Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book, the author develops and compares these theories, emphasizing their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to $K$-theory, and the Chern-Connes character for $K$-theory and $K$-homology. The cyclic homology theories studied in this text require a good deal of functional analysis in bornological vector spaces, which is supplied in the first chapters. The focal points here are the relationship with inductive systems and the functional calculus in non-commutative bornological algebras. Some chapters are more elementary and independent of the rest of the book and will be of interest to researchers and students working on functional analysis and its applications.
Author(s): Ralf Meyer
Series: EMS Tracts in Mathematics
Publisher: European Mathematical Society
Year: 2007
Language: English
Pages: 368
Preface......Page 5
Contents......Page 7
Introduction......Page 9
1 Bornological vector spaces and inductive systems......Page 17
Basic definitions......Page 21
Some functional analysis......Page 30
Constructions with bornological vector spaces......Page 37
Categories of inductive systems......Page 61
Dissecting bornological vector spaces......Page 67
Metrisability and the approximation property......Page 75
2 Relations between entire, analytic, and local cyclic homology......Page 82
Several definitions......Page 84
Comparison of analytic and local cyclic homology......Page 90
The local homotopy category of chain complexes......Page 92
Some counterexamples with compact Lie groups......Page 106
3 The spectral radius of bounded subsets and its applications......Page 113
The spectral radius......Page 114
Locally multiplicative bornological algebras......Page 116
Analytically nilpotent bornological algebras......Page 123
Isoradial homomorphisms......Page 125
Examples of isoradial homomorphisms......Page 130
Isoradial hulls of subalgebras......Page 139
Passage to inductive systems......Page 143
4 Periodic cyclic homology via pro-nilpotent extensions......Page 151
Pro-algebras......Page 153
Homotopy invariance of periodic cyclic homology......Page 163
Excision in periodic cyclic homology......Page 167
Exterior products......Page 181
5 Analytic cyclic homology and analytically nilpotent extensions......Page 187
Lanilcurs and the analytic tensor algebra......Page 188
Analytic cyclic homology via analytic tensor algebras......Page 201
Basic properties......Page 205
Excision in analytic and local cyclic homology......Page 220
6 Local homotopy invariance and isoradial subalgebras......Page 248
Local homotopy equivalences......Page 249
Approximate local homotopy equivalences......Page 255
Application to isoradial homomorphisms......Page 258
Local and approximate local homotopy category......Page 262
7 The Chern–Connes character......Page 267
The bivariant Chern–Connes character......Page 268
The Universal Coefficient Theorem......Page 274
The character for idempotents and invertibles......Page 278
Finitely summable Fredholm modules......Page 281
The character for general Fredholm modules......Page 287
Chain complexes over additive categories......Page 299
Basic constructions with algebras and modules......Page 306
Non-commutative differential forms......Page 316
Fedosov type products......Page 322
Homological algebra for modules......Page 329
Hochschild homology and cyclic homology......Page 336
Biprojective algebras......Page 346
Bibliography......Page 351
Notation and Symbols......Page 357
Index......Page 363
