AN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRY
1 Introduction 1 2 Noncommutative Spaces and Algebras of Functions 7 21 Algebras 7 22 Commutative Spaces 11 23 Noncommutative Spaces 13 231 The Jacobson (or Hull-Kernel) Topology 14 24 Compact Operators 18 25 Real Algebras and Jordan Algebras 19 3 Projective Systems of Noncommutative Lattices 21 31 The Topological Approximation 21 32 Order and Topology 23 33 How to Recover the Space Being Approximated 30 34 Noncommutative Lattices 35 341 The Space PrimA as a Poset 36 342 AF-Algebras 36 343 From Bratteli Diagrams to Noncommutative Lattices 43 344 From Noncommutative Lattices to Bratteli Diagrams 45 35 How to Recover the Algebra Being Approximated 56 36 Operator Valued Functions on Noncommutative Lattices 57 4 Modules as Bundles 59 41 Modul es 60 42 Projective Modules of Finite Type 62 43 Hermitian Structures over Projective Modules 65 44 The Algebra of Endomorphisms of a Module 66 45 More Bimodules of Various Kinds 67 5 A Few Elements of K-Theory 69 51 The Group K0 69 52 The K-Theory of the Penrose Tiling 73 53 Higher-Order K-Groups 80 6 The Spectral Calculus 83 61 Infinitesimals 83 62 The Dixmier Trace 84 63 Wodzicki Residue and Connes’ Trace Theorem 89 64 SpectralT riples 93 65 The Canonical Triple over a Manifold 95 66 Distance and Integralfor a SpectralT riple 99 67 A Two-Point Space 100 68 RealSp ectralT riples 101 69 Products and Equivalence of Spectral Triples 102 7 Noncommutative Differential Forms 105 71 UniversalDifferen tialF orms 105 711 The UniversalAl gebra of Ordinary Functions 110 72 Connes’ DifferentialF orms 111 721 The UsualExterior Algebra 113 722 The Two-Point Space Again 117 73 Scalar Product for Forms 119 8 Connections on Modules 123 81 Abelian Gauge Connections 123 811 UsualEl ectromagnetism 125 82 UniversalConnections 125 83 Connections Compatible with Hermitian Structures 129 84 The Action of the Gauge Group 130 85 Connections on Bimodules 131 9 Field Theories on Modules 133 91 Yang-Mills Models 133 911 UsualGauge Theory 136 912 Yang-Mills on a Two-Point Space 137 92 The Bosonic Part of the Standard Model 139 93 The Bosonic SpectralAction 140 94 Fermionic Models 147 941 Fermionic Models on a Two-Point Space 148 942 The Standard Model 149 95 The Fermionic SpectralAction 149 10 Gravity Models 151 101 Gravity `a la Connes-Dixmier-Wodzicki 151 102 SpectralGra vity 153 103 Linear Connections 157 1031 UsualEinstein Gravity 161 104 Other Gravity Models 162 11 Quantum Mechanical Models on Noncommutative Lattices 165