Preface ........................................................ ix
1. The elementary algebra of K-theory ........................... 1
1.1. Projective modules, idempotents, and vector bundles ..... 2
1.1.1. General properties ............................... 4
1.1.2. Similarity of idempotents ........................ 5
1.1.3. Relationship to vector bundles ................... 5
1.2. Passage to K-theory ..................................... 8
1.2.1. Euler characteristics of finite projective
complexes ........................................ 9
1.2.2. Definition of К0 for non-unital rings ........... 10
1.3. Exactness properties of K-theory ....................... 12
1.3.1. Half-exactness of К0 ............................ 12
1.3.2. Invertible elements and the index map ........... 13
1.3.3. Nilpotent extensions and local rings ............ 15
2. Functional calculus and topological K-theory ................ 19
2.1. Bornological analysis .................................. 19
2.1.1. Spaces of continuous maps ....................... 22
2.1.2. Bornological tensor products .................... 23
2.1.3. Local Banach algebras and functional calculus ... 24
2.2. Homotopy invariance and exact sequences for local Banach
algebras ............................................... 27
2.2.1. Homotopy invariance of K-theory ................. 28
2.2.2. Higher K-theory ................................. 30
2.2.3. The Puppe exact sequence ........................ 31
2.2.4. The Mayer-Vietoris sequence ..................... 32
2.2.5. Projections and idempotents in C*-algebras ...... 34
2.3. Invariance of K-theory for isoradial subalgebras ....... 36
2.3.1. Isoradial homomorphisms ......................... 36
2.3.2. Nearly idempotent elements ...................... 38
2.3.3. The invariance results .......................... 39
2.3.4. Continuity and stability ........................ 41
3. Homotopy invariance of stabilised algebraic K-theory ........ 45
3.1. Ingredients in the proof ............................... 46
3.1.1. Split-exact functors and quasi-homomorphisms .... 46
3.1.2. Inner automorphisms and stability ............... 49
3.1.3. A convenient stabilisation ...................... 51
3.1.4. Holder continuity ............................... 53
3.2. The homotopy invariance result ......................... 54
3.2.1. A key lemma ..................................... 54
3.2.2. The main results ................................ 57
3.2.3. Weak versus full stability ...................... 60
4. Bott periodicity ............................................ 63
4.1. Toeplitz algebras ...................................... 63
4.2. The proof of Bott periodicity .......................... 65
4.3. Some K-theory computations ............................. 69
4.3.1. The Atiyah-Hirzebruch spectral sequence ......... 72
5. The K-theory of crossed products ............................ 75
5.1. Crossed products for a single automorphism ............. 75
5.1.1. Crossed Toeplitz algebras ....................... 77
5.2. The Pimsner-Voiculescu exact sequence .................. 79
5.2.1. Some consequences of the Pimsner-Voiculescu
Theorem ......................................... 83
5.3. A glimpse of the Baum-Connes conjecture ................ 83
5.3.1. Toeplitz cones .................................. 88
5.3.2. Proof of the decomposition theorem .............. 89
6. Towards bivariant K-theory: how to classify extensions ...... 91
6.1. Some tricks with smooth homotopies ..................... 91
6.2. Tensor algebras and classifying maps for extensions .... 94
6.3. The suspension-stable homotopy category ................ 99
6.3.1. Behaviour for infinite direct sums ............. 105
6.3.2. An alternative approach ........................ 106
6.4. Exact triangles in the suspension-stable homotopy
category .............................................. 108
6.5. Long exact sequences in triangulated categories ....... 113
6.6. Long exact sequences in the suspension-stable homotopy
category .............................................. 116
6.7. The universal property of the suspension-stable homotopy
category .............................................. 119
7. Bivariant K-theory for bornological algebras ............... 123
7.1. Some tricks with stabilisations ....................... 124
7.1.1. Comparing stabilisations ....................... 124
7.1.2. A general class of stabilisations .............. 125
7.1.3. Smooth stabilisations everywhere ............... 128
7.2. Definition and basic properties ....................... 129
7.3. Bott periodicity and related results .................. 132
7.4. K-theory versus bivariant K-theory .................... 135
7.4.1. Comparison with other topological K-theories ... 137
7.5. The Weyl algebra ...................................... 139
8. A survey of bivariant K-theories ........................... 141
8.1. K-Theory with coefficients ............................ 143
8.2. Algebraic dual K-theory ............................... 146
8.3. Homotopy-theoretic KK-theory .......................... 148
8.4. Brown-Douglas-Fillmore extension theory ............... 149
8.5. Bivariant K-theories for C*-algebras .................. 152
8.5.1. Adapting our machinery ......................... 152
8.5.2. Another variant related to E-theory ............ 156
8.5.3. Comparison with Kasparov's definition .......... 157
8.5.4. Some remarks on the Kasparov product ........... 164
8.6. Equivariant bivariant K-theories ...................... 171
9. Algebras of continuous trace, twisted K-theory ............. 173
9.1. Algebras of continuous trace .......................... 173
9.2. Twisted K-theory ...................................... 182
10. Crossed products by  and Connes' Thom Isomorphism ........ 185
10.1. Crossed products and Takai Duality .................. 185
10.2. Connes' Thom Isomorphism Theorem .................... 189
10.2.1. Connes' original proof ...................... 189
10.2.2. Another proof ............................... 191
11. Applications to physics ................................... 195
11.1. K-theory in physics ................................. 195
11.2. T-duality ........................................... 197
12. Some connections with index theory ........................ 203
12.1. Pseudo-differential operators ....................... 204
12.1.1. Definition of pseudo-differential
operators ................................... 204
12.1.2. Index problems from pseudo-differential
operators ................................... 207
12.1.3. The Dolbeault operator ...................... 208
12.2. The index theorem of Baum, Douglas, and Taylor ...... 210
12.2.1. Toeplitz operators .......................... 210
12.2.2. A formula for the boundary map .............. 212
12.2.3. Application to the Dolbeault operator ....... 215
12.3. The index theorems of Kasparov and Atiyah-Singer .... 216
12.3.1. The Thom isomorphism and the Dolbeault
operator .................................... 220
12.3.2. The Dolbeault element and the topological
index map ................................... 222
13. Localisation of triangulated categories ................... 225
13.1. Examples of localisations ........................... 229
13.1.1. The Universal Coefficient Theorem ........... 230
13.1.2. The Baum-Connes assembly map via
localisation ................................ 233
13.2. The Octahedral Axiom ................................ 234
Bibliography .................................................. 241
Notation and Symbols .......................................... 249
Index ......................................................... 257
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