| C*-Algebras and Operator Theory | p. 1 |
| Bounded Operators and Functional Calculus | p. 1 |
| Positive Operators and the Strong Operator Topology | p. 6 |
| C*-Algebras | p. 8 |
| The GNS Construction | p. 12 |
| Representations of Commutative C*-Algebras | p. 14 |
| Abstract C*-Algebras | p. 16 |
| Ideals and Quotients | p. 17 |
| Unbounded Operators | p. 19 |
| Exercises | p. 22 |
| Notes | p. 27 |
| Index Theory and Extensions | p. 29 |
| Fredholm Operators and the Calkin Algebra | p. 29 |
| The Essential Spectrum | p. 32 |
| The Toeplitz Extension | p. 34 |
| Essentially Normal Operators | p. 36 |
| C*-Algebra Extensions | p. 38 |
| Extensions and the Calkin Algebra | p. 40 |
| The Extension Semigroup | p. 41 |
| Geometric Examples of Extensions | p. 44 |
| Exercises | p. 49 |
| Notes | p. 54 |
| Completely Positive Maps | p. 55 |
| Completely Positive Maps | p. 55 |
| Quasicentral Approximate Units | p. 57 |
| Nuclearity | p. 60 |
| Voiculescu's Theorem | p. 63 |
| Block-Diagonal Maps | p. 65 |
| Proof of Voiculescu's Theorem | p. 68 |
| Property T and Ext | p. 71 |
| Kasparov's Technical Theorem | p. 76 |
| Exercises | p. 79 |
| Notes | p. 83 |
| K-Theory | p. 85 |
| The Group K[subscript 0](A) | p. 85 |
| K[subscript 0] for Non-Unital Algebras | p. 90 |
| Relative K-Theory and Excision | p. 92 |
| Homotopy | p. 97 |
| Higher K-Theory | p. 98 |
| Inner Automorphisms | p. 101 |
| Products | p. 103 |
| Another Description of K[subscript 1] | p. 106 |
| Bott Periodicity | p. 110 |
| Exercises | p. 113 |
| Notes | p. 120 |
| Duality Theory | p. 123 |
| Extension Groups and Dual C*-Algebras | p. 123 |
| K-Homology | p. 125 |
| Relative K-Homology | p. 130 |
| Excision in K-Homology | p. 133 |
| Example: the Theta Curve | p. 137 |
| Exercises | p. 138 |
| Notes | p. 139 |
| Coarse Geometry and K-Homology | p. 141 |
| Coarse Structures | p. 141 |
| Coarse Geometry of Cones | p. 145 |
| The C*-Algebra of a Coarse Space | p. 147 |
| K-Theory for Metric Coarse Structures | p. 152 |
| K-Theory for Topological Coarse Structures | p. 157 |
| The Homotopy Invariance of K-Homology | p. 160 |
| Exercises | p. 162 |
| Notes | p. 165 |
| The Brown-Douglas-Fillmore Theorem | p. 167 |
| Generalized Homology Theories | p. 168 |
| The Index Pairing | p. 170 |
| Steenrod Homology Theory | p. 180 |
| The Cluster Axiom for K-Homology | p. 183 |
| The Brown-Douglas-Fillmore Theorem | p. 186 |
| The Universal Coefficient Theorem | p. 188 |
| Exercises | p. 196 |
| Notes | p. 198 |
| Kasparov's K-Homology | p. 199 |
| Fredholm Modules | p. 199 |
| The Kasparov Groups | p. 204 |
| Normalization of Fredholm Modules | p. 208 |
| Kasparov Theory and Duality | p. 212 |
| Relative K-Homology | p. 215 |
| Schrodinger Pairs | p. 219 |
| The Index Pairing | p. 223 |
| Exercises | p. 233 |
| Notes | p. 237 |
| The Kasparov Product | p. 239 |
| The Product of Fredholm Operators | p. 239 |
| The Definition of the Kasparov Product | p. 243 |
| Index One Operators and Homotopy Invariance | p. 248 |
| Stability | p. 252 |
| Bott Periodicity | p. 253 |
| Boundary Maps and the Kasparov Product | p. 259 |
| The Kasparov Product and the Index Pairing | p. 264 |
| Exercises | p. 265 |
| Notes | p. 267 |
| Elliptic Differential Operators | p. 269 |
| First-Order Differential Operators | p. 269 |
| Symmetric and Selfadjoint Differential Operators | p. 271 |
| Wave Operators | p. 274 |
| Ellipticity | p. 279 |
| Elliptic Operators on Open Manifolds | p. 284 |
| The Homology Class of a Selfadjoint Operator | p. 286 |
| Elliptic Operators and the Kasparov Product | p. 290 |
| The Homology Class of a Symmetric Operator | p. 293 |
| Exercises | p. 297 |
| Notes | p. 303 |
| Index Theory | p. 305 |
| Dirac Operators | p. 306 |
| Spin[superscript c]-Manifolds | p. 311 |
| Even-Dimensional Spin[superscript c]-Manifolds | p. 318 |
| Index Theory for Hypersurfaces | p. 320 |
| The Index Theorem for Spin[superscript c]-Manifolds | p. 326 |
| Toeplitz Index Theorems | p. 329 |
| Index Theory on Strongly Pseudoconvex Domains | p. 333 |
| Exercises | p. 341 |
| Notes | p. 345 |
| Higher Index Theory | p. 347 |
| Metrics of Positive Scalar Curvature | p. 347 |
| Non-Positive Sectional Curvature | p. 350 |
| Coarse Geometry and Assembly Maps | p. 352 |
| Scaleable Spaces and the Baum-Connes Conjecture | p. 357 |
| Equivariant Assembly | p. 363 |
| The Descent Principle | p. 369 |
| Exercises | p. 373 |
| Notes | p. 375 |
| Gradings | p. 377 |
| Graded Vector Spaces and Algebras | p. 377 |
| Graded Tensor Products | p. 378 |
| Multigradings | p. 379 |
| Hermitian Modules and K-Theory | p. 380 |
| Graded Hermitian Modules | p. 384 |
| Notes | p. 385 |
| Real K-Homology | p. 387 |
| Real C*-Algebras | p. 387 |
| K-Theory for Real C*-Algebras | p. 388 |
| K-Homology for Real C*-Algebras | p. 389 |
| Notes | p. 390 |
| References | p. 391 |
| Index | p. 401 |
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