic cohomology and its relation with K-theory and K-homology, noncommutative index theory, and noncommutative quotients. Much has happened in the field between these two editions which unfortunately cannot be dealt with here without substantially increasing the number of pages. For an introduction to the circle of ideas relating number theory and algebraic geometry to noncommutative geometry I refer to Connes 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’s monograph [55]. How to use this book: Noncommutative geometry draws on many ideas and techniques from different areas of mathematics which are usually not all mastered by a graduate student. Given this, I would like to say a few words about using this book as a textbook for a graduate course. My experience is that it is not possible to cover all the material in one term and one really needs a full year course for that. One can teach a one term course based on Chapter 3, cyclic cohomology. Here the instructor needs to fill in background material on homological algebra that I have assumed and is not discussed in the text: resolutions, derived functors, long exact sequences, and spectral sequences. I have also assumed basic knowledge of algebraic topology based on differential forms and de Rham cohomology. Alternatively a one term course can be taught based on Chapter 4, Connes–Chern character. This chapter is more analysis based and assumes viii Preface to the second edition basic notions of functional analysis and operator theory. The instructor can add extra material on K-theory, K-homology, spectral triples, pseudodifferential operators, and the index theorem. I think basic ideas of Chapter 1 on duality theorems, and Chapter 2 on noncommutative quotients should be incorporated in any introductory course on the subject. It is a great pleasure to thank Manfred Karbe and Irene Zimmermann at the EMS Publishing House for their care, support, and advice, in different stages of the produc- tion of this book. I would also like to thank all those who wrote in with encouraging words of support, or critical comments and suggestions. Masoud Khalkhali London, Ontario, September 2013 Contents Preface to the second edition vii Introduction xi 1 Examples of algebra-geometry correspondences 1 1.1 Locally compact spaces and commutative C -algebras ........ 1 1.1.1 The spectrum ........................... 2 1.1.2 The Gelfand transform ...................... 3 1.1.3 Noncommutative spaces ..................... 7 1.1.4 Noncommutative spaces from groups .............. 9 1.1.5 Noncommutative tori ....................... 14 1.2 Vector bundles, finite projective modules, and idempotents ...... 18 1.3 Affine varieties and finitely generated commutative reduced algebras . 24 1.4 Affine schemes and commutative rings ................. 26 1.5 Compact Riemann surfaces and algebraic function fields ....... 28 1.6 Sets and Boolean algebras ....................... 29 1.7 From groups to Hopf algebras and quantum groups .......... 30 1.7.1 Symmetry in noncommutative geometry ............ 39 2 Noncommutative quotients 47 2.1 Groupoids ................................ 47 2.2 Groupoid algebras ............................ 52 2.3 Morita equivalence ........................... 64 2.4 Morita equivalence for C -algebras .................. 73 2.5 Noncommutative quotients ....................... 79 2.6 Sources of noncommutative spaces ................... 86 3 Cyclic cohomology 87 3.1 Hochschild cohomology ........................ 89 3.2 Hochschild cohomology as a derived functor .............. 95 3.3 Deformation theory ........................... 102 3.4 Topological algebras .......................... 112 3.5 Examples: Hochschild (co)homology ................. 115 3.6 Cyclic cohomology ........................... 124 3.7 Connes’ long exact sequence ...................... 136 3.8 Connes’ spectral sequence ....................... 140 3.9 Cyclic modules ............................. 143 3.10 Examples: cyclic cohomology ..................... 148 x Contents 3.11 Hopf cyclic cohomology ........................ 154 4 Connes–Chern character 163 4.1 Connes–Chern character in K-theory .................. 163 4.1.1 Basic K-theory .......................... 164 4.1.2 Pairing with cyclic cohomology ................. 166 4.1.3 Noncommutative Chern–Weil theory .............. 172 4.1.4 The Gauss–Bonnet theorem and scalar curvature in noncommutative geometry .................... 176 4.2 Connes–Chern character in K-homology ................ 177 4.3 Algebras stable under holomorphic functional calculus ........ 194 4.4 A final word: basic noncommutative geometry in a nutshell ...... 199 Appendices 201 A Gelfand–Naimark theorems 201 A.1 Gelfand’s theory of commutative Banach algebras ........... 201 A.2 States and the GNS construction .................... 205 B Compact operators, Fredholm operators, and abstract index theory 212 C Projective modules 219 D Equivalence of categories 221 Bibliography 223 Index 235 Introduction One of the major advances of science in the 20th century was the discovery of a math- ematical formulation of quantum mechanics by Heisenberg in 1925 [103].1 From a mathematical point of view, this transition from classical mechanics to quantum me- chanics amounts to, among other things, passing from the commutative algebra of classical observables to the noncommutative algebra of quantum mechanical observ- ables. To understand this better we recall that in classical mechanics an observable of a system (e.g. energy, position, momentum, etc.) is a function on a manifold called the phase space of the system. Classical observables can therefore be multiplied in a pointwise manner and this multiplication is obviously commutative. Immediately after Heisenberg’s work, ensuing papers by Dirac [73] and Born–Heisenberg–Jordan [16], made it clear that a quantum mechanical observable is a (selfadjoint) linear operator on a Hilbert space, called the state space of the system. These operators can again be multiplied with composition as their multiplication, but this operation is not necessarily commutative any longer.2 In fact Heisenberg’s commutation relation h pq qp D 1 2i shows that position and momentum operators do not commute and this in turn can be shown to be responsible for the celebrated uncertainty principle of Heisenberg. Thus, to get a more accurate description of nature one is more or less forced to replace the commutative algebra of
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