This book provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course on the subject or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry, and homological algebra at a first-year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes-Chern characters in $K$-theory and $K$-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well.
Author(s): Masoud Khalkhali
Series: Ems Series of Lectures in Mathematics
Publisher: European Mathematical Society
Year: 2009
Language: English
Pages: 240
Contents......Page 8
Introduction......Page 10
Locally compact spaces and commutative C*-algebras......Page 18
Vector bundles, finite projective modules, and idempotents......Page 32
Affine varieties and finitely generated commutative reduced algebras......Page 38
Affine schemes and commutative rings......Page 41
Compact Riemann surfaces and algebraic function fields......Page 42
Sets and Boolean algebras......Page 43
From groups to Hopf algebras and quantum groups......Page 44
Groupoids......Page 60
Groupoid algebras......Page 65
Morita equivalence......Page 77
Morita equivalence for C*-algebras......Page 86
Noncommutative quotients......Page 92
Sources of noncommutative spaces......Page 99
Cyclic cohomology......Page 100
Hochschild cohomology......Page 102
Hochschild cohomology as a derived functor......Page 108
Deformation theory......Page 115
Topological algebras......Page 125
Examples: Hochschild (co)homology......Page 128
Cyclic cohomology......Page 137
Connes' long exact sequence......Page 149
Connes' spectral sequence......Page 153
Cyclic modules......Page 156
Examples: cyclic cohomology......Page 161
Connes–Chern character in K-theory......Page 167
Connes–Chern character in K-homology......Page 180
Algebras stable under holomorphic functional calculus......Page 197
A final word: basic noncommutative geometry in a nutshell......Page 201
A.1 Gelfand’s theory of commutative Banach algebras......Page 203
A.2 States and the GNS construction......Page 207
B Compact operators, Fredholm operators, and abstract index theory......Page 214
C Projective modules......Page 221
D Equivalence of categories......Page 223
Bibliography......Page 226
Index......Page 236
