Basic Noncommutative Geometry : Second Edition

This text provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes–Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well. Two new sections have been added to this second edition: one concerns the Gauss–Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative two torus, and the second is a brief introduction to Hopf cyclic cohomology. The bibliography has been extended and some new examples are presented. Preface to the second edition 7 Contents 9 Introduction 11 1 Examples of algebra-geometry correspondences 19 1.1 Locally compact spaces and commutative C*-algebras 19 1.1.1 The spectrum 20 1.1.2 The Gelfand transform 21 1.1.3 Noncommutative spaces 25 1.1.4 Noncommutative spaces from groups 27 1.1.5 Noncommutative tori 32 1.2 Vector bundles, finite projective modules, and idempotents 36 1.3 Affine varieties and finitely generated commutative reduced algebras 42 1.4 Affine schemes and commutative rings 44 1.5 Compact Riemann surfaces and algebraic function fields 46 1.6 Sets and Boolean algebras 47 1.7 From groups to Hopf algebras and quantum groups 48 1.7.1 Symmetry in noncommutative geometry 57 2 Noncommutative quotients 65 2.1 Groupoids 65 2.2 Groupoid algebras 70 2.3 Morita equivalence 82 2.4 Morita equivalence for C*-algebras 91 2.5 Noncommutative quotients 97 2.6 Sources of noncommutative spaces 104 3 Cyclic cohomology 105 3.1 Hochschild cohomology 107 3.2 Hochschild cohomology as a derived functor 113 3.3 Deformation theory 120 3.4 Topological algebras 130 3.5 Examples: Hochschild (co)homology 133 3.6 Cyclic cohomology 142 3.7 Connes' long exact sequence 154 3.8 Connes' spectral sequence 158 3.9 Cyclic modules 161 3.10 Examples: cyclic cohomology 166 3.11 Hopf cyclic cohomology 172 4 Connes–Chern character 181 4.1 Connes–Chern character in K-theory 181 4.1.1 Basic K-theory 182 4.1.2 Pairing with cyclic cohomology 184 4.1.3 Noncommutative Chern–Weil theory 190 4.1.4 The Gauss–Bonnet theorem and scalar curvature in noncommutative geometry 194 4.2 Connes–Chern character in K-homology 195 4.3 Algebras stable under holomorphic functional calculus 212 A final word: basic 4.4 noncommutative geometry in a nutshell 217 A Gelfand–Naimark theorems 219 A.1 Gelfand's theory of commutative Banach algebras 219 A.2 States and the GNS construction 223 B Compact operators, Fredholm operators, and abstract index theory 230 C Projective modules 237 D Equivalence of categories 239 Bibliography 241 Index 253 This text provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes-Chern characters in -theory and -homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well. Two new sections have been added to the second edition: the first new section concerns the Gauss-Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative two torus, and the second new section is a brief introduction to Hopf cyclic cohomology. The bibliography has been extended and some new examples are presented "Basic Noncommutative Geometry provides an introduction to noncommutative geometry and some of its applications. The book can be used either as a textbook for a graduate course on the subject or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes-Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well."--Publisher's description. "Basic Noncommutative Geometry provides an introduction to noncommutative geometry and some of its applications. The book can be used either as a textbook for a graduate course on the subject or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes-Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well."-- Publisher's description Examples of algebra-geometry correspondences Noncommutative quotients Cyclic cohomology Connes-Chern character Appendices: Gelfand-Naimark theorems Compact operators, Fredholm operators, and abstract index theory Projective modules Equivalence of categories.