1. Introduction ................................................. 9
2. The K-functor in various situations ......................... 11
2.1. The Grothendieck group ................................. 11
2.2. The K-theory of spaces ................................. 13
2.2.1. The Serre-Swan Theorem .......................... 13
2.2.2. The group K0(X) ................................. 15
2.2.3. Reduced K-theory ................................ 16
2.2.4. Homotopy invariance and excision ................ 17
2.2.5. More on the Serre-Swan Theorem .................. 21
2.2.6. Higher K-groups and Bott periodicity ............ 23
2.3. Algebraic K-theory ..................................... 26
2.3.1. K0(Λ) and projective Λ-modules .................. 26
2.3.2. K1(Λ) and the Whitehead Lemma ................... 28
2.3.3. Relative K-groups and Milnor's construction ..... 29
2.3.4. Relative K-groups and excision .................. 32
2.3.5. Quillen's higher K-theory ....................... 34
2.4. K-theory of C*-algebras ................................ 35
2.4.1. Algebraic K-theory of C*-algebras ............... 38
2.5. K-homology ............................................. 39
3. Cyclic homology ............................................. 44
3.1. Noncommutative differential forms ...................... 44
3.1.1. Differential forms and the Karoubi operator ..... 44
3.1.2. The harmonic decomposition ...................... 49
3.1.3. Cyclic homology ................................. 51
3.1.4. The nonunital case .............................. 55
3.2. Deformations of the algebra ΩA ......................... 56
3.2.1. Universal algebra extensions .................... 56
3.2.2. The Cuntz algebra QA ............................ 59
3.2.3. The Cuntz algebra and KK-theory ................. 61
3.3. Cochains and characters ................................ 63
3.4. Supertraces ............................................ 65
4. Chern characters ............................................ 67
4.1. K-theory and Chern classes ............................. 67
4.2. Algebraic Chern-Weil theory ............................ 68
4.3. Higher traces .......................................... 72
4.4. Even higher traces and K0(A) ........................... 73
4.5. Toeplitz operators ..................................... 75
4.6. Odd higher traces and K1(A) ............................ 77
5. Excision in cyclic homology and algebraic K-theory .......... 82
5.1. The Loday-Quillen theorem .............................. 82
5.2. Excision in cyclic homology ............................ 88
5.3. Excision in algebraic K-theory ......................... 94
6. Further examples ............................................ 97
6.1. Derivations and cyclic homology ........................ 97
6.2. Cyclic and reduced cyclic homology .................... 101
6.3. Canonical classes in reduced cyclic cohomology ........ 102
6.4. Reduced cyclic cohomology of the Weyl algebra ......... 105
6.5. Excision in periodic cyclic cohomology ................ 106
7. Entire cyclic cohomology of Banach algebras ................ 111
7.1. Topology on ΩA and QA ................................. 111
7.2. Simplicial normalization .............................. 114
7.2.1. A homotopy of superalgebras .................... 115
7.2.2. Homotopy of supertraces ........................ 117
7.3. Homotopy invariance of entire cyclic cohomology ....... 119
7.4. Derivations and entire cyclic cohomology .............. 121
7.5. The JLO cocycle ....................................... 122
Bibliography .................................................. 125
Index ......................................................... 128
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