An Introduction to Noncommutative Spaces and their Geometries: Characterization of the Shallow Subsurface Implications for Urban Infrastructure and Environmental Assessment

2.1 Algebras......Page 14
2.2 Commutative Spaces......Page 18
2.3 Noncommutative Spaces......Page 20
2.3.1 The Jacobson (or Hull-Kernel) Topology......Page 21
2.4 Compact Operators......Page 25
2.5 Real Algebras and Jordan Algebras......Page 26
3.1 The Topological Approximation......Page 28
3.2 Order and Topology......Page 30
3.3 How to Recover the Space Being Approximated......Page 37
3.4 Noncommutative Lattices......Page 42
3.4.2 AF-Algebras......Page 43
3.4.3 From Bratteli Diagrams to Noncommutative Lattices......Page 50
3.4.4 From Noncommutative Lattices to Bratteli Diagrams......Page 52
3.5 How to Recover the Algebra Being Approximated......Page 63
3.6 Operator Valued Functions on Noncommutative Lattices......Page 64
4.1 Modules......Page 67
4.2 Projective Modules of Finite Type......Page 69
4.3 Hermitian Structures over Projective Modules......Page 72
4.4 The Algebra of Endomorphisms of a Module......Page 73
4.5 More Bimodules of Various Kinds......Page 74
5.1 The Group K_0......Page 76
5.2 The K-Theory of the Penrose Tiling......Page 80
5.3 Higher-Order K-Groups......Page 87
6.1 Infinitesimals......Page 89
6.2 The Dixmier Trace......Page 90
6.3 Wodzicki Residue and Connes’ Trace Theorem......Page 95
6.4 Spectral Triples......Page 99
6.5 The Canonical Triple over a Manifold......Page 101
6.6 Distance and Integral for a Spectral Triple......Page 105
6.7 A Two-Point Space......Page 106
6.8 Real Spectral Triples......Page 107
6.9 Products and Equivalence of Spectral Triples......Page 108
7.1 Universal Di.erential Forms......Page 110
7.1.1 The Universal Algebra of Ordinary Functions......Page 115
7.2 Connes’ Di.erential Forms......Page 116
7.2.1 The Usual Exterior Algebra......Page 118
7.2.2 The Two-Point Space Again......Page 122
7.3 Scalar Product for Forms......Page 124
8.1 Abelian Gauge Connections......Page 127
8.2 Universal Connections......Page 129
8.3 Connections Compatible with Hermitian Structures......Page 133
8.4 The Action of the Gauge Group......Page 134
8.5 Connections on Bimodules......Page 135
9.1 Yang-Mills Models......Page 137
9.1.1 Usual Gauge Theory......Page 140
9.1.2 Yang-Mills on a Two-Point Space......Page 141
9.2 The Bosonic Part of the Standard Model......Page 143
9.3 The Bosonic Spectral Action......Page 144
9.4 Fermionic Models......Page 151
9.4.1 Fermionic Models on a Two-Point Space......Page 152
9.5 The Fermionic Spectral Action......Page 153
10.1 Gravity `a la Connes-Dixmier-Wodzicki......Page 155
10.2 Spectral Gravity......Page 157
10.3 Linear Connections......Page 161
10.3.1 Usual Einstein Gravity......Page 165
10.4 Other Gravity Models......Page 166
A.1 Basic Notions of Topology......Page 173
A.2 The Gel’fand-Naimark-Segal Construction......Page 176
A.3 Hilbert Modules......Page 179
A.4 Strong Morita Equivalence......Page 185
A.5 Partially Ordered Sets......Page 187
A.6 Pseudodi.erential Operators......Page 190